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Arithmetic patterns

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Difficulty: 1 – 5

5156743

Compare Sequence A and Sequence B. Sequence A: \(7, 17, 27, 37, 47, 57\) Sequence B: \(207, 217, 227, 237, 247, 257\) a) What do all the numbers in both sequences have in common in the ones place? b) How much greater is each number in Sequence B than the corresponding number in Sequence A?

Hints

- Compare the last digit of each number in both sequences. - Subtract one pair of corresponding terms. - Compare the hundreds places in the two sequences.

Solution

1. Every number in both sequences has \(7\) in the ones place. 2. Subtract corresponding terms: for example, \(207-7=200\) and \(217-17=200\). 3. Each term in Sequence B is \(200\) greater than the term in the same position in Sequence A.

Answer

a) Every number has \(7\) in the ones place. b) Each number in Sequence B is \(200\) greater than the corresponding number in Sequence A.

5156813

List all the even whole numbers strictly between \(777\) and \(793\).

Hints

- Recall which digits can appear in the ones place of an even number. - Find the first even number after \(777\). - Count forward by twos.

Solution

1. The first even number greater than \(777\) is \(778\). 2. Add \(2\) repeatedly to generate the remaining even numbers. 3. The last even number less than \(793\) is \(792\).

Answer

\(778, 780, 782, 784, 786, 788, 790, 792\)

5156823

List the odd whole numbers from \(85\) through \(105\) in order.

Hints

- Recall which digits can appear in the ones place of an odd number. - Find the odd number that comes after \(85\). - Count forward by twos and pay attention to the change from \(99\) to \(101\).

Solution

1. Start with the odd number \(85\). 2. Add \(2\) each time to find the next odd number. 3. Continue through the change from \(99\) to \(101\) until you reach \(105\).

Answer

\(85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105\)

5156913

Find the rule and continue the sequence. \(225, 250, 275, \ldots, \ldots, \ldots, 375\)

Hints

- Find the difference between consecutive terms. - Determine how much must be added to \(225\) to reach \(250\). - Check whether that difference stays the same. - Continue using the same change until you reach \(375\).

Solution

1. The difference between the first two terms is \(250-225=25\), and \(275-250=25\). 2. The rule is to add \(25\) each time. 3. The missing terms are \(275+25=300\), \(300+25=325\), and \(325+25=350\). 4. The next term is \(350+25=375\), which matches the given endpoint.

Answer

The missing numbers are \(300, 325,\) and \(350\). Rule: add \(25\) each time.

5157633

Study the positive multiples of \(5\) and \(10\) through \(100\). a) Which digits can appear in the ones place of positive multiples of \(5\)? b) Which digit always appears in the ones place of positive multiples of \(10\)? c) Which numbers through \(100\) are multiples of both \(5\) and \(10\)?

Hints

- List the first several positive multiples of \(5\) and \(10\). - Look only at the last digit of each multiple. - Which numbers appear in both lists?

Solution

1. The positive multiples of \(5\) are \(5, 10, 15, 20, 25, 30, \ldots\). Their ones digits alternate between \(5\) and \(0\). 2. Every positive multiple of \(10\) has a ones digit of \(0\). 3. Every multiple of \(10\) is also a multiple of \(5\). Through \(100\), the common multiples are \(10, 20, 30, 40, 50, 60, 70, 80, 90\), and \(100\).

Answer

a) \(0\) and \(5\) b) \(0\) c) \(10, 20, 30, 40, 50, 60, 70, 80, 90, 100\)

5157733

Find the products in each pair. Explain how the two facts are related. a) \(3 \times 4\) and \(6 \times 4\) b) \(10 \times 7\) and \(5 \times 7\) c) \(4 \times 8\) and \(8 \times 8\)

Hints

- Compare the first factors in each pair. - How can the known product help you find the other product? - What happens to a product when one factor is doubled or halved and the other factor stays the same?

Solution

1. \(3 \times 4 = 12\). Doubling the first factor from \(3\) to \(6\) doubles the product, so \(6 \times 4 = 24\). 2. \(10 \times 7 = 70\). Halving the first factor from \(10\) to \(5\) halves the product, so \(5 \times 7 = 35\). 3. \(4 \times 8 = 32\). Doubling the first factor from \(4\) to \(8\) doubles the product, so \(8 \times 8 = 64\).

Answer

a) \(3 \times 4 = 12\) and \(6 \times 4 = 24\); the first factor and the product are doubled. b) \(10 \times 7 = 70\) and \(5 \times 7 = 35\); the first factor and the product are halved. c) \(4 \times 8 = 32\) and \(8 \times 8 = 64\); the first factor and the product are doubled.

5157763

Compare the 3s facts \(3, 6, 9, \ldots\) with the 6s facts \(6, 12, 18, \ldots\). Find three different pairs of facts that have the same product. Write each pair in this form: \((\square) \times 3 = \square\) \((\square) \times 6 = \square\)

Hints

- List the 3s products and check which also appear in the 6s facts. - To keep the product the same when the factor \(3\) becomes \(6\), what must happen to the other factor? - How are \(3\) and \(6\) related?

Solution

1. Common products in the 3s and 6s facts include \(6, 12, 18, 24\), and \(30\). 2. For product \(6\), \(2 \times 3 = 6\) and \(1 \times 6 = 6\). 3. For product \(12\), \(4 \times 3 = 12\) and \(2 \times 6 = 12\). 4. For product \(18\), \(6 \times 3 = 18\) and \(3 \times 6 = 18\). 5. Other correct pairs are possible.

Answer

One possible set is: \(2 \times 3 = 6\) and \(1 \times 6 = 6\) \(4 \times 3 = 12\) and \(2 \times 6 = 12\) \(6 \times 3 = 18\) and \(3 \times 6 = 18\)

5158583

Continue the sequence backward. Write the next eight terms after \(712, 710, 708, \ldots\).

Hints

- Find how much each term decreases. - Think about the number that is \(2\) less than \(700\). - Check that you wrote exactly eight new terms.

Solution

1. Each term is \(2\) less than the term before it. 2. Starting from \(708\), subtract \(2\) eight times. 3. The sequence passes from \(700\) to \(698\).

Answer

\(706, 704, 702, 700, 698, 696, 694, 692\)

5158723

Find the products in each pair. What pattern do you notice? a) \(1 \times 9\) and \(0 \times 9\) b) \(14 \times 1\) and \(14 \times 0\) c) \(1 \times 1\) and \(0 \times 1\) d) \(0 \times 25\) and \(1 \times 25\)

Hints

- What happens when you take exactly one group of a quantity? - What happens when you take zero groups of a quantity? - In each product with a factor of \(1\), which number remains unchanged?

Solution

1. The products are: a) \(1 \times 9 = 9\) and \(0 \times 9 = 0\); b) \(14 \times 1 = 14\) and \(14 \times 0 = 0\); c) \(1 \times 1 = 1\) and \(0 \times 1 = 0\); d) \(0 \times 25 = 0\) and \(1 \times 25 = 25\). 2. Multiplying by \(1\) leaves the other factor unchanged, while multiplying by \(0\) gives a product of \(0\).

Answer

a) \(9\) and \(0\) b) \(14\) and \(0\) c) \(1\) and \(0\) d) \(0\) and \(25\) Pattern: A number multiplied by \(1\) stays the same, and a number multiplied by \(0\) has a product of \(0\).

5158813

Write each number sequence. Do not include the endpoints. a) Multiples of \(100\) strictly between \(0\) and \(1000\) b) Multiples of \(10\) strictly between \(460\) and \(540\) c) Numbers in steps of \(50\) strictly between \(700\) and \(950\)

Hints

- Determine the step size in each part. - “Strictly between” means that the endpoints are not included. - Add the same step repeatedly until the next term would reach or pass the upper endpoint.

Solution

1. Count by \(100\) from \(100\) through \(900\). 2. Count by \(10\) from \(470\) through \(530\). 3. Count by \(50\) from \(750\) through \(900\).

Answer

a) \(100, 200, 300, 400, 500, 600, 700, 800, 900\) b) \(470, 480, 490, 500, 510, 520, 530\) c) \(750, 800, 850, 900\)

5158903

Continue each sequence through the stated ending value. a) \(670, 680, 690, \ldots, 750\) b) \(312, 310, 308, \ldots, 296\) c) \(450, 500, 550, \ldots, 800\)

Hints

- Compare the first two terms in each sequence. - Decide whether the terms increase or decrease. - Find the amount of the repeated change. - In b), pay attention to the tens and hundreds transition.

Solution

1. Sequence a) increases by \(10\), giving \(700, 710, 720, 730, 740,\) and \(750\). 2. Sequence b) decreases by \(2\), giving \(306, 304, 302, 300, 298,\) and \(296\). 3. Sequence c) increases by \(50\), giving \(600, 650, 700, 750,\) and \(800\).

Answer

a) \(670, 680, 690, 700, 710, 720, 730, 740, 750\) b) \(312, 310, 308, 306, 304, 302, 300, 298, 296\) c) \(450, 500, 550, 600, 650, 700, 750, 800\)

5158913

Fill in the missing terms. a) \(125, 150, \underline{\qquad}, 200, 225, \underline{\qquad}\) b) \(840, 820, \underline{\qquad}, 780, \underline{\qquad}, 740\)

Hints

- Find the change between two adjacent known terms. - Check whether the same change works throughout the sequence. - Verify each missing term using the next known term.

Solution

1. Sequence a) increases by \(25\). The missing terms are \(150+25=175\) and \(225+25=250\). 2. Sequence b) decreases by \(20\). The missing terms are \(820-20=800\) and \(780-20=760\).

Answer

a) \(175\) and \(250\) b) \(800\) and \(760\)

5158923

Find the rule and continue each sequence until it has six terms. a) \(15, 30, 45, \ldots\) b) \(910, 880, 850, \ldots\)

Hints

- Find the change from the first term to the second term. - Check whether the same change works from the second term to the third. - State the repeated rule in words. - Count the terms carefully so that each sequence has six terms.

Solution

1. In a), each term is \(15\) greater than the previous term. Continue with \(60, 75,\) and \(90\). 2. In b), each term is \(30\) less than the previous term. Continue with \(820, 790,\) and \(760\).

Answer

a) Rule: add \(15\). Sequence: \(15, 30, 45, 60, 75, 90\) b) Rule: subtract \(30\). Sequence: \(910, 880, 850, 820, 790, 760\)

5159113

Before calculating, notice which place value changes from one expression to the next. Then find each sum. \(300 + 400\) \(350 + 400\) \(350 + 420\) \(358 + 420\) \(358 + 421\)

Hints

- Compare each expression with the one above it. - Focus on the place value that changed. - Use the previous sum to update the next one.

Solution

1. \(300 + 400 = 700\). 2. The first addend increases by \(50\), so \(350 + 400 = 750\). 3. The second addend increases by \(20\), so \(350 + 420 = 770\). 4. The first addend increases by \(8\), so \(358 + 420 = 778\). 5. The second addend increases by \(1\), so \(358 + 421 = 779\).

Answer

\(300 + 400 = 700\) \(350 + 400 = 750\) \(350 + 420 = 770\) \(358 + 420 = 778\) \(358 + 421 = 779\)

5159123

Calculate the sequence. Watch which place value changes from one expression to the next. \(800 - 300\) \(870 - 300\) \(870 - 340\) \(879 - 340\) \(879 - 345\)

Hints

- Compare each expression with the previous one. - Identify whether the first number or the number being subtracted changed and by how much. - Adjust the previous difference instead of starting over.

Solution

1. \(800 - 300 = 500\). 2. The first number increases by \(70\), so \(870 - 300 = 570\). 3. The number being subtracted increases by \(40\), so \(870 - 340 = 530\). 4. The first number increases by \(9\), so \(879 - 340 = 539\). 5. The number being subtracted increases by \(5\), so \(879 - 345 = 534\).

Answer

\(800 - 300 = 500\) \(870 - 300 = 570\) \(870 - 340 = 530\) \(879 - 340 = 539\) \(879 - 345 = 534\)

5159133

Find each difference. Pay attention to how the numbers change step by step. \(600 - 400\) \(630 - 400\) \(630 - 410\) \(635 - 410\) \(635 - 418\)

Hints

- Compare each line with the one before it. - When the number being subtracted grows, the difference decreases by the same amount. - For the last expression, start from \(635 - 410\) and subtract \(8\) more.

Solution

1. \(600 - 400 = 200\). 2. The first number increases by \(30\), so \(630 - 400 = 230\). 3. The number being subtracted increases by \(10\), so \(630 - 410 = 220\). 4. The first number increases by \(5\), so \(635 - 410 = 225\). 5. The number being subtracted increases by \(8\), so \(635 - 418 = 217\).

Answer

\(600 - 400 = 200\) \(630 - 400 = 230\) \(630 - 410 = 220\) \(635 - 410 = 225\) \(635 - 418 = 217\)

5159273

Complete the subtraction sequence. What pattern do you notice? a) \(1000 - 20\) b) \(1000 - 40\) c) \(1000 - 60\)

Hints

- Calculate all three differences. - Compare the numbers being subtracted. - Relate the change in the number being subtracted to the change in the difference.

Solution

1. \(1000 - 20 = 980\). 2. \(1000 - 40 = 960\). 3. \(1000 - 60 = 940\). 4. The number being subtracted increases by \(20\) each time, so the difference decreases by \(20\) each time.

Answer

a) \(980\) b) \(960\) c) \(940\) The differences decrease by \(20\) each time.

5159593

Calculate each sequence. Notice which place value changes in the result from one line to the next. a) \(300 + 400\) \(320 + 400\) \(320 + 450\) \(326 + 450\) \(326 + 451\) b) \(800 - 500\) \(870 - 500\) \(870 - 530\) \(879 - 530\) \(879 - 534\)

Hints

- Compare each expression with the previous one. - Identify whether the change is in hundreds, tens, or ones. - Update the previous result instead of recalculating from the beginning.

Solution

1. For a), the sums are \(700\), \(720\), \(770\), \(776\), and \(777\). The changes move from tens to ones. 2. For b), the differences are \(300\), \(370\), \(340\), \(349\), and \(345\). Again, the later changes affect the ones place.

Answer

a) \(700\), \(720\), \(770\), \(776\), \(777\) b) \(300\), \(370\), \(340\), \(349\), \(345\)

5165263

Calculate each set. Use the relationships among the three expressions. a) \(450 + 30\), \(450 + 6\), \(450 + 36\) b) \(720 + 5\), \(720 + 40\), \(720 + 45\)

Hints

- Notice what changes from the first expression to the second. - Use the first two expressions to combine both changes in the third. - Keep track of whether you are adding tens or ones.

Solution

1. a) \(450 + 30 = 480\) and \(450 + 6 = 456\). Since \(36 = 30 + 6\), \(450 + 36 = 486\). 2. b) \(720 + 5 = 725\) and \(720 + 40 = 760\). Since \(45 = 40 + 5\), \(720 + 45 = 765\).

Answer

a) \(480\), \(456\), \(486\) b) \(725\), \(760\), \(765\)

5165273

Solve each set, using the first two expressions to help with the third. a) \(960 - 40\), \(960 - 8\), \(960 - 48\) b) \(390 - 70\), \(390 - 2\), \(390 - 72\)

Hints

- Subtract in steps: remove the tens first, then the ones. - Watch what happens to the tens place when you subtract the ones. - The third expression combines the two separate subtractions.

Solution

1. a) \(960 - 40 = 920\) and \(960 - 8 = 952\). Since \(48 = 40 + 8\), \(960 - 48 = 912\). 2. b) \(390 - 70 = 320\) and \(390 - 2 = 388\). Since \(72 = 70 + 2\), \(390 - 72 = 318\).

Answer

a) \(920\), \(952\), \(912\) b) \(320\), \(388\), \(318\)

5165303

Count backward by the stated amount and complete each sequence. a) By \(5\)s: \(720, 715, 710, \ldots, 680\) b) By \(50\)s: \(550, 500, 450, \ldots, 200\)

Hints

- Counting backward means each term is smaller than the term before it. - Subtract the stated amount each time.

Solution

1. In a), subtract \(5\) repeatedly after \(710\): \(705, 700, 695, 690, 685, 680\). 2. In b), subtract \(50\) repeatedly after \(450\): \(400, 350, 300, 250, 200\).

Answer

a) \(720, 715, 710, 705, 700, 695, 690, 685, 680\) b) \(550, 500, 450, 400, 350, 300, 250, 200\)

5204063

Study the addition pattern: \(350 + 120 = 470\) \(360 + 120 = 480\) \(370 + 120 = 490\) a) What pattern do you notice in the first addends, \(350\), \(360\), and \(370\)? b) How does the sum change from one equation to the next? c) Use \(370 + 120 = 490\) to find \(400 + 120\) without starting over.

Hints

- Compare each first addend with the one before it. - Decide whether the sum changes by the same amount. - Find how much greater \(400\) is than \(370\).

Solution

1. The first addend increases by \(10\) each time. 2. Because the second addend stays the same, the sum also increases by \(10\) each time. 3. The first addend changes from \(370\) to \(400\), an increase of \(30\). 4. Increase the known sum by \(30\): \(490 + 30 = 520\).

Answer

a) The first addend increases by \(10\) each time. b) The sum increases by \(10\) each time. c) \(400 + 120 = 520\).

5204193

Describe how the difference in a subtraction equation changes in each case. a) The first number increases by \(65\). b) The number being subtracted increases by \(65\).

Hints

- Test each change with an equation such as \(100 - 20 = 80\). - What happens when the amount you start with increases? - What happens when the amount you subtract increases?

Solution

1. Increasing the first number by \(65\) increases the distance between the two numbers by \(65\), so the difference increases by \(65\). 2. Increasing the number being subtracted by \(65\) means \(65\) more is subtracted, so the difference decreases by \(65\).

Answer

a) The difference increases by \(65\). b) The difference decreases by \(65\).

5204713

Anna has saved \(\$350\). She buys a skateboard for \(\$120\) and puts the remaining money back into her savings jar. a) How much money does Anna put in the jar? b) How much would she put in the jar if she had originally saved \(\$100\) more? c) Starting with \(\$350\), how much would she put in the jar if the skateboard cost \(\$20\) more?

Hints

- Subtract the skateboard cost from the amount saved. - For part b), decide how starting with more money changes the remainder. - For part c), decide how spending more changes the remainder. - Use your answer from part a) instead of starting over.

Solution

1. Subtract the cost from the amount saved: \(\$350 - \$120 = \$230\). 2. If Anna started with \(\$100\) more, the remainder would also be \(\$100\) more: \(\$230 + \$100 = \$330\). 3. If the skateboard cost \(\$20\) more, the remainder would be \(\$20\) less: \(\$230 - \$20 = \$210\).

Answer

a) \(\$230\) b) \(\$330\) c) \(\$210\)

5204733

a) A baker makes \(150\) rolls in the morning and sells \(65\) by noon. How many rolls remain? b) A second baker makes \(20\) fewer rolls than the first baker and also sells \(65\). Use your answer from part a) to find how many rolls remain for the second baker.

Hints

- Compare the two starting amounts. - Both bakers sell the same number of rolls. - Use the difference in their starting amounts to adjust the first answer.

Solution

1. The first baker has \(150 - 65 = 85\) rolls left. 2. The second baker starts with \(20\) fewer rolls but sells the same number, so the second remainder is also \(20\) less. 3. The second baker has \(85 - 20 = 65\) rolls left.

Answer

a) \(85\) rolls remain. b) \(65\) rolls remain.

5204763

The difference between two numbers is \(350\). The number being subtracted increases by \(60\), while the first number stays the same. How does the difference change, and what is its new value?

Hints

- Identify the number being subtracted. - Think about what happens when more is subtracted. - Test the pattern with a simple equation such as \(10 - 2 = 8\).

Solution

1. When the number being subtracted increases, more is subtracted, so the difference decreases. 2. The difference decreases by the same amount, \(60\). 3. The new difference is \(350 - 60 = 290\).

Answer

The difference decreases by \(60\), so its new value is \(290\).

5204853

Explore how changing one number affects a subtraction result. a) Find each difference: \(400 - 100 = \dots\) \(450 - 100 = \dots\) \(500 - 100 = \dots\) b) What pattern do you notice when the first number increases by \(50\)? c) Start again with \(400 - 100\). How does the difference change if the number being subtracted increases by \(50\)?

Hints

- Calculate the three differences first. - Compare consecutive results as the first number increases. - In part c), first find the new number being subtracted after increasing \(100\) by \(50\). - Decide whether subtracting more makes the difference greater or smaller.

Solution

1. The differences are \(400 - 100 = 300\), \(450 - 100 = 350\), and \(500 - 100 = 400\). 2. Each time the first number increases by \(50\), the difference also increases by \(50\). 3. Increasing the number being subtracted in the original equation gives \(400 - 150 = 250\). 4. Compared with \(300\), the difference decreases by \(50\).

Answer

a) \(300\), \(350\), \(400\) b) The difference increases by \(50\). c) The difference decreases by \(50\), from \(300\) to \(250\).

5204903

The difference between two numbers is \(240\). The smaller number increases by \(60\). What is the new difference?

Hints

- Think of the difference as the distance between two numbers on a number line. - What happens to that distance when the smaller number moves closer to the larger number? - Test the pattern with \(10 - 4 = 6\).

Solution

1. In the subtraction model, the smaller number is the number being subtracted. 2. Increasing the number being subtracted by \(60\) decreases the difference by \(60\). 3. The new difference is \(240 - 60 = 180\).

Answer

The new difference is \(180\).

5204943

A class wants to save \(\$450\) for a field trip. The class has already saved \(\$280\). a) How much more money does the class need? b) How much would the class still need if it had already saved \(\$30\) more? c) How much would the class still need if the field trip cost \(\$30\) more?

Hints

- Subtract the amount saved from the goal. - For part b), decide how saving more changes the remaining amount. - For part c), decide how a higher goal changes the remaining amount.

Solution

1. The amount still needed is \(\$450 - \$280 = \$170\). 2. Saving \(\$30\) more reduces the amount still needed by \(\$30\): \(\$170 - \$30 = \$140\). 3. Increasing the cost by \(\$30\) increases the amount still needed by \(\$30\): \(\$170 + \$30 = \$200\).

Answer

a) \(\$170\) b) \(\$140\) c) \(\$200\)

5204983

Luke has \(360\) trading cards and gives \(140\) to his younger brother. Maya has \(390\) trading cards and gives \(170\) to her sister. Who has more cards left? Explain your answer.

Hints

- Find how many cards Luke has left. - Find how many cards Maya has left. - Compare the two starting amounts and the two amounts given away. - What happens to a difference when both numbers increase by the same amount?

Solution

1. Luke has \(360 - 140 = 220\) cards left. 2. Maya has \(390 - 170 = 220\) cards left. 3. They have the same number of cards left. 4. Maya started with \(30\) more cards but also gave away \(30\) more cards, so the difference stays the same.

Answer

They each have \(220\) cards left.

5205063

Anya finds \(500 - 140\). Ben finds \(500 - 180\). Who gets the greater difference? Explain without calculating both differences completely. By how much do the two differences differ?

Hints

- Compare the first numbers in the two equations. - What happens to a difference when the number being subtracted increases? - Find the difference between \(180\) and \(140\).

Solution

1. Both equations have the same first number, \(500\). 2. Ben subtracts \(180\), which is \(40\) more than \(140\). 3. Subtracting more produces a smaller difference, so Anya's difference is greater. 4. The two differences differ by \(180 - 140 = 40\).

Answer

Anya gets the greater difference. Her result is \(40\) greater than Ben's.

5205133

Maria finds \(580 - 230 = 350\). Describe how the difference changes in each case. a) Only the first number increases by \(40\). b) Only the number being subtracted increases by \(40\).

Hints

- Think about what happens when you start with more but subtract the same amount. - Think about what happens when you subtract more from the same amount. - Compare each new result with \(350\).

Solution

1. Increasing the first number gives \(620 - 230 = 390\), so the difference increases by \(40\). 2. Increasing the number being subtracted gives \(580 - 270 = 310\), so the difference decreases by \(40\).

Answer

a) The difference increases by \(40\), to \(390\). b) The difference decreases by \(40\), to \(310\).

5207533

Find each sum. What do you notice when you compare the results? a) \(260 + 380\) b) \(450 + 190\) c) \(170 + 470\) d) \(540 + 100\)

Hints

- Find all four sums. - Compare the results. - Look for how a change in one addend is balanced by a change in the other.

Solution

1. The sums are \(260 + 380 = 640\), \(450 + 190 = 640\), \(170 + 470 = 640\), and \(540 + 100 = 640\). 2. Each pair of addends has a total of \(640\). When one addend changes, the other changes by the opposite amount, preserving the sum.

Answer

a) \(640\) b) \(640\) c) \(640\) d) \(640\) All four sums are equal.

5215043

The difference between two numbers is \(380\). The number being subtracted decreases by \(40\). What is the new difference?

Hints

- Think about what happens when you subtract less. - Test the pattern with \(10 - 5 = 5\). - Decide whether the difference changes in the same or opposite direction as the number being subtracted.

Solution

1. Decreasing the number being subtracted means less is subtracted. 2. Therefore, the difference increases by \(40\). 3. The new difference is \(380 + 40 = 420\).

Answer

The new difference is \(420\).

5225233

Decide whether the following statement about consecutive whole numbers is true or false. Explain your reasoning. The sum of two consecutive whole numbers is always even.

Hints

- Test the statement with small pairs of consecutive whole numbers. - Which number in each pair is even, and which is odd? - What happens when an even number and an odd number are added?

Solution

One of two consecutive whole numbers is even and the other is odd. The sum of an even number and an odd number is odd. For example, \(4+5=9\). Therefore, the statement is false.

Answer

The statement is false. The sum of two consecutive whole numbers is always odd.

5351413

Five points are marked on the number line. Determine the numbers represented by \(A\), \(B\), \(C\), \(D\), and \(E\).

Figure for problem 535141

Hints

- Find the value between two labeled ticks. - Count how many equal intervals divide that distance. - Determine the value of one small interval. - Count forward from the nearest labeled value to each marker.

Solution

1. The distance from \(0\) to \(200\) is divided into \(10\) equal intervals, so each small interval represents \(20\). 2. Point \(A\) is \(6\) intervals to the right of \(0\), which is \(120\), so \(A = 120\). 3. Point \(B\) is \(7\) intervals to the right of \(200\), or \(140\) more, so \(B = 340\). 4. Point \(C\) is \(5\) intervals to the right of \(400\), or \(100\) more, so \(C = 500\). 5. Point \(D\) is \(8\) intervals to the right of \(600\), or \(160\) more, so \(D = 760\). 6. Point \(E\) is \(5\) intervals to the right of \(800\), or \(100\) more, so \(E = 900\).

Answer

\(A = 120\), \(B = 340\), \(C = 500\), \(D = 760\), \(E = 900\)

5352483

a) Complete the number wall. b) Add the three numbers in the bottom row. Then add the middle bottom-row number one more time. What do you notice?

Figure for problem 535248

Hints

- Each brick is the sum of the two bricks directly below it. - Complete the wall from bottom to top. - Compare the requested sum with the top brick.

Solution

1. The middle row is \(15 + 22 = 37\) and \(22 + 13 = 35\). 2. The top is \(37 + 35 = 72\). 3. The bottom-row sum is \(15 + 22 + 13 = 50\). 4. Adding the middle number again gives \(50 + 22 = 72\), which equals the top brick.

Answer

a) The middle row is \(37\), \(35\), and the top is \(72\). b) \(15 + 22 + 13 + 22 = 72\). The result equals the number in the top brick.

5352783

Complete the number wall. Each brick is the sum of the two bricks directly below it. What is the top brick?

Figure for problem 535278

Hints

- Each brick is the sum of the two bricks below it. - Start with the bottom row and work upward. - Add neighboring bottom bricks to find each brick above.

Solution

1. Find the second-row bricks: \(12 + 15 = 27\) and \(15 + 20 = 35\). 2. Find the top: \(27 + 35 = 62\).

Answer

Second row: \(27\), \(35\) Top: \(62\)

5352803

Complete the number wall with the larger numbers. Each brick is the sum of the two bricks directly below it.

Figure for problem 535280

Hints

- Use what you know about adding hundreds. - The number-wall rule stays the same for larger numbers. - Check that the two middle bricks add to the top.

Solution

1. Find the second-row bricks: \(120 + 250 = 370\) and \(250 + 380 = 630\). 2. Find the top: \(370 + 630 = 1000\).

Answer

Second row: \(370\), \(630\) Top: \(1000\)

5354283

Complete the number wall. Each brick is the sum of the two adjacent bricks directly below it.

Figure for problem 535428

Hints

- Begin with the bottom row. - Add each neighboring pair to find the brick above it.

Solution

1. The second row is \(4 + 2 = 6\), \(2 + 5 = 7\), and \(5 + 3 = 8\). 2. The third row is \(6 + 7 = 13\) and \(7 + 8 = 15\). 3. The top brick is \(13 + 15 = 28\).

Answer

Second row: \(6\), \(7\), \(8\) Third row: \(13\), \(15\) Top: \(28\)

5354293

Complete the number wall. All values stay within \(1000\).

Figure for problem 535429

Hints

- Add neighboring bottom values to complete the next row. - Recheck each sum before moving upward.

Solution

1. The second row is \(100 + 50 = 150\), \(50 + 120 = 170\), and \(120 + 130 = 250\). 2. The third row is \(150 + 170 = 320\) and \(170 + 250 = 420\). 3. The top brick is \(320 + 420 = 740\).

Answer

Second row: \(150\), \(170\), \(250\) Third row: \(320\), \(420\) Top: \(740\)

5354363

Which number wall has the greater top value? Complete both walls and compare.

Figure for problem 535436

Hints

- Complete the middle row of each wall first. - Then compare the two top values.

Solution

1. In wall a), the second row is \(5 + 5 = 10\) and \(5 + 5 = 10\), so the top is \(10 + 10 = 20\). 2. In wall b), the second row is \(4 + 7 = 11\) and \(7 + 4 = 11\), so the top is \(11 + 11 = 22\). 3. Since \(22 > 20\), wall b) has the greater top value.

Answer

Wall b) has the greater top value: \(22\), compared with \(20\) for wall a).

5354373

Complete both number walls. What do you notice about their top values?

Figure for problem 535437

Hints

- Calculate each wall carefully to the top. - Compare the final values.

Solution

1. In wall 1, the second row is \(200 + 100 = 300\) and \(100 + 200 = 300\), so the top is \(300 + 300 = 600\). 2. In wall 2, the second row is \(220 + 90 = 310\) and \(90 + 200 = 290\), so the top is \(310 + 290 = 600\). 3. Both walls have the same top value.

Answer

Both top values are \(600\).

5373503

Array B was made from array A without changing the number of dots in each row. Describe the change and use array A to calculate the number of dots in array B.

Figure for problem 537350

Hints

- Compare the number of rows in the two arrays. - Use the product for array A instead of recounting every dot.

Solution

1. Array A has \(4\) rows of \(6\) dots, so \(4 \times 6 = 24\). 2. Array B has \(8\) rows of \(6\) dots. The number of rows doubled from \(4\) to \(8\). 3. Therefore, the product also doubles: \(8 \times 6 = 2 \times 24 = 48\).

Answer

The number of rows doubled. Array B contains \(8 \times 6 = 48\) dots.

5373513

Exactly half of the rows in the large array are used to make the small array. What multiplication equation represents the small array? Explain using halving.

Figure for problem 537351

Hints

- Halve the number of rows first. - Halve the product of the large array to check.

Solution

1. The large array represents \(6 \times 8 = 48\). 2. Half of \(6\) rows is \(3\) rows. 3. The small array therefore represents \(3 \times 8 = 24\), which is half of \(48\).

Answer

The small array represents \(3 \times 8 = 24\). The number of rows is halved from \(6\) to \(3\), so the product is halved from \(48\) to \(24\).

5381453

The bar chart shows the number of math puzzles solved each day. Describe the pattern from Monday through Friday.

Figure for problem 538145

Hints

- Read the bar values in order. - Look for a change that repeats. - Check the pattern using more than one pair of bars.

Solution

1. The bar values are \(10, 15, 20, 25,\) and \(30\). 2. Each value is \(5\) greater than the value before it.

Answer

The number of math puzzles solved increases by \(5\) each day.

5381463

The height of each bar should increase by \(4\). Which bar has the wrong value, and what should its value be?

Figure for problem 538146

Hints

- Compare each bar’s value with the value before it. - Add \(4\) to predict each next value. - Check whether the final bar fits after correcting the error.

Solution

1. The pattern begins \(8, 12, 16\), increasing by \(4\) each time. 2. The fourth value should be \(16+4=20\), but the chart shows \(16\). 3. The fifth value, \(24\), fits the intended pattern after the corrected fourth value.

Answer

Bar \(4\) is incorrect. Its value should be \(20\).

5381723

The bars follow a regular pattern. If the pattern continues on Friday, what value should Friday’s bar have?

Figure for problem 538172

Hints

- Read the bar values in order. - Find the change that repeats from day to day. - Apply the same change to Thursday’s value.

Solution

1. The values increase by \(4\) each day: \(12, 16, 20, 24\). 2. The next value is \(24+4=28\).

Answer

Friday’s bar should have a value of \(28\).

5157153

Solve the related subtraction problems. What pattern do you notice? a) \(500 - 142\) b) \(500 - 242\) c) \(500 - 342\)

Hints

- Compare the numbers being subtracted in the three expressions. - Use the first difference to predict the next one. - Think about what happens when the number being subtracted increases by \(100\).

Solution

1. For a), \(500 - 142 = 358\). 2. In b), the number being subtracted is \(100\) greater than in a), so the difference is \(100\) less: \(358 - 100 = 258\). 3. In c), the number being subtracted increases by another \(100\), so the difference decreases by another \(100\): \(258 - 100 = 158\).

Answer

a) \(358\) b) \(258\) c) \(158\) The differences decrease by \(100\) each time.

5157253

Use the digit cards \(4\), \(5\), \(6\), and \(7\) to make two two-digit addends. Use each digit exactly once. a) Put \(4\) and \(5\) in the tens places. What two addition equations can you make, and what is their sum? b) Put \(4\) and \(6\) in the tens places. What two equations can you make, and what is their sum? c) Put \(4\) and \(7\) in the tens places. What is the sum? d) What pattern do you notice in the sums?

Hints

- Decide which digits are in the tens places and which are in the ones places. - After choosing the tens digits, place the remaining two digits in the ones places in both possible orders. - Compare consecutive sums to find the change.

Solution

1. For a), the ones digits are \(6\) and \(7\): \(46 + 57 = 103\) and \(47 + 56 = 103\). 2. For b), the ones digits are \(5\) and \(7\): \(45 + 67 = 112\) and \(47 + 65 = 112\). 3. For c), the ones digits are \(5\) and \(6\): \(45 + 76 = 121\) and \(46 + 75 = 121\). 4. The sums increase by \(9\): \(112 - 103 = 9\) and \(121 - 112 = 9\).

Answer

a) \(46 + 57 = 103\) and \(47 + 56 = 103\) b) \(45 + 67 = 112\) and \(47 + 65 = 112\) c) \(45 + 76 = 121\) and \(46 + 75 = 121\) d) The sums are \(103\), \(112\), and \(121\), increasing by \(9\) each time.

5157263

Use the digits \(1\), \(3\), \(4\), and \(6\) exactly once in each addition equation. a) Find the sums: \(13 + 46\) \(16 + 43\) \(31 + 64\) \(34 + 61\) b) Why do the first two equations have the same sum? c) Why are the last two sums much greater than the first two?

Hints

- Calculate all four sums first. - Compare the tens digits and ones digits in equations with equal sums. - Consider how a digit's value changes when it moves from the ones place to the tens place.

Solution

1. The sums are \(13 + 46 = 59\), \(16 + 43 = 59\), \(31 + 64 = 95\), and \(34 + 61 = 95\). 2. In the first two equations, the same digits occupy the tens places and the same digits occupy the ones places. Only the ones digits switch addends, so the total does not change. 3. In the last two equations, the larger digits occupy the tens places. A digit in the tens place has ten times its value in the ones place, making the total greater.

Answer

a) \(59\), \(59\), \(95\), \(95\) b) The same two digits are in the tens places and the same two digits are in the ones places; only their order between addends changes. c) The larger digits are in the tens places in the last two equations.

5157283

Study this pattern made by reversing the digits of each number. \(41 - 14 = 27\) \(52 - 25 = 27\) \(63 - 36 = 27\) a) Continue the pattern with the next two equations. b) What do you notice about the differences? c) Change the first equation to \(42 - 24\). How does the difference change? Find two more reversed-digit subtraction equations with the same new difference.

Hints

- Observe how both digits change from one line to the next. - Compare the difference between the two digits in each original number. - For part c), look for other two-digit numbers whose tens digit is \(2\) greater than the ones digit.

Solution

1. Increasing both digits by \(1\) gives \(74 - 47 = 27\) and \(85 - 58 = 27\). 2. The difference stays \(27\) because the tens digit is always \(3\) greater than the ones digit. 3. \(42 - 24 = 18\). A digit difference of \(2\) gives the same result, so examples include \(53 - 35 = 18\) and \(64 - 46 = 18\).

Answer

a) \(74 - 47 = 27\) and \(85 - 58 = 27\) b) Every difference is \(27\). c) The new difference is \(18\). Examples: \(53 - 35 = 18\) and \(64 - 46 = 18\).

5157293

Investigate how the difference between two digits affects a reversed-digit subtraction. \(21 - 12 = 9\) because the digits differ by \(1\). \(31 - 13 = 18\) because the digits differ by \(2\). \(41 - 14 = 27\) because the digits differ by \(3\). a) Use the pattern to find \(61 - 16\). b) Predict \(81 - 18\), then verify it. c) Write a reversed-digit subtraction equation with difference \(45\).

Hints

- Compare each digit difference with the corresponding multiple of \(9\). - Notice how the result changes when the digit difference increases by \(1\). - For part c), find two digits that differ by \(5\).

Solution

1. The subtraction result is \(9\) times the difference between the digits. 2. For \(61 - 16\), the digits differ by \(5\), so \(5 \times 9 = 45\). 3. For \(81 - 18\), the digits differ by \(7\), so the prediction is \(7 \times 9 = 63\). Direct subtraction confirms \(81 - 18 = 63\). 4. To obtain \(45\), the digits must differ by \(5\). One example is \(72 - 27 = 45\).

Answer

a) \(45\) b) \(63\); indeed, \(81 - 18 = 63\). c) One example is \(72 - 27 = 45\).

5157593

Square products form a diagonal in a multiplication table. a) Find \(4 \times 4\), \(5 \times 5\), and \(6 \times 6\). b) Find the products in the neighboring table entries \(5 \times 4\) and \(5 \times 6\). Also find \(4 \times 6\) and \(6 \times 4\). c) Compare \(5 \times 5\) with \(4 \times 6\). What do you notice? Check whether the same relationship holds for \(4 \times 4\) and \(3 \times 5\).

Hints

- A square fact has two equal factors. - Locate the neighboring entries around the square fact. - Evaluate the products and compare their difference.

Solution

1. The square products are \(4 \times 4 = 16\), \(5 \times 5 = 25\), and \(6 \times 6 = 36\). 2. The neighboring products are \(5 \times 4 = 20\) and \(5 \times 6 = 30\). Also, \(4 \times 6 = 24\) and \(6 \times 4 = 24\). 3. The square product \(5 \times 5 = 25\) is \(1\) greater than \(4 \times 6 = 24\). 4. The same relationship holds because \(4 \times 4 = 16\) is \(1\) greater than \(3 \times 5 = 15\).

Answer

a) \(16, 25, 36\) b) \(5 \times 4 = 20\), \(5 \times 6 = 30\), \(4 \times 6 = 24\), \(6 \times 4 = 24\) c) In both cases, the square product is \(1\) greater than the product of the numbers immediately before and after the square factor.

5157773

Complete the table so that the two multiplication facts in each row have the same product. <table> <tr> <td>5s facts</td> <td>Product</td> <td>10s facts</td> </tr> <tr> <td>\(2 \times 5 =\)</td> <td>\(10\)</td> <td>\(\square \times 10 =\)</td> </tr> <tr> <td>\(\square \times 5 =\)</td> <td>\(20\)</td> <td>\(2 \times 10 =\)</td> </tr> <tr> <td>\(6 \times 5 =\)</td> <td>\(\square\)</td> <td>\(\square \times 10 =\)</td> </tr> <tr> <td>\(\square \times 5 =\)</td> <td>\(40\)</td> <td>\(4 \times 10 =\)</td> </tr> </table>

Hints

- Start with the product in the middle column. - How many groups of \(10\) make that product? - Once the product is known, find the missing factor in the other fact.

Solution

1. In the first row, \(1 \times 10 = 10\), so the missing factor is \(1\). 2. In the second row, \(4 \times 5 = 20\), so the missing factor is \(4\). 3. In the third row, \(6 \times 5 = 30\). The matching 10s fact is \(3 \times 10 = 30\). 4. In the fourth row, \(8 \times 5 = 40\), so the missing factor is \(8\).

Answer

Row 1: \(1 \times 10\) Row 2: \(4 \times 5\) Row 3: product \(30\); \(3 \times 10\) Row 4: \(8 \times 5\)

5157783

Find the missing factor that makes both sides equal. a) \(1 \times 4 = \square \times 2\) b) \(2 \times 4 = \square \times 2\) c) \(3 \times 4 = \square \times 2\) d) \(4 \times 4 = \square \times 2\) e) \(5 \times 4 = \square \times 2\) What pattern do you notice in the missing factors?

Hints

- Evaluate the left side first. - Then ask which number multiplied by \(2\) gives that product. - Compare each first factor on the left with its missing factor.

Solution

1. The left-side products are \(4, 8, 12, 16\), and \(20\). 2. The matching 2s facts are \(2 \times 2 = 4\), \(4 \times 2 = 8\), \(6 \times 2 = 12\), \(8 \times 2 = 16\), and \(10 \times 2 = 20\). 3. The missing factor is always twice the factor multiplied by \(4\) on the left.

Answer

a) \(2\) b) \(4\) c) \(6\) d) \(8\) e) \(10\) Pattern: Each missing factor is twice the first factor on the left.

5157833

Investigate the relationship between the 5s facts and the 10s facts. First, evaluate: \(2 \times 5 = \square\) \(2 \times 10 = \square\) What happens to the product when you multiply the same number by \(10\) instead of \(5\)? Write two more examples that show your observation.

Hints

- Evaluate the first two facts and compare their products. - How are \(5\) and \(10\) related? - Test the relationship with another factor.

Solution

1. \(2 \times 5 = 10\) and \(2 \times 10 = 20\). 2. The product \(20\) is twice the product \(10\). 3. Because \(10\) is twice \(5\), multiplying the same number by \(10\) gives twice the product of multiplying it by \(5\). 4. For example, \(3 \times 5 = 15\) and \(3 \times 10 = 30\); also, \(4 \times 5 = 20\) and \(4 \times 10 = 40\).

Answer

The product doubles. Possible examples: \(3 \times 5 = 15\) and \(3 \times 10 = 30\) \(4 \times 5 = 20\) and \(4 \times 10 = 40\)

5157843

Find each missing factor so that the two products are equal. a) \(4 \times 4 = \square \times 8\) b) \(6 \times 4 = \square \times 8\) c) \(8 \times 4 = \square \times 8\) d) \(10 \times 4 = \square \times 8\) What rule relates the factors when the products stay equal?

Hints

- Evaluate the left side of each equation. - Which 8s fact has the same product? - Compare the first factors on the two sides.

Solution

1. The left-side products are \(16, 24, 32\), and \(40\). 2. Dividing each product by \(8\) gives the missing factors \(2, 3, 4\), and \(5\). 3. The factor paired with \(8\) is half the factor paired with \(4\). Equivalently, when one factor doubles from \(4\) to \(8\), the other factor must be halved to keep the product equal.

Answer

a) \(4 \times 4 = 2 \times 8\) b) \(6 \times 4 = 3 \times 8\) c) \(8 \times 4 = 4 \times 8\) d) \(10 \times 4 = 5 \times 8\) Rule: Doubling one factor and halving the other keeps the product the same.

5157863

Lucas says, “If I know a product in the 3s facts, I can find the matching product in the 6s facts by doubling.” Test his claim using the factors \(4\) and \(7\). Is Lucas correct? Explain briefly.

Hints

- Evaluate each pair of facts. - Compare the product in the 3s facts with the product in the 6s facts. - How are \(3\) and \(6\) related?

Solution

1. For factor \(4\), \(4 \times 3 = 12\). Doubling \(12\) gives \(24\), and \(4 \times 6 = 24\). 2. For factor \(7\), \(7 \times 3 = 21\). Doubling \(21\) gives \(42\), and \(7 \times 6 = 42\). 3. Lucas is correct because \(6\) is twice \(3\). With the other factor unchanged, doubling one factor doubles the product.

Answer

Yes. \(4 \times 3 = 12\) and \(4 \times 6 = 24\); \(7 \times 3 = 21\) and \(7 \times 6 = 42\). Because \(6\) is twice \(3\), the matching products double.

5158123

Solve each pair. Use related multiplication facts if helpful. a) \(16 \div 2 = \square\) and \(16 \div 4 = \square\) b) \(20 \div 2 = \square\) and \(20 \div 4 = \square\) c) \(40 \div 4 = \square\) and \(40 \div 8 = \square\) What pattern do you notice in the quotients? Explain.

Hints

- Compare the divisors in each pair. - Then compare the two quotients. - How are doubling and halving connected here? - Use multiplication to check the quotients.

Solution

1. The quotients are: a) \(16 \div 2 = 8\) and \(16 \div 4 = 4\); b) \(20 \div 2 = 10\) and \(20 \div 4 = 5\); c) \(40 \div 4 = 10\) and \(40 \div 8 = 5\). 2. In each pair, the dividend stays the same and the second divisor is twice the first divisor. 3. Because each division is exact, doubling the divisor halves the quotient.

Answer

a) \(8\) and \(4\) b) \(10\) and \(5\) c) \(10\) and \(5\) Pattern: With the same dividend, doubling the divisor halves the quotient in these exact division facts.

5158133

Find each quotient. a) \(8 \div 2 = \square\) and \(16 \div 2 = \square\) b) \(12 \div 3 = \square\) and \(24 \div 3 = \square\) c) \(15 \div 5 = \square\) and \(30 \div 5 = \square\) How does the quotient change when the dividend is doubled? Explain.

Hints

- What happens to the dividend in each pair? - Evaluate all the quotients before comparing them. - What rule describes the change in the quotient?

Solution

1. The quotients are: a) \(8 \div 2 = 4\) and \(16 \div 2 = 8\); b) \(12 \div 3 = 4\) and \(24 \div 3 = 8\); c) \(15 \div 5 = 3\) and \(30 \div 5 = 6\). 2. In each pair, the dividend doubles while the divisor stays the same. 3. The quotient also doubles.

Answer

a) \(4\) and \(8\) b) \(4\) and \(8\) c) \(3\) and \(6\) The quotient doubles when the dividend doubles and the divisor stays the same.

5158143

Evaluate and compare the quotients in each pair. a) \(6 \div 3\) and \(12 \div 6\) b) \(10 \div 2\) and \(20 \div 4\) c) \(14 \div 2\) and \(28 \div 4\) Why are the two quotients in each pair equal? Explain using doubling.

Hints

- Evaluate every quotient first. - Compare how the dividend changes and how the divisor changes. - What happens when both numbers in a division equation are doubled?

Solution

1. The quotients are: a) \(6 \div 3 = 2\) and \(12 \div 6 = 2\); b) \(10 \div 2 = 5\) and \(20 \div 4 = 5\); c) \(14 \div 2 = 7\) and \(28 \div 4 = 7\). 2. In each pair, both the dividend and the divisor are doubled. 3. Doubling both numbers in a division equation keeps the quotient unchanged.

Answer

a) \(2\) and \(2\) b) \(5\) and \(5\) c) \(7\) and \(7\) The quotient stays the same because both the dividend and divisor are doubled.

5158823

Continue each sequence or fill in its missing terms. a) \(775, 800, 825, \ldots, \ldots, \ldots, \ldots, \ldots\) (Continue by \(25\)s through \(950\).) b) \(105, 100, 95, \ldots, \ldots, \ldots, \ldots, \ldots\) (Continue backward by \(5\)s through \(70\).) c) \(340, 360, \underline{\qquad}, 400, \underline{\qquad}, 440, \underline{\qquad}\) (Fill in the missing terms in the sequence that increases by \(20\).)

Hints

- Decide whether each sequence increases or decreases. - Find the constant change between consecutive terms. - Continue by adding or subtracting that same amount.

Solution

1. For a), add \(25\) repeatedly after \(825\): \(850, 875, 900, 925, 950\). 2. For b), subtract \(5\) repeatedly after \(95\): \(90, 85, 80, 75, 70\). 3. For c), add \(20\) each time. The missing terms are \(380, 420,\) and \(460\).

Answer

a) \(850, 875, 900, 925, 950\) b) \(90, 85, 80, 75, 70\) c) \(380, 420, 460\)

5158833

Find the requested numbers. Do not include the endpoints. a) Which even whole numbers are strictly between \(894\) and \(912\)? b) Which odd whole numbers are strictly between \(395\) and \(413\)? c) Which multiples of \(10\) are strictly between \(672\) and \(728\)?

Hints

- Use the ones digit to identify even and odd numbers. - A multiple of \(10\) has \(0\) in the ones place. - Check each number against the stated endpoints.

Solution

1. Even numbers have \(0, 2, 4, 6,\) or \(8\) in the ones place. Counting by twos gives \(896\) through \(910\). 2. Odd numbers have \(1, 3, 5, 7,\) or \(9\) in the ones place. Counting by twos gives \(397\) through \(411\). 3. Multiples of \(10\) have \(0\) in the ones place. The requested numbers are \(680\) through \(720\) in steps of \(10\).

Answer

a) \(896, 898, 900, 902, 904, 906, 908, 910\) b) \(397, 399, 401, 403, 405, 407, 409, 411\) c) \(680, 690, 700, 710, 720\)

5159183

Continue the pattern with two more subtraction expressions. Find every difference and describe the pattern. \(542 - 199\) \(552 - 198\) \(562 - 197\)

Hints

- Track the change in the first number from line to line. - Track the change in the number being subtracted. - Combine those two changes to predict the next difference.

Solution

1. \(542 - 199 = 542 - 200 + 1 = 343\). 2. \(552 - 198 = 552 - 200 + 2 = 354\). 3. \(562 - 197 = 562 - 200 + 3 = 365\). 4. The first number increases by \(10\) while the number being subtracted decreases by \(1\), so each difference increases by \(11\). 5. The next expressions are \(572 - 196 = 376\) and \(582 - 195 = 387\).

Answer

\(542 - 199 = 343\) \(552 - 198 = 354\) \(562 - 197 = 365\) \(572 - 196 = 376\) \(582 - 195 = 387\) The differences increase by \(11\) each time.

5159193

Find the differences. Why do all three results stay the same? \(735 - 399\) \(736 - 400\) \(737 - 401\)

Hints

- Compare the first numbers in consecutive expressions. - Compare the numbers being subtracted in the same way. - Think about the distance between two numbers when both increase equally.

Solution

1. \(735 - 399 = 735 - 400 + 1 = 336\). 2. \(736 - 400 = 336\). 3. \(737 - 401 = 336\). 4. From one expression to the next, both the first number and the number being subtracted increase by \(1\). Increasing both numbers by the same amount keeps their difference constant.

Answer

\(735 - 399 = 336\) \(736 - 400 = 336\) \(737 - 401 = 336\) The difference stays the same because both numbers increase by \(1\) each time.

5159603

Compare each pair. Calculate the expression on the left, then use it to find the result on the right. a) \(450 + 300\) and \(450 + 320\) b) \(680 - 200\) and \(680 - 240\) c) \(512 + 100\) and \(512 + 107\) d) \(975 - 400\) and \(975 - 406\)

Hints

- Find the change between the two expressions in each pair. - Decide whether that change makes the second result greater or less. - Adjust the first result by the same amount.

Solution

1. a) \(450 + 300 = 750\). The second addend is \(20\) greater, so the next sum is \(750 + 20 = 770\). 2. b) \(680 - 200 = 480\). The second expression subtracts \(40\) more, so its result is \(480 - 40 = 440\). 3. c) \(512 + 100 = 612\). Adding \(7\) more gives \(612 + 7 = 619\). 4. d) \(975 - 400 = 575\). Subtracting \(6\) more gives \(575 - 6 = 569\).

Answer

a) \(750\) and \(770\) b) \(480\) and \(440\) c) \(612\) and \(619\) d) \(575\) and \(569\)

5161123

Study this subtraction pattern. \(959 - 595\) \(848 - 484\) \(737 - 373\) a) Find the three differences. b) What do you notice? c) Write and solve the next two expressions in the pattern.

Hints

- Track how each digit changes from one line to the next. - Compare the three calculated differences. - Continue the same digit pattern for part c).

Solution

1. \(959 - 595 = 364\), \(848 - 484 = 364\), and \(737 - 373 = 364\). 2. Every difference is \(364\). 3. Each digit in both numbers decreases by \(1\) from one line to the next. 4. The next expressions are \(626 - 262 = 364\) and \(515 - 151 = 364\).

Answer

a) All three differences are \(364\). b) The difference stays constant. c) \(626 - 262 = 364\) and \(515 - 151 = 364\)

5161143

Continue the pattern. The numbers are palindromes, meaning they read the same from left to right and right to left. \(979 - 121 = 858\) \(868 - 121 = 747\) \(757 - 121 = 636\) a) Write the next three equations. b) Describe the pattern in the results.

Hints

- Determine how the first number changes from line to line. - Notice that the number being subtracted remains fixed. - Look for a shared property of \(858\), \(747\), and \(636\).

Solution

1. The first number decreases by \(111\) each time, while \(121\) stays fixed. 2. Continue with \(646 - 121 = 525\), \(535 - 121 = 414\), and \(424 - 121 = 303\). 3. The results are also palindromes and decrease by \(111\) each time.

Answer

a) \(646 - 121 = 525\), \(535 - 121 = 414\), and \(424 - 121 = 303\) b) The results are palindromes that decrease by \(111\) each time.

5161303

Build a number chain using this rule: 1. Start with three different digits. 2. Make the greatest and least three-digit arrangements of those digits. 3. Subtract the least number from the greatest number. 4. Repeat using the digits of the result. 5. Stop when a result repeats. Start with the digits \(8\), \(1\), and \(4\). How many subtraction steps occur before a result repeats? Record the full chain.

Hints

- Rearrange the digits again after every subtraction. - Put the greatest digit in the hundreds place for the greatest number. - Stop as soon as a result has appeared before. - Check each subtraction carefully, especially when regrouping is needed.

Solution

1. Arrange \(8,4,1\): \(841 - 148 = 693\). 2. Arrange \(9,6,3\): \(963 - 369 = 594\). 3. Arrange \(9,5,4\): \(954 - 459 = 495\). 4. Repeating the rule with \(9,5,4\) gives \(954 - 459 = 495\) again. 5. The result repeats on the fourth subtraction step.

Answer

\(841 - 148 = 693\) \(963 - 369 = 594\) \(954 - 459 = 495\) \(954 - 459 = 495\) The first repeated result appears on step \(4\).

5165413

Calculate each pair of multiplication facts and compare the products. a) \(4 \times 2\) and \(4 \times 4\) b) \(6 \times 2\) and \(6 \times 4\) c) \(9 \times 2\) and \(9 \times 4\) What pattern do you notice? What happens to the product when the second factor is doubled?

Hints

- Identify what stays the same in each pair and what changes. - Compare \(2\) with \(4\). - Calculate both products in each pair before describing the pattern.

Solution

1. Part a: \(4 \times 2 = 8\) and \(4 \times 4 = 16\). 2. Part b: \(6 \times 2 = 12\) and \(6 \times 4 = 24\). 3. Part c: \(9 \times 2 = 18\) and \(9 \times 4 = 36\). 4. In each pair, the second factor doubles from \(2\) to \(4\), and the product also doubles.

Answer

a) \(8\) and \(16\) b) \(12\) and \(24\) c) \(18\) and \(36\) When one factor is doubled and the other factor stays the same, the product doubles.

5165423

Continue each pattern and find the products. Pattern A: \(3 \times 6 = 18\) \(4 \times 6 = 24\) \(5 \times 6 = \square\) \(6 \times 6 = \square\) Pattern B: \(9 \times 8 = 72\) \(8 \times 8 = 64\) \(7 \times 8 = \square\) \(6 \times 8 = \square\) How does the product change at each step in Pattern A and in Pattern B?

Hints

- How does the first factor change in each pattern? - Which set of multiplication facts is used in Pattern A? - Which set of multiplication facts is used in Pattern B? - Can you find the next product by adding or subtracting one equal group?

Solution

1. In Pattern A, \(5 \times 6 = 30\) and \(6 \times 6 = 36\). The first factor increases by \(1\), so the product increases by \(6\) each step. 2. In Pattern B, \(7 \times 8 = 56\) and \(6 \times 8 = 48\). The first factor decreases by \(1\), so the product decreases by \(8\) each step.

Answer

Pattern A: \(30, 36\); the product increases by \(6\) each step. Pattern B: \(56, 48\); the product decreases by \(8\) each step.

5175483

In a magic square, every row, every column, and both diagonals have the same sum. Complete the square. <table border="1" style="text-align:center;"> <tr><td>250</td><td>130</td><td>220</td></tr> <tr><td> </td><td>200</td><td> </td></tr> <tr><td>180</td><td> </td><td> </td></tr> </table>

Hints

- Find the sum of the completed first row. - Every row, column, and diagonal must have that same sum. - Next choose a row or column with only one missing number. - Check the final column and both diagonals.

Solution

1. The first row gives the common sum: \(250 + 130 + 220 = 600\). 2. In the first column, the missing number is \(600 - 250 - 180 = 170\). 3. In the second column, the missing number is \(600 - 130 - 200 = 270\). 4. In the second row, the remaining number is \(600 - 170 - 200 = 230\). 5. In the third row, the remaining number is \(600 - 180 - 270 = 150\). 6. Check the third column: \(220 + 230 + 150 = 600\). 7. Check the diagonals: \(250 + 200 + 150 = 600\) and \(220 + 200 + 180 = 600\).

Answer

The completed square is: <table border="1" style="text-align:center;"> <tr><td>250</td><td>130</td><td>220</td></tr> <tr><td>170</td><td>200</td><td>230</td></tr> <tr><td>180</td><td>270</td><td>150</td></tr> </table>

5181453

Continue the pattern and fill in the missing factors. \(7 \times 4 = (7 \times 2) + (7 \times 2)\) \(7 \times 5 = (7 \times 2) + (7 \times 3)\) \(7 \times 6 = (7 \times \dots) + (7 \times \dots)\) \(7 \times 7 = (7 \times \dots) + (7 \times \dots)\) \(7 \times 8 = (7 \times \dots) + (7 \times \dots)\)

Hints

- Look at how the two smaller factors change from one line to the next. - Split each second factor as evenly as possible. - For an even number, the two addends are equal. For an odd number, they differ by \(1\).

Solution

1. The second factor is split into two addends that are equal or differ by \(1\). 2. Since \(6 = 3 + 3\), \(7 \times 6 = 7 \times 3 + 7 \times 3\). 3. Since \(7 = 3 + 4\), \(7 \times 7 = 7 \times 3 + 7 \times 4\). 4. Since \(8 = 4 + 4\), \(7 \times 8 = 7 \times 4 + 7 \times 4\).

Answer

\(7 \times 6 = (7 \times 3) + (7 \times 3)\) \(7 \times 7 = (7 \times 3) + (7 \times 4)\) \(7 \times 8 = (7 \times 4) + (7 \times 4)\)

5203933

In an addition expression, the first addend increases by \(210\) and the second addend decreases by \(130\). How does the sum change overall?

Hints

- Consider each change separately. - One change raises the sum and the other lowers it. - Combine the two changes to find the net effect.

Solution

1. Increasing the first addend increases the sum by \(210\). 2. Decreasing the second addend decreases the sum by \(130\). 3. The net change is \(210 - 130 = 80\). 4. Therefore the sum increases by \(80\).

Answer

The sum increases by \(80\).

5203943

A student's new sum is \(500\) greater than the original sum. The student increased the first addend by \(320\). By how much was the second addend increased?

Hints

- Restate what changed in the original sum. - Identify the unknown change you need to find. - Consider what the sum would do if only the first addend increased by \(320\). - Write down the total change and the known addend change. - The changes in the two addends combine to make the total change in the sum.

Solution

1. The total increase in the sum is \(500\). 2. The first addend accounts for an increase of \(320\). 3. The remaining increase is \(500 - 320 = 180\), so the second addend increased by \(180\).

Answer

The second addend increased by \(180\).

5204073

Start with an addition equation. Increase the first addend by \(15\) and decrease the second addend by \(15\). What happens to the sum? Test your idea with your own example, such as \(100 + 100\), and explain why it works.

Hints

- Choose addends that are easy to work with. - Find the original sum and the new sum. - Think about how the two changes balance each other.

Solution

1. Start with \(100 + 100 = 200\). 2. Increase the first addend: \(100 + 15 = 115\). 3. Decrease the second addend: \(100 - 15 = 85\). 4. The new equation is \(115 + 85 = 200\). 5. The sum stays the same because adding \(15\) to one addend and subtracting \(15\) from the other produces a net change of \(0\).

Answer

The sum stays the same. For example, \(100 + 100 = 200\) and \(115 + 85 = 200\).

5204163

The sum in an addition equation is \(450\). a) How does the sum change if you add \(80\) to the first addend? b) How does the sum change if you subtract \(30\) from the second addend? c) What is the new sum if both changes happen? Explain your reasoning.

Hints

- Think about how changing one addend changes the sum. - Decide whether each change makes the sum greater or less. - Combine the two changes into one net change.

Solution

1. Adding \(80\) to one addend increases the sum by \(80\). 2. Subtracting \(30\) from one addend decreases the sum by \(30\). 3. The combined change is \(80 - 30 = 50\). 4. The new sum is \(450 + 50 = 500\).

Answer

a) The sum increases by \(80\). b) The sum decreases by \(30\). c) The new sum is \(500\) because the net change is an increase of \(50\).

5204173

Consider \(300 + 200 = 500\). How does the sum change if both addends decrease by \(40\)? Find the new sum without recalculating the entire addition equation, and explain your reasoning.

Hints

- Decide how changing one addend affects the sum. - The same change is made to both addends. How many times does it affect the sum? - Combine the two decreases before changing the original sum.

Solution

1. Decreasing the first addend by \(40\) decreases the sum by \(40\). 2. Decreasing the second addend by \(40\) decreases the sum by another \(40\). 3. The total decrease is \(40 + 40 = 80\). 4. The new sum is \(500 - 80 = 420\).

Answer

The sum decreases by \(80\), so the new sum is \(420\).

5204183

In an addition equation, the first addend increases by \(120\), and the second addend decreases by \(50\). How does the sum change overall?

Hints

- Decide how each change affects the sum. - Treat the increase and decrease as opposite changes. - Find the net change.

Solution

1. Increasing the first addend by \(120\) increases the sum by \(120\). 2. Decreasing the second addend by \(50\) decreases the sum by \(50\). 3. The net change is \(120 - 50 = 70\). 4. Therefore, the sum increases by \(70\).

Answer

The sum increases by \(70\).

5204243

How does the difference in a subtraction equation change if the first number increases by \(50\) and the number being subtracted decreases by \(50\)?

Hints

- Think about what happens when you start with more. - Think about what happens when you subtract less. - Test the changes with an equation such as \(200 - 100\).

Solution

1. Increasing the first number by \(50\) increases the difference by \(50\). 2. Decreasing the number being subtracted by \(50\) also increases the difference by \(50\), because less is subtracted. 3. The total increase is \(50 + 50 = 100\).

Answer

The difference increases by \(100\).

5204313

The sum of two numbers is \(480\). a) The first number increases by \(30\). What is the new sum? b) Starting with the sum from part a), the second number then decreases by \(30\). Find the resulting sum and compare it with the original sum of \(480\).

Hints

- Decide how increasing one addend affects the sum. - Then decide how decreasing the other addend by the same amount affects it. - You may test the pattern with smaller numbers.

Solution

1. Increasing one addend by \(30\) increases the sum by \(30\): \(480 + 30 = 510\). 2. Decreasing the other addend by \(30\) decreases the new sum by \(30\): \(510 - 30 = 480\). 3. The final sum equals the original sum because the two changes cancel each other.

Answer

a) The new sum is \(510\). b) The resulting sum is \(480\), the same as the original sum.

5204353

Start with \(430 + 120 = 550\). Increase the first addend by \(50\) and decrease the second addend by \(80\). Find the new sum and describe how it compares with the original sum.

Hints

- Think about how changing one addend affects a sum. - Find each changed addend. - Add the changed numbers. - Compare the new sum with \(550\), or combine the increase and decrease into one net change.

Solution

1. The new first addend is \(430 + 50 = 480\). 2. The new second addend is \(120 - 80 = 40\). 3. The new sum is \(480 + 40 = 520\). 4. The net change is \(50 - 80 = -30\), so the new sum is \(30\) less than \(550\).

Answer

The new sum is \(520\). It is \(30\) less than the original sum.

5204363

Luke starts with \(600 - 250 = 350\). He increases \(600\) by \(40\), but he wants the difference to remain \(350\). What must he do to \(250\)? Explain your reasoning.

Hints

- Think of the difference as the distance between two numbers on a number line. - If one number moves \(40\) units to the right, what must happen to the other number to keep the distance unchanged? - Test the idea with a simpler equation such as \(10 - 5 = 5\).

Solution

1. The new first number is \(600 + 40 = 640\). 2. To keep the difference at \(350\), solve \(640 - x = 350\). 3. The new number being subtracted is \(640 - 350 = 290\). 4. Since \(290 - 250 = 40\), the number being subtracted must also increase by \(40\). 5. Adding the same amount to both numbers keeps their difference unchanged.

Answer

He must increase \(250\) by \(40\), making it \(290\). Then \(640 - 290 = 350\).

5204493

In an addition equation, the first addend increases by \(80\). How must the second addend change so that the sum is only \(50\) greater than the original sum?

Hints

- Compare the increase of \(80\) with the target increase of \(50\). - Decide whether the second addend must increase or decrease. - Test your idea with an equation such as \(100 + 100 = 200\).

Solution

1. The target change in the sum is an increase of \(50\). 2. Increasing the first addend by \(80\) would increase the sum by \(80\). 3. The increase is \(80 - 50 = 30\) too great. 4. Therefore, the second addend must decrease by \(30\).

Answer

The second addend must decrease by \(30\).

5204723

Start with \(670 - 230 = 440\). a) Add \(40\) to the first number. What is the new difference, and how does it compare with \(440\)? b) Starting again with the original equation, add \(40\) to the number being subtracted. What is the new difference, and how does it compare with \(440\)? c) What happens to the difference if you add \(100\) to both numbers in the original equation? Explain.

Hints

- Find each new difference and compare it with \(440\). - Think about how changing the first number differs from changing the number being subtracted. - On a number line, what happens to the distance if both numbers move the same amount?

Solution

1. Increasing the first number gives \(710 - 230 = 480\), which is \(40\) greater than \(440\). 2. Increasing the number being subtracted gives \(670 - 270 = 400\), which is \(40\) less than \(440\). 3. Increasing both numbers gives \(770 - 330 = 440\). The difference stays the same because the distance between the two numbers does not change.

Answer

a) The new difference is \(480\), which is \(40\) greater. b) The new difference is \(400\), which is \(40\) less. c) The difference remains \(440\).

5204743

Describe how the difference changes. a) Find \(780 - 150\). b) Decrease the first number by \(50\). How does the new difference compare with the answer to part a)? c) Start again with the equation in part a). Increase the number being subtracted by \(20\). How does the new difference compare with the answer to part a)?

Hints

- What happens to the difference when the amount you start with decreases? - What happens when the amount you subtract increases? - Try to determine each change without recomputing from scratch.

Solution

1. The original difference is \(780 - 150 = 630\). 2. Decreasing the first number gives \(730 - 150 = 580\). The difference decreases by \(50\). 3. Increasing the number being subtracted gives \(780 - 170 = 610\). The difference decreases by \(20\).

Answer

a) \(630\) b) The difference decreases by \(50\), to \(580\). c) The difference decreases by \(20\), to \(610\).

5204753

The sum of two numbers is \(420\). The first addend increases by \(50\), and the second addend decreases by \(30\). What is the new sum?

Hints

- Decide how each change affects the sum. - Apply the two changes one at a time. - You may also combine them into one net change.

Solution

1. Increasing the first addend by \(50\) increases the sum to \(420 + 50 = 470\). 2. Decreasing the second addend by \(30\) decreases the sum to \(470 - 30 = 440\).

Answer

The new sum is \(440\).

5204863

Anna finds \(630 - 210 = 420\). Ben wants a different subtraction equation with the same difference, \(420\). He decreases the first number by \(30\), from \(630\) to \(600\). How must Ben change the number being subtracted so that the difference remains \(420\)? State whether he must increase or decrease it.

Hints

- Write Ben's equation with \(600\) as the first number and \(420\) as the difference. - Find the missing number being subtracted. - Compare it with \(210\).

Solution

1. Let \(x\) be the new number being subtracted: \(600 - x = 420\). 2. Solve for the number being subtracted: \(x = 600 - 420 = 180\). 3. Compare it with the original number being subtracted: \(210 - 180 = 30\). 4. Ben must decrease the number being subtracted by \(30\). Decreasing both numbers by the same amount keeps the difference unchanged.

Answer

Ben must decrease the number being subtracted by \(30\), from \(210\) to \(180\).

5204913

The difference in a subtraction equation is \(300\). The first number decreases by \(50\). How must the number being subtracted change so that the difference remains \(300\)?

Hints

- Test the situation with an equation such as \(400 - 100 = 300\). - After decreasing the first number, decide whether you must subtract more or less. - Remember that changing both numbers by the same amount keeps their difference unchanged.

Solution

1. Decreasing the first number by \(50\) would decrease the difference from \(300\) to \(250\). 2. To restore the difference to \(300\), the difference must increase by \(50\). 3. Decreasing the number being subtracted by \(50\) increases the difference by \(50\). 4. Therefore, both numbers must decrease by the same amount.

Answer

The number being subtracted must also decrease by \(50\).

5204953

A water tank contains \(650\,\text{L}\). A garden uses \(380\,\text{L}\). a) How many liters remain in the tank? b) Use your answer from part a). How many liters would remain if the garden used \(20\,\text{L}\) less? c) How many liters would remain if the tank had contained \(20\,\text{L}\) more at the start?

Hints

- First find how much water remains. - Decide how using less water changes the remainder. - Decide how starting with more water changes the remainder. - Adjust the answer from part a) instead of starting over.

Solution

1. The amount remaining is \(650 - 380 = 270\), so \(270\,\text{L}\) remain. 2. Using \(20\,\text{L}\) less increases the remainder by \(20\,\text{L}\): \(270 + 20 = 290\). 3. Starting with \(20\,\text{L}\) more also increases the remainder by \(20\,\text{L}\): \(270 + 20 = 290\).

Answer

a) \(270\,\text{L}\) b) \(290\,\text{L}\) c) \(290\,\text{L}\)

5204993

Study the subtraction pattern: \(540 - 210 = 330\) \(550 - 220 = 330\) \(560 - 230 = 330\) a) Write the next equation in the pattern. b) How do the first number and the number being subtracted change in each step? c) Why does the difference stay the same? Explain the rule in your own words.

Hints

- Compare the first numbers from one line to the next. - Compare the numbers being subtracted from one line to the next. - Describe the rule when both numbers in a subtraction increase by the same amount. - Think of the difference as the distance between the two numbers.

Solution

1. Both numbers increase by \(10\), so the next equation is \(570 - 240 = 330\). 2. In each step, the first number and the number being subtracted both increase by \(10\). 3. Adding the same amount to both numbers does not change the distance between them, so the difference remains unchanged.

Answer

a) \(570 - 240 = 330\) b) Both numbers increase by \(10\). c) The difference stays the same because both numbers change by the same amount.

5205143

The difference in a subtraction equation is \(150\). The first number increases by \(30\), and the number being subtracted decreases by \(10\). What is the new difference? Explain your reasoning.

Hints

- Decide how increasing the first number affects the difference. - Decide how decreasing the number being subtracted affects the difference. - Apply the two changes one at a time.

Solution

1. Increasing the first number by \(30\) increases the difference to \(150 + 30 = 180\). 2. Decreasing the number being subtracted by \(10\) means less is subtracted, so the difference increases by another \(10\). 3. The new difference is \(180 + 10 = 190\).

Answer

The new difference is \(190\).

5211793

Fill in the empty cells so that every row and every column has the same sum. <table border="1" style="width:150px; text-align:center;"> <tr><td>25</td><td>45</td><td>20</td></tr> <tr><td> </td><td>30</td><td> </td></tr> <tr><td>30</td><td> </td><td> </td></tr> </table>

Hints

- Find the sum of the completed first row. - Use that same sum for every row and column. - Start with a row or column that has only one empty cell. - Check all three columns at the end.

Solution

1. The completed first row has sum \(25 + 45 + 20 = 90\). 2. In the first column, the missing number is \(90 - 25 - 30 = 35\). 3. In the second column, the missing number is \(90 - 45 - 30 = 15\). 4. In the second row, the last number is \(90 - 35 - 30 = 25\). 5. In the third row, the last number is \(90 - 30 - 15 = 45\). 6. The third column checks: \(20 + 25 + 45 = 90\).

Answer

The completed square is: <table border="1" style="width:150px; text-align:center;"> <tr><td>25</td><td>45</td><td>20</td></tr> <tr><td>35</td><td>30</td><td>25</td></tr> <tr><td>30</td><td>15</td><td>45</td></tr> </table>

5211803

Fill in the empty cells so that every row and every column has the same sum. <table border="1" style="width:150px; text-align:center;"> <tr><td>150</td><td>250</td><td> </td></tr> <tr><td>350</td><td>200</td><td>50</td></tr> <tr><td> </td><td>150</td><td> </td></tr> </table>

Hints

- Find the sum of the completed second row. - Use that same sum for every row and column. - Start where only one number is missing.

Solution

1. The completed second row has sum \(350 + 200 + 50 = 600\). 2. The top-right number is \(600 - 150 - 250 = 200\). 3. The bottom-left number is \(600 - 150 - 350 = 100\). 4. The bottom-right number is \(600 - 100 - 150 = 350\). 5. The third column checks: \(200 + 50 + 350 = 600\).

Answer

The completed square is: <table border="1" style="width:150px; text-align:center;"> <tr><td>150</td><td>250</td><td>200</td></tr> <tr><td>350</td><td>200</td><td>50</td></tr> <tr><td>100</td><td>150</td><td>350</td></tr> </table>

5214363

Compare how the results change. a) Find \(270 + 80\) and \(270 + 90\). How much greater is the second sum? b) Find \(640 - 50\) and \(660 - 50\). Why is the second difference greater than the first?

Hints

- Identify which number changes in each pair of expressions. - Decide whether that change makes the result greater or less. - Use the size of the change to compare the results. - Compare the expressions closely before calculating from scratch.

Solution

1. \(270 + 80 = 350\) and \(270 + 90 = 360\). 2. The second addend increases by \(10\), so the sum also increases by \(10\). 3. \(640 - 50 = 590\) and \(660 - 50 = 610\). 4. The first number in the second subtraction is \(20\) greater while the number being subtracted is unchanged, so the second difference is \(20\) greater.

Answer

a) The sums are \(350\) and \(360\). The second is \(10\) greater. b) The differences are \(590\) and \(610\). The second is \(20\) greater because its first number is \(20\) greater.

5214783

Start with \(120 + 230 = 350\). Answer without recomputing the entire sum each time. a) Anton increases the first addend by \(40\). How does the sum change? b) Starting again with the original equation, Bea decreases the second addend by \(30\). How does the sum change? c) Starting again with the original equation, both addends increase by \(20\). How does the sum change, and what is the new sum?

Hints

- Consider how changing one addend changes the sum. - Apply each change to the known sum of \(350\). - For part c), combine the changes to both addends.

Solution

1. Increasing one addend by \(40\) increases the sum by \(40\), to \(350 + 40 = 390\). 2. Decreasing one addend by \(30\) decreases the sum by \(30\), to \(350 - 30 = 320\). 3. Increasing both addends by \(20\) increases the sum by \(20 + 20 = 40\), so the new sum is \(390\).

Answer

a) The sum increases by \(40\), to \(390\). b) The sum decreases by \(30\), to \(320\). c) The sum increases by \(40\), to \(390\).

5214793

Luke claims, “If I increase the first addend by \(20\) and decrease the second addend by \(20\), the sum always stays the same.” a) Test Luke's claim using \(340 + 160 = 500\). What are the new addends, and what is their sum? b) Explain why the sum stays the same. c) What would happen to \(500\) if both addends increased by \(20\) instead?

Hints

- Change each addend as described and find the new sum. - Compare the increase in one addend with the decrease in the other. - For part c), combine the two increases.

Solution

1. The changed addends are \(340 + 20 = 360\) and \(160 - 20 = 140\). 2. Their sum is \(360 + 140 = 500\), so the claim works for this example. 3. The increase of \(20\) and decrease of \(20\) cancel, producing a net change of \(0\). 4. If both addends increase by \(20\), the sum increases by \(40\), from \(500\) to \(540\).

Answer

a) The new addends are \(360\) and \(140\), and their sum is \(500\). b) The changes cancel because one addend increases by the same amount that the other decreases. c) The sum would increase by \(40\), to \(540\).

5214933

How does a sum change if the first addend increases by \(120\) and the second addend decreases by \(150\)?

Hints

- Decide how each change affects the sum. - Treat the increase and decrease as opposite changes. - Find the net change.

Solution

1. Increasing the first addend increases the sum by \(120\). 2. Decreasing the second addend decreases the sum by \(150\). 3. The net change is \(120 - 150 = -30\). 4. Therefore, the sum decreases by \(30\).

Answer

The sum decreases by \(30\).

5214943

How does a difference change if the first number decreases by \(40\) and the number being subtracted also decreases by \(15\)?

Hints

- Think of subtraction as the first number minus the number being subtracted. - Decide how decreasing the first number affects the difference. - Decide how decreasing the number being subtracted affects the difference. - Test the two changes in order with a simple example such as \(100 - 50\).

Solution

1. Decreasing the first number by \(40\) decreases the difference by \(40\). 2. Decreasing the number being subtracted by \(15\) increases the difference by \(15\), because less is subtracted. 3. The net change is \(-40 + 15 = -25\). 4. Therefore, the difference decreases by \(25\).

Answer

The difference decreases by \(25\).

5215053

Start with \(500 - 150 = 350\). a) Increase the first number by \(20\). What is the new difference? b) Then also increase the number being subtracted by \(20\). How does the resulting difference compare with the original difference of \(350\)?

Hints

- Decide how increasing the first number affects the difference. - Then decide how increasing the number being subtracted affects it. - Think of the difference as the distance between two numbers.

Solution

1. Increasing the first number by \(20\) increases the difference to \(350 + 20 = 370\). 2. Increasing the number being subtracted by \(20\) then decreases the difference by \(20\). 3. The resulting equation is \(520 - 170 = 350\), so the difference returns to its original value.

Answer

a) The new difference is \(370\). b) The difference is \(350\), the same as the original difference.

5215133

Start with \(500 - 200 = 300\). Find the difference in each case. a) The first number increases by \(40\). b) The number being subtracted increases by \(40\). c) Both numbers increase by \(40\).

Hints

- Consider each change separately. - Increasing the first number and increasing the number being subtracted affect the difference in opposite ways. - What happens to the distance between two numbers when both increase by the same amount?

Solution

1. Increasing the first number gives \(540 - 200 = 340\), so the difference increases by \(40\). 2. Increasing the number being subtracted gives \(500 - 240 = 260\), so the difference decreases by \(40\). 3. Increasing both numbers gives \(540 - 240 = 300\). The difference stays the same because both numbers change by the same amount.

Answer

a) \(340\) b) \(260\) c) \(300\)

5215143

Two numbers have a difference of \(150\). The greater number decreases by \(20\), and the lesser number increases by \(10\). What is the new difference?

Hints

- Picture the two numbers as points on a number line. - Decide how moving the greater number toward the lesser number changes the distance. - Then decide how moving the lesser number toward the greater number changes it. - In each step, decide whether the gap becomes larger or smaller.

Solution

1. Decreasing the greater number by \(20\) reduces the difference to \(150 - 20 = 130\). 2. Increasing the lesser number by \(10\) reduces the difference by another \(10\). 3. The new difference is \(130 - 10 = 120\).

Answer

The new difference is \(120\).

5319633

The three number walls form a pattern. 1. For each wall, find the missing numbers in the middle row and the top brick. 2. Compare the top bricks in a), b), and c). What happens when the middle number in the bottom row increases by \(5\)? Explain why.

Figure for problem 531963

Hints

- Add adjacent bricks to fill the middle row. - Add the two middle-row bricks to find the top. - Compare the bottom middle numbers and note how much they increase. - Compare the top numbers and look for a relationship. - Determine how many times the bottom middle number contributes to the top.

Solution

1. In a), the middle row is \(10 + 15 = 25\) and \(15 + 12 = 27\), so the top brick is \(25 + 27 = 52\). 2. In b), the middle row is \(10 + 20 = 30\) and \(20 + 12 = 32\), so the top brick is \(30 + 32 = 62\). 3. In c), the middle row is \(10 + 25 = 35\) and \(25 + 12 = 37\), so the top brick is \(35 + 37 = 72\). 4. The top bricks increase by \(10\). The bottom middle number is used in both middle-row sums, so increasing it by \(5\) increases the top by \(2 \times 5 = 10\).

Answer

1. a) Middle row: \(25\), \(27\); top: \(52\) b) Middle row: \(30\), \(32\); top: \(62\) c) Middle row: \(35\), \(37\); top: \(72\) 2. Each increase of \(5\) in the bottom middle number increases the top by \(10\), because that number contributes to both bricks in the middle row.

5319653

Complete the number wall. Each brick is the sum of the two bricks directly below it.

Figure for problem 531965

Hints

- Look for a group of three bricks where two values are known. - When the upper brick and one lower brick are known, subtract to find the other lower brick. - Enter each new value before moving to nearby bricks. - Use the top brick to check your completed wall.

Solution

1. Find the second brick in the bottom row: \(19 - 7 = 12\). 2. Find the fourth brick in the bottom row: \(22 - 7 = 15\). 3. Find the left brick in the next row: \(8 + 12 = 20\). 4. Find the two bricks in the third row: \(20 + 19 = 39\) and \(19 + 22 = 41\). 5. Check the top: \(39 + 41 = 80\).

Answer

Bottom row: \(8\), \(12\), \(7\), \(15\) Second row: \(20\), \(19\), \(22\) Third row: \(39\), \(41\) Top: \(80\)

5319673

Complete the number wall. Each brick is the sum of the two bricks directly below it. Give the missing numbers in this order: - bottom row, from left to right; - second row, from left to right; - third row, from left to right.

Figure for problem 531967

Hints

- Start with the brick labeled \(12\) and the two bricks below it. - When an upper brick and one lower brick are known, subtract to find the other lower brick. - Work upward where possible, then use the top brick to work backward on the other side.

Solution

1. The middle brick labeled \(12\) is above the unknown bottom brick and \(4\), so the unknown is \(12 - 4 = 8\). 2. The left brick in the second row is \(5 + 8 = 13\). 3. The left brick in the third row is \(13 + 12 = 25\). 4. Use the top to find the right brick in the third row: \(53 - 25 = 28\). 5. Find the right brick in the second row: \(28 - 12 = 16\). 6. Find the rightmost bottom brick: \(16 - 4 = 12\).

Answer

Bottom-row blanks: \(8\), \(12\) Second-row blanks: \(13\), \(16\) Third-row blanks: \(25\), \(28\)

5319753

Complete both number walls. Then compare the bottom-left bricks and the top bricks in a) and b). What do you notice? Explain your finding.

Figure for problem 531975

Hints

- Start with groups of bricks in which two neighboring values are already known. - Use addition to find a brick above two known bricks. - Use subtraction when an upper brick and one brick below it are known. - After completing both walls, compare the bottom-left bricks and the top bricks.

Solution

1. In a), the bottom-left brick is \(32 - 15 = 17\), the right brick in the middle row is \(15 + 24 = 39\), and the top is \(32 + 39 = 71\). 2. In b), the bottom-left brick is \(35 - 15 = 20\), the right brick in the middle row is \(15 + 24 = 39\), and the top is \(35 + 39 = 74\). 3. The bottom-left brick increases by \(3\), from \(17\) to \(20\), and the top also increases by \(3\), from \(71\) to \(74\).

Answer

a) Bottom row: \(17\), \(15\), \(24\); middle row: \(32\), \(39\); top: \(71\) b) Bottom row: \(20\), \(15\), \(24\); middle row: \(35\), \(39\); top: \(74\) The bottom-left brick and the top brick both increase by \(3\).

5319853

Lara builds a three-row number wall with \(6\), \(8\), and \(12\) in the bottom row. a) Complete the wall. What number is in the top brick? b) Lara increases each bottom-row number by \(2\). Complete the new wall. How much greater is its top brick? c) Find a rule. How much greater would the top brick be if each bottom-row number increased by \(5\)?

Figure for problem 531985

Hints

- Each brick is the sum of the two bricks directly below it. - Complete the original and changed walls, then compare their tops. - Compare the increase in each bottom brick with the increase in the top. - Determine which bottom brick contributes more than once to the top.

Solution

1. The middle row is \(6 + 8 = 14\) and \(8 + 12 = 20\), so the top is \(14 + 20 = 34\). 2. The new bottom row is \(8\), \(10\), \(14\). The middle row is \(18\), \(24\), and the top is \(18 + 24 = 42\). It is \(42 - 34 = 8\) greater. 3. If every bottom number increases by the same amount, the top increases by four times that amount: the outer numbers contribute once each, and the middle number contributes twice. An increase of \(5\) in each bottom number increases the top by \(4 \times 5 = 20\).

Answer

a) Middle row: \(14\), \(20\); top: \(34\) b) New bottom row: \(8\), \(10\), \(14\); middle row: \(18\), \(24\); top: \(42\). The top is \(8\) greater. c) The top would be \(20\) greater.

5319933

Complete the number wall. Each brick is the sum of the two bricks directly below it. What numbers belong in the second and fourth positions of the bottom row?

Figure for problem 531993

Hints

- Start with the middle brick in the second row and the \(9\) below it. - Subtract a known lower brick from the brick above to find the other lower brick. - Work across the wall, then use the top brick to work backward.

Solution

1. The middle brick in the second row is \(21\), and one brick below it is \(9\). The other bottom brick is \(21 - 9 = 12\). 2. The left brick in the second row is \(4 + 12 = 16\). 3. The left brick in the third row is \(16 + 21 = 37\). 4. Use the top to find the right brick in the third row: \(70 - 37 = 33\). 5. Find the right brick in the second row: \(33 - 21 = 12\). 6. Find the rightmost bottom brick: \(12 - 9 = 3\).

Answer

Second bottom brick: \(12\) Fourth bottom brick: \(3\)

5320013

Complete the number wall. Each brick is the sum of the two bricks directly below it. What number belongs in the top brick?

Figure for problem 532001

Hints

- Look for groups of three bricks with two known values. - Add two lower bricks to find the brick above them. - Subtract a known lower brick from an upper brick to find the other lower brick. - Find the missing bottom bricks first.

Solution

1. Find the second bottom brick: \(13 - 5 = 8\). 2. Find the rightmost bottom brick: \(15 - 6 = 9\). 3. Find the middle brick in the second row: \(8 + 6 = 14\). 4. Find the third-row bricks: \(13 + 14 = 27\) and \(14 + 15 = 29\). 5. Find the top: \(27 + 29 = 56\).

Answer

\(56\)

5320073

Complete the number wall. Each brick is the sum of the two bricks directly below it. What number belongs in the middle brick of the second row from the bottom?

Figure for problem 532007

Hints

- Each pair of neighboring bricks adds to the brick above. - When an upper brick and one lower brick are known, subtract to find the other lower brick. - Begin with a group of three bricks that has two known values.

Solution

1. Find the second bottom brick: \(13 - 5 = 8\). 2. Find the third bottom brick: \(18 - 6 = 12\). 3. The requested brick is above \(8\) and \(12\): \(8 + 12 = 20\). 4. Check the upper rows: \(13 + 20 = 33\), \(20 + 18 = 38\), and \(33 + 38 = 71\).

Answer

\(20\)

5320213

Complete each number wall. Every brick is the sum of the two bricks directly below it.

Figure for problem 532021

Hints

- Each brick equals the sum of the two bricks below it. - If an upper brick and one lower brick are known, subtract to find the other lower brick. - Start where two of the three connected values are known. - Use each new value to solve the next connected group.

Solution

1. Wall a): \(23 - 8 = 15\), \(12 + 15 = 27\), and \(27 + 23 = 50\). 2. Wall b): \(39 - 14 = 25\), \(71 - 39 = 32\), and \(32 - 14 = 18\). 3. Wall c): \(7 + 22 = 29\), \(70 - 29 = 41\), and \(41 - 22 = 19\).

Answer

a) Bottom middle: \(15\); second-row left: \(27\); top: \(50\) b) Bottom left: \(25\); second-row right: \(32\); bottom right: \(18\) c) Second-row left: \(29\); second-row right: \(41\); bottom right: \(19\)

5320343

Complete the number wall. Each brick is the sum of the two bricks directly below it.

Figure for problem 532034

Hints

- Find a group of three connected bricks with only one missing value. - Begin with the brick labeled \(230\) and the \(90\) below it. - Use each new value to solve neighboring bricks. - Add when moving upward and subtract when moving downward. - Check the completed wall from bottom to top.

Solution

1. Find the second bottom brick: \(230 - 90 = 140\). 2. Find the left brick in the second row: \(110 + 140 = 250\). 3. Find the right brick in the second row: \(490 - 230 = 260\). 4. Find the rightmost bottom brick: \(260 - 90 = 170\). 5. Find the left brick in the third row: \(250 + 230 = 480\). 6. Find the top: \(480 + 490 = 970\).

Answer

Bottom row: \(110\), \(140\), \(90\), \(170\) Second row: \(250\), \(230\), \(260\) Third row: \(480\), \(490\) Top: \(970\)

5320353

Complete the number wall. Each brick is the sum of the two bricks directly below it. Give these values: - the left brick in the second row; - the middle brick in the bottom row; - the top brick.

Figure for problem 532035

Hints

- Start with the right brick in the second row and the two bricks below it. - When an upper brick and one lower brick are known, subtract to find the other lower brick. - Then add upward.

Solution

1. Find the middle bottom brick: \(35 - 23 = 12\). 2. Find the left brick in the second row: \(15 + 12 = 27\). 3. Find the top: \(27 + 35 = 62\).

Answer

Left brick in the second row: \(27\) Middle bottom brick: \(12\) Top brick: \(62\)

5320993

Complete the number wall. Each brick is the sum of the two bricks directly below it. What number belongs in the top brick?

Figure for problem 532099

Hints

- Look for connected groups with two known values. - Use addition to find an upper brick when both lower bricks are known. - Use subtraction to find a lower brick when the upper brick and the other lower brick are known. - Continue until you reach the top.

Solution

1. Find the right brick in the second row: \(14 + 7 = 21\). 2. Find the middle brick in the second row: \(45 - 21 = 24\). 3. Find the missing bottom brick: \(24 - 14 = 10\). 4. Find the left brick in the second row: \(10 + 10 = 20\). 5. Find the left brick in the third row: \(20 + 24 = 44\). 6. Find the top: \(44 + 45 = 89\).

Answer

\(89\)

5352343

Complete the number wall. Each brick is the sum of the two bricks directly below it. Decide where you can begin.

Figure for problem 535234

Hints

- If an upper brick and one lower brick are known, subtract to find the other lower brick. - Look for a connected group with only one missing value. - Work upward or downward depending on the given information.

Solution

1. Find the left brick in the second row: \(63 - 35 = 28\). 2. Find the middle bottom brick: \(35 - 21 = 14\). 3. Find the left bottom brick: \(28 - 14 = 14\).

Answer

Bottom row: \(14\), \(14\), \(21\) Second row: \(28\), \(35\) Top: \(63\)

5352363

The two number walls differ in only one bottom-row brick. a) Complete both walls. b) What do you notice? If the middle brick in the bottom row increases by \(1\), how much does the top increase?

Figure for problem 535236

Hints

- Complete both walls before comparing them. - Determine which two middle-row bricks use the bottom middle number. - Explain why its change reaches the top twice.

Solution

1. In a), the middle row is \(8 + 10 = 18\) and \(10 + 12 = 22\), so the top is \(18 + 22 = 40\). 2. In b), the middle row is \(8 + 11 = 19\) and \(11 + 12 = 23\), so the top is \(19 + 23 = 42\). 3. The bottom middle brick increases by \(1\), while the top increases by \(42 - 40 = 2\).

Answer

a) In a), the middle row is \(18\), \(22\), and the top is \(40\). In b), the middle row is \(19\), \(23\), and the top is \(42\). b) Increasing the bottom middle brick by \(1\) increases the top by \(2\).

5352793

Complete the number wall. Each brick is the sum of the two bricks directly below it.

Figure for problem 535279

Hints

- Use subtraction to find a missing lower brick from an upper brick and the other lower brick. - Ask what number plus \(14\) equals \(30\). - Once the lower row is complete, add upward.

Solution

1. Find the left bottom brick: \(30 - 14 = 16\). 2. Find the right bottom brick: \(26 - 14 = 12\). 3. Find the top: \(30 + 26 = 56\).

Answer

Bottom row: \(16\), \(14\), \(12\) Top: \(56\)

5352813

Complete the large number wall. Each brick is the sum of the two bricks directly below it.

Figure for problem 535281

Hints

- Find a connected group with only one missing value. - Most bricks can be found by adding upward, but one step requires subtraction.

Solution

1. Find the missing bottom brick: \(150 - 80 = 70\). 2. Complete the second row: \(50 + 70 = 120\) and \(80 + 60 = 140\). 3. Complete the third row: \(120 + 150 = 270\) and \(150 + 140 = 290\). 4. Find the top: \(270 + 290 = 560\).

Answer

Bottom row: \(50\), \(70\), \(80\), \(60\) Second row: \(120\), \(150\), \(140\) Third row: \(270\), \(290\) Top: \(560\)

5352833

Complete the number wall. Pay attention to when you must add and when you must subtract.

Figure for problem 535283

Hints

- Begin with a connected group that already has two known values. - You may need to find a higher brick before you can fill a lower blank.

Solution

1. Find the second bottom brick: \(90 - 30 = 60\). 2. Find the left brick in the second row: \(40 + 60 = 100\). 3. Find the right brick in the second row: \(170 - 90 = 80\). 4. Find the rightmost bottom brick: \(80 - 30 = 50\). 5. Find the left brick in the third row: \(100 + 90 = 190\). 6. Find the top: \(190 + 170 = 360\).

Answer

Bottom row: \(40\), \(60\), \(30\), \(50\) Second row: \(100\), \(90\), \(80\) Third row: \(190\), \(170\) Top: \(360\)

5352863

Study the pattern in this number wall, then fill in every blank. Each brick is the sum of the two bricks directly below it.

Figure for problem 535286

Hints

- Look for repeated values and symmetry in the given bricks. - When the upper brick and one lower brick are known, subtract to find the other lower brick.

Solution

1. Find the first and third bricks in the bottom row: \(150 - 50 = 100\) in each case. 2. The middle brick in the second row is \(50 + 100 = 150\), so all three bricks in that row are \(150\). 3. Each brick in the third row is \(150 + 150 = 300\). 4. The top brick is \(300 + 300 = 600\).

Answer

Bottom row: \(100\), \(50\), \(100\), \(50\) Second row: \(150\), \(150\), \(150\) Third row: \(300\), \(300\) Top: \(600\)

5352913

Complete this four-level number wall. Start with the given bricks and work carefully toward every blank.

Figure for problem 535291

Hints

- Begin with the given \(250\) and the \(150\) directly below it. - Use the bottom bricks to complete the second row. - Then add neighboring bricks row by row as you move upward.

Solution

1. The missing bottom brick is \(250 - 150 = 100\). 2. The left brick in the second row is \(50 + 100 = 150\). 3. The right brick in the second row is \(150 + 80 = 230\). 4. The third row is \(150 + 250 = 400\) and \(250 + 230 = 480\). 5. The top brick is \(400 + 480 = 880\).

Answer

Bottom row: \(50\), \(100\), \(150\), \(80\) Second row: \(150\), \(250\), \(230\) Third row: \(400\), \(480\) Top: \(880\)

5353063

Fill in the missing values in the number wall. Remember that each brick equals the sum of the two adjacent bricks directly below it.

Figure for problem 535306

Hints

- Start with a group of three connected bricks that has only one blank. - Add two neighboring lower bricks to find the brick above them. - If the upper brick and one lower brick are known, subtract to find the other lower brick.

Solution

1. Find the two missing bottom bricks: \(15 - 5 = 10\) and \(15 - 7 = 8\). 2. The middle brick in the second row is \(10 + 8 = 18\). 3. The two bricks in the third row are \(15 + 18 = 33\) and \(18 + 15 = 33\). 4. The top brick is \(33 + 33 = 66\).

Answer

Bottom row: \(5\), \(10\), \(8\), \(7\) Second row: \(15\), \(18\), \(15\) Third row: \(33\), \(33\) Top: \(66\)

5353073

Complete the number wall to the top. What repeating pattern appears in the bottom row?

Figure for problem 535307

Hints

- Use the given \(250\) to find a neighboring bottom brick. - The two third-row bricks must add to \(1000\). - Compare the completed bottom-row values for a repeating pattern.

Solution

1. The second bottom brick is \(250 - 100 = 150\). 2. The left brick in the second row is \(100 + 150 = 250\), so the left brick in the third row is \(250 + 250 = 500\). 3. The right brick in the third row is \(1000 - 500 = 500\). 4. The right brick in the second row is \(500 - 250 = 250\). 5. The fourth bottom brick is \(250 - 100 = 150\). 6. The bottom row alternates between \(100\) and \(150\).

Answer

Bottom row: \(100\), \(150\), \(100\), \(150\) Second row: \(250\), \(250\), \(250\) Third row: \(500\), \(500\) Top: \(1000\) The bottom-row pattern is alternating \(100, 150, 100, 150\).

5353103

Complete this five-row number wall. Each brick is the sum of the two bricks directly below it.

Figure for problem 535310

Hints

- First use subtraction to fill the two blanks in the bottom row. - Then work upward one brick at a time. - Each upper brick is the sum of the two bricks directly below it.

Solution

1. Complete the bottom row: \(7 - 4 = 3\) and \(7 - 5 = 2\), giving \(2, 3, 4, 2, 5\). 2. Complete the second row: \(2 + 3 = 5\) and \(4 + 2 = 6\), giving \(5, 7, 6, 7\). 3. Add upward for the third row: \(5 + 7 = 12\), \(7 + 6 = 13\), and \(6 + 7 = 13\). 4. The fourth row is \(12 + 13 = 25\) and \(13 + 13 = 26\). 5. The top brick is \(25 + 26 = 51\).

Answer

Bottom row: \(2\), \(3\), \(4\), \(2\), \(5\) Second row: \(5\), \(7\), \(6\), \(7\) Third row: \(12\), \(13\), \(13\) Fourth row: \(25\), \(26\) Top: \(51\)

5353223

Complete the number wall. Then increase each of the three bottom-row numbers by \(3\). How much greater is the new top brick?

Figure for problem 535322

Hints

- Complete the original wall first. - Build a second wall after increasing every bottom-row number by \(3\). - Compare the two top bricks.

Solution

1. The original middle row is \(1 + 4 = 5\) and \(4 + 2 = 6\), so the original top is \(5 + 6 = 11\). 2. The new bottom row is \(4\), \(7\), \(5\). 3. The new middle row is \(4 + 7 = 11\) and \(7 + 5 = 12\), so the new top is \(11 + 12 = 23\). 4. The new top is \(23 - 11 = 12\) greater.

Answer

The original top is \(11\), and the new top is \(23\). The new top is \(12\) greater.

5353233

Complete the number wall. What happens to the top brick if only the middle bottom-row brick increases by \(5\)? What happens instead if only the two outer bottom-row bricks each increase by \(5\)?

Figure for problem 535323

Hints

- Try the two changes separately. - Determine how many times each bottom-row value contributes to the top.

Solution

1. The original middle row is \(5 + 10 = 15\) and \(10 + 5 = 15\), so the top is \(30\). 2. Increasing only the middle bottom brick by \(5\) gives bottom row \(5\), \(15\), \(5\). The new top is \((5 + 15) + (15 + 5) = 40\), an increase of \(10\). 3. Increasing only the two outer bricks by \(5\) gives bottom row \(10\), \(10\), \(10\). The new top is \(20 + 20 = 40\), also an increase of \(10\).

Answer

In both cases, the top brick increases by \(10\).

5353573

Complete every blank in the number wall. Use each known brick to decide where to add and where to subtract.

Figure for problem 535357

Hints

- Begin with a known brick in the second row and the known bottom brick beneath it. - Check the completed wall by adding from bottom to top.

Solution

1. Find the second bottom brick: \(220 - 120 = 100\). 2. Find the last bottom brick: \(200 - 120 = 80\). 3. The left brick in the second row is \(150 + 100 = 250\). 4. The third row is \(250 + 220 = 470\) and \(220 + 200 = 420\). 5. The top brick is \(470 + 420 = 890\).

Answer

Bottom row: \(150\), \(100\), \(120\), \(80\) Second row: \(250\), \(220\), \(200\) Third row: \(470\), \(420\) Top: \(890\)

5353583

Complete the number wall. What value must go in the bottom-left brick?

Figure for problem 535358

Hints

- Use the \(32\) and the \(15\) beneath it to find the bottom-left value. - Once the bottom row is complete, add upward.

Solution

1. The bottom-left value is \(32 - 15 = 17\). 2. The right brick in the second row is \(15 + 12 = 27\). 3. The top brick is \(32 + 27 = 59\).

Answer

The bottom-left brick is \(17\), the right brick in the second row is \(27\), and the top brick is \(59\).

5353663

Fill in all the missing bricks in the number wall.

Figure for problem 535366

Hints

- Begin at the lower left by using \(8\) and \(14\). - After the bottom row is complete, add adjacent bricks to move upward.

Solution

1. Find the missing bottom brick: \(14 - 8 = 6\). 2. Complete the second row: \(6 + 7 = 13\) and \(7 + 10 = 17\). 3. Complete the third row: \(14 + 13 = 27\) and \(13 + 17 = 30\). 4. The top brick is \(27 + 30 = 57\).

Answer

Bottom row: \(8\), \(6\), \(7\), \(10\) Second row: \(14\), \(13\), \(17\) Third row: \(27\), \(30\) Top: \(57\)

5353683

Calculate the five-row number wall all the way to the top. Every result is less than \(1000\).

Figure for problem 535368

Hints

- Add each pair of neighboring bricks to find the brick directly above them. - Work upward one complete row at a time.

Solution

1. The second row is \(50 + 60 = 110\), \(60 + 40 = 100\), \(40 + 70 = 110\), and \(70 + 30 = 100\). 2. The third row is \(110 + 100 = 210\), \(100 + 110 = 210\), and \(110 + 100 = 210\). 3. The fourth row is \(210 + 210 = 420\) and \(210 + 210 = 420\). 4. The top brick is \(420 + 420 = 840\).

Answer

The top value is \(840\).

5353803

Complete the number wall. Decide whether to add or subtract at each step.

Figure for problem 535380

Hints

- Find a connected set of three bricks with only one unknown value. - When the upper brick and one lower brick are known, subtract.

Solution

1. The bottom-left brick is \(25 - 15 = 10\). 2. The right brick in the second row is \(60 - 25 = 35\). 3. The bottom-right brick is \(35 - 15 = 20\).

Answer

Bottom row: \(10\), \(15\), \(20\) Second row: \(25\), \(35\) Top: \(60\)

5354103

Complete the number wall. Each brick is the sum of the two neighboring bricks directly below it.

Figure for problem 535410

Hints

- Find connected groups with exactly one missing value. - If the upper brick and one lower brick are known, subtract to find the other lower brick. - Continue step by step from each completed group.

Solution

1. The missing bottom values are \(7 - 4 = 3\) and \(7 - 4 = 3\), so the bottom row is \(2, 3, 4, 3, 2\). 2. The outside blanks in the second row are \(2 + 3 = 5\) and \(3 + 2 = 5\), giving \(5, 7, 7, 5\). 3. The third row is \(5 + 7 = 12\), \(7 + 7 = 14\), and \(7 + 5 = 12\). 4. The fourth row is \(12 + 14 = 26\) and \(14 + 12 = 26\). 5. The top is \(26 + 26 = 52\).

Answer

Bottom row: \(2\), \(3\), \(4\), \(3\), \(2\) Second row: \(5\), \(7\), \(7\), \(5\) Third row: \(12\), \(14\), \(12\) Fourth row: \(26\), \(26\) Top: \(52\)

5354133

Find every missing value in this four-row number wall.

Figure for problem 535413

Hints

- Begin by working backward from the known \(20\) to the bottom row. - Later, use the top value to find the missing brick in the third row.

Solution

1. The second bottom brick is \(20 - 12 = 8\). 2. The left brick in the second row is \(5 + 8 = 13\). 3. The left brick in the third row is \(13 + 20 = 33\). 4. The right brick in the third row is \(72 - 33 = 39\). 5. The right brick in the second row is \(39 - 20 = 19\). 6. The last bottom brick is \(19 - 12 = 7\).

Answer

Bottom row: \(5\), \(8\), \(12\), \(7\) Second row: \(13\), \(20\), \(19\) Third row: \(33\), \(39\) Top: \(72\)

5354143

Complete the missing values in the number wall.

Figure for problem 535414

Hints

- When an upper brick and one lower brick are known, subtract to find the other lower brick. - Check the completed wall by adding from the bottom to the top.

Solution

1. The second bottom brick is \(110 - 50 = 60\). 2. The left brick in the second row is \(40 + 60 = 100\). 3. The left brick in the third row is \(100 + 110 = 210\). 4. The right brick in the third row is \(440 - 210 = 230\). 5. The right brick in the second row is \(230 - 110 = 120\). 6. The last bottom brick is \(120 - 50 = 70\).

Answer

Bottom row: \(40\), \(60\), \(50\), \(70\) Second row: \(100\), \(110\), \(120\) Third row: \(210\), \(230\) Top: \(440\)

5354323

Use inverse operations to fill the lower blanks in this number wall.

Figure for problem 535432

Hints

- Work downward by subtracting when an upper brick is known. - Begin with a connected group that has only one blank.

Solution

1. The left brick in the second row is \(750 - 420 = 330\). 2. The middle bottom brick is \(330 - 180 = 150\). 3. The right bottom brick is \(420 - 150 = 270\).

Answer

Bottom row: \(180\), \(150\), \(270\) Second row: \(330\), \(420\) Top: \(750\)

5354383

Fill the blanks in this four-row number wall. You may work upward by adding or downward by subtracting.

Figure for problem 535438

Hints

- Combine addition and subtraction as you move between rows. - Check every connected group of three bricks when you finish.

Solution

1. The left brick in the third row is \(40 - 22 = 18\). 2. The middle brick in the second row is \(18 - 8 = 10\). 3. The second bottom brick is \(8 - 3 = 5\). 4. The third bottom brick is \(10 - 5 = 5\). 5. The right brick in the second row is \(22 - 10 = 12\). 6. Check the last bottom brick: \(12 - 5 = 7\).

Answer

Bottom row: \(3\), \(5\), \(5\), \(7\) Second row: \(8\), \(10\), \(12\) Third row: \(18\), \(22\) Top: \(40\)

5354393

Complete the number wall. All values are no greater than \(1000\).

Figure for problem 535439

Hints

- Use the known \(270\) and \(150\) to fill the bottom-row blank. - Then add upward through the wall.

Solution

1. The third bottom value is \(270 - 150 = 120\). 2. The outside bricks in the second row are \(100 + 150 = 250\) and \(120 + 80 = 200\). 3. The third row is \(250 + 270 = 520\) and \(270 + 200 = 470\). 4. The top is \(520 + 470 = 990\).

Answer

Bottom row: \(100\), \(150\), \(120\), \(80\) Second row: \(250\), \(270\), \(200\) Third row: \(520\), \(470\) Top: \(990\)

5363243

These are subtraction walls. For each brick, use the rule \(\text{upper brick} = \text{lower-left brick} - \text{lower-right brick}\). Find the top brick in a) and b). What happens to the top when the middle number in the bottom row increases by \(1\)?

Figure for problem 536324

Hints

- Complete both walls using the stated subtraction rule. - Compare the two top bricks. - Track how the bottom middle number affects both bricks in the middle row.

Solution

1. In a), the middle row is \(50 - 15 = 35\) and \(15 - 5 = 10\), so the top is \(35 - 10 = 25\). 2. In b), the middle row is \(50 - 16 = 34\) and \(16 - 5 = 11\), so the top is \(34 - 11 = 23\). 3. When the bottom middle number increases by \(1\), the top decreases from \(25\) to \(23\), a decrease of \(2\).

Answer

The top is \(25\) in a) and \(23\) in b). Increasing the bottom middle number by \(1\) decreases the top by \(2\).

5373903

A growing dot pattern forms arrays with dimensions \(2 \times 3\), \(3 \times 4\), and \(4 \times 5\). Describe how the dimensions change, find the number of dots in the next array, and determine how many dots are added from the third array to the next one.

Figure for problem 537390

Hints

- Increase both dimensions by \(1\). - Compare the number of dots in the third array with the number in the next array.

Solution

1. Both the number of rows and the number of columns increase by \(1\) each time. 2. The next array has dimensions \(5 \times 6\), so it contains \(5 \times 6 = 30\) dots. 3. The third array contains \(4 \times 5 = 20\) dots, so the increase is \(30 - 20 = 10\) dots.

Answer

The rows and columns each increase by \(1\). The next array is \(5 \times 6\) with \(30\) dots, an increase of \(10\) dots.

5381643

Which bar has a height exactly halfway between the heights of the other two bars?

Figure for problem 538164

Hints

- Read the three values in order. - Compare the change from A to B with the change from B to C. - Equal changes show the middle value.

Solution

1. The difference from A to B is \(16 - 8 = 8\). 2. The difference from B to C is \(24 - 16 = 8\). 3. Because the differences are equal, B is exactly halfway between A and C.

Answer

Bar B is exactly halfway between Bars A and C.

5381873

Check the statement: “From a) to b), every bar increased by the same amount.”

Figure for problem 538187

Hints

- Compare the same bar in both graphs. - Find all three increases. - Decide whether the increases are equal.

Solution

1. A increased by \(15 - 10 = 5\). 2. B increased by \(25 - 20 = 5\). 3. C increased by \(35 - 30 = 5\). 4. Every bar increased by \(5\), so the statement is true.

Answer

The statement is true. Every bar increased by \(5\).

5381883

Compare the order of the bar heights in Graphs a) and b). 1) How does the order change? 2) Which bar keeps the same value, and what is that value?

Figure for problem 538188

Hints

- Order the three bars in each graph from least to greatest. - Compare the two orders. - Check whether any bar keeps the same value.

Solution

1. In a), \(A < B < C\). 2. In b), \(A > B > C\). 3. The order is reversed, and B stays at \(20\) in both graphs.

Answer

1) The order is reversed. 2) Bar B stays at \(20\).

5157273

Use the digit cards \(2\), \(4\), \(5\), and \(8\) to make two two-digit addends. Use every digit exactly once. a) How should you arrange the digits to make the least possible sum? Find the sum. b) How should you arrange the digits to make the greatest possible sum? Find the sum. c) How many different sums are possible?

Hints

- The tens places contribute more to the sum than the ones places. - Choose the two tens digits systematically and record the resulting sum. - Switching the two ones digits between the addends does not change the sum.

Solution

1. To make the least sum, place the two smallest digits in the tens places. For example, \(25 + 48 = 73\) or \(28 + 45 = 73\). 2. To make the greatest sum, place the two largest digits in the tens places. For example, \(52 + 84 = 136\) or \(54 + 82 = 136\). 3. The six possible unordered pairs of tens digits are \(2,4\), \(2,5\), \(2,8\), \(4,5\), \(4,8\), and \(5,8\). They produce the distinct sums \(73\), \(82\), \(109\), \(100\), \(127\), and \(136\). Therefore there are \(6\) different sums.

Answer

a) Least sum: \(73\), such as \(25 + 48\) b) Greatest sum: \(136\), such as \(52 + 84\) c) There are \(6\) different sums: \(73\), \(82\), \(100\), \(109\), \(127\), and \(136\).

5157303

You can reverse a three-digit number by switching its hundreds and ones digits while keeping the tens digit fixed. For example, \(451\) becomes \(154\). a) \(421 - 124\) b) \(623 - 326\) c) \(825 - 528\) d) What do you notice about the results and their digits?

Hints

- Subtract carefully by place value. - Compare the tens digits of the results. - Add the hundreds and ones digits in each result. - Think about why different starting numbers can produce the same difference.

Solution

1. \(421 - 124 = 297\). 2. \(623 - 326 = 297\). 3. \(825 - 528 = 297\). 4. Every result is \(297\). Its tens digit is \(9\), and its hundreds and ones digits add to \(2 + 7 = 9\).

Answer

a) \(297\) b) \(297\) c) \(297\) d) All three results are \(297\). The tens digit is \(9\), and the outer digits add to \(9\).

5161133

Compare the results of these two subtraction sets. Set A: \(654 - 456\) \(765 - 567\) \(876 - 678\) Set B: \(642 - 246\) \(753 - 357\) \(864 - 468\) a) Find every difference. b) How do the results in Set A compare with those in Set B?

Hints

- Calculate all three expressions in Set A first and compare them. - Repeat for Set B. - Check whether one constant result is a multiple of the other.

Solution

1. Set A gives \(654 - 456 = 198\), \(765 - 567 = 198\), and \(876 - 678 = 198\). 2. Set B gives \(642 - 246 = 396\), \(753 - 357 = 396\), and \(864 - 468 = 396\). 3. Each set has a constant difference. Also, \(396 = 2 \times 198\), so every result in Set B is twice the corresponding result in Set A.

Answer

a) Set A: \(198\), \(198\), \(198\); Set B: \(396\), \(396\), \(396\) b) Each Set B result is twice the Set A result.

5161313

Two students start number chains using the same rule: arrange three digits to make the greatest and least possible numbers, subtract, and repeat with the digits of the result. Maya starts with \(7,2,1\). Theo starts with \(3,0,8\). a) Find each student's first subtraction. b) Continue each chain until it reaches \(495\). c) Compare the two chains.

Hints

- Work on one chain at a time. - With a zero, the least arrangement may begin with \(0\), as in \(038\), which has value \(38\). - Look for a result that appears in both chains.

Solution

1. Maya: \(721 - 127 = 594\). Theo: the least arrangement is \(038\), whose value is \(38\), so \(830 - 38 = 792\). 2. Maya continues with \(954 - 459 = 495\), reaching \(495\) in \(2\) steps. 3. Theo continues: \(972 - 279 = 693\), then \(963 - 369 = 594\), then \(954 - 459 = 495\). 4. Both chains eventually reach \(594\) and then \(495\), although Theo's chain is longer.

Answer

a) Maya: \(721 - 127 = 594\); Theo: \(830 - 38 = 792\) b) Maya: \(594 \rightarrow 495\). Theo: \(792 \rightarrow 693 \rightarrow 594 \rightarrow 495\). c) Both chains meet at \(594\) and end at \(495\).

5175493

In a number triangle, each side number is the sum of the two corner numbers next to that side. The three side numbers are \(220\), \(340\), and \(280\). a) Find the sum of the three corner numbers. Use the fact that the sum of the three side numbers is twice the sum of the three corner numbers. b) Find the three corner numbers.

Hints

- Add the three side numbers, then use the stated relationship to find the corner-number sum. - A side number gives the sum of two corners, so subtract it from the total of all three corners to find the opposite corner. - Check by adding the corner numbers in pairs.

Solution

1. Add the side numbers: \(220 + 340 + 280 = 840\). 2. The corner-number sum is half of \(840\): \(840 \div 2 = 420\). 3. The corner opposite the side labeled \(340\) is \(420 - 340 = 80\). 4. The corner opposite the side labeled \(280\) is \(420 - 280 = 140\). 5. The corner opposite the side labeled \(220\) is \(420 - 220 = 200\). 6. Check: \(80 + 140 = 220\), \(140 + 200 = 340\), and \(200 + 80 = 280\).

Answer

a) \(420\) b) \(80\), \(140\), and \(200\)

5205073

The difference in a subtraction equation is \(300\). The first number increases by \(50\). How must the number being subtracted change so that the new difference is \(320\)? Explain your reasoning.

Hints

- First determine the difference after only the first number increases. - Compare that intermediate difference with \(320\). - To make a difference smaller, should the number being subtracted increase or decrease? - Test the same idea with a simple subtraction such as \(10 - 5 = 5\).

Solution

1. Increasing the first number by \(50\) would increase the difference to \(300 + 50 = 350\). 2. The target difference, \(320\), is \(350 - 320 = 30\) less than \(350\). 3. To decrease the difference by \(30\), increase the number being subtracted by \(30\).

Answer

The number being subtracted must increase by \(30\).

5319663

Complete the large number wall. Each brick is the sum of the two bricks directly below it.

Figure for problem 531966

Hints

- Find a place where an upper brick and one brick below it are known. - Use subtraction to find missing bricks in lower rows. - Use each new value to solve neighboring bricks. - Then add upward to complete the upper rows.

Solution

1. Find the second bottom brick: \(222 - 128 = 94\). 2. Find the third bottom brick: \(209 - 62 = 147\). 3. Find the middle brick in the second row: \(94 + 147 = 241\). 4. Find the third-row bricks: \(222 + 241 = 463\) and \(241 + 209 = 450\). 5. Find the top: \(463 + 450 = 913\).

Answer

Bottom row: \(128\), \(94\), \(147\), \(62\) Second row: \(222\), \(241\), \(209\) Third row: \(463\), \(450\) Top: \(913\)

5319763

Complete both number walls. Each brick is the sum of the two bricks directly below it.

Figure for problem 531976

Hints

- Start at the top. If an upper brick and one brick below it are known, subtract to find the other lower brick. - Continue one row at a time. - Look for groups of three bricks with two known values. - Check each result by adding upward.

Solution

1. Wall a): \(500 - 240 = 260\); \(260 - 160 = 100\); \(160 - 60 = 100\); \(100 - 60 = 40\); \(240 - 100 = 140\); and \(140 - 40 = 100\). 2. Wall b): \(600 - 330 = 270\); \(330 - 130 = 200\); \(270 - 130 = 140\); \(200 - 120 = 80\); \(130 - 80 = 50\); and \(140 - 50 = 90\).

Answer

a) Bottom row: \(100\), \(60\), \(40\), \(100\); second row: \(160\), \(100\), \(140\); third row: \(260\), \(240\); top: \(500\) b) Bottom row: \(120\), \(80\), \(50\), \(90\); second row: \(200\), \(130\), \(140\); third row: \(330\), \(270\); top: \(600\)

5319943

Complete the number wall. Each brick is the sum of the two bricks directly below it. What numbers belong in the first, third, and fourth positions of the bottom row?

Figure for problem 531994

Hints

- Start at the top. Use the top and one known brick below it to find the other brick in that row. - Continue working downward one row at a time. - When an upper brick and one lower brick are known, subtract to find the other lower brick.

Solution

1. Find the left brick in the third row: \(670 - 308 = 362\). 2. Find the middle brick in the second row: \(362 - 189 = 173\). 3. Find the third bottom brick: \(173 - 101 = 72\). 4. Find the right brick in the second row: \(308 - 173 = 135\). 5. Find the fourth bottom brick: \(135 - 72 = 63\). 6. Find the first bottom brick: \(189 - 101 = 88\).

Answer

First bottom brick: \(88\) Third bottom brick: \(72\) Fourth bottom brick: \(63\)

5320363

Use backward reasoning to complete the number wall. Each brick is the sum of the two bricks directly below it. Find these values: - the right brick in the third row from the bottom, next to \(430\); - the left brick in the second row from the bottom; - the second brick from the left in the bottom row.

Figure for problem 532036

Hints

- Begin at the top and find the missing brick next to \(430\). - Use subtraction when working from an upper brick to a missing lower brick. - Use each new value to continue downward. - Check by adding from the bottom upward.

Solution

1. Find the right brick below the top: \(905 - 430 = 475\). 2. Find the left brick in the second row: \(430 - 230 = 200\). 3. Find the second bottom brick: \(200 - 120 = 80\). 4. The remaining values are \(230 - 80 = 150\), \(475 - 230 = 245\), and \(245 - 150 = 95\).

Answer

Right brick in the third row: \(475\) Left brick in the second row: \(200\) Second bottom brick: \(80\)

5321003

Complete the number wall. Each brick is the sum of the two bricks directly below it. What number belongs in the rightmost position of the bottom row?

Figure for problem 532100

Hints

- Start with a connected group that has two known values. - When an upper brick and one lower brick are known, subtract to find the other lower brick. - Enter each new value before choosing the next step. - You may need to work backward from the top.

Solution

1. Find the second bottom brick: \(250 - 140 = 110\). 2. Find the left brick in the second row: \(80 + 110 = 190\). 3. Find the left brick in the third row: \(190 + 250 = 440\). 4. Find the right brick in the third row: \(900 - 440 = 460\). 5. Find the right brick in the second row: \(460 - 250 = 210\). 6. Find the rightmost bottom brick: \(210 - 140 = 70\).

Answer

\(70\)

5352843

Complete the number wall. The values are less familiar, but the same rule applies: each brick is the sum of the two bricks directly below it.

Figure for problem 535284

Hints

- The number-wall rule does not change when the values are less familiar. - Use written side calculations if the arithmetic is difficult to do mentally.

Solution

1. Find the third bottom brick: \(196 - 104 = 92\). 2. Complete the second row: \(123 + 85 = 208\) and \(85 + 92 = 177\). 3. Complete the third row: \(208 + 177 = 385\) and \(177 + 196 = 373\). 4. Find the top: \(385 + 373 = 758\).

Answer

Bottom row: \(123\), \(85\), \(92\), \(104\) Second row: \(208\), \(177\), \(196\) Third row: \(385\), \(373\) Top: \(758\)

5353083

Complete the number wall. Some blanks require you to work backward by subtracting.

Figure for problem 535308

Hints

- If you know a sum and one addend, subtract to find the other addend. - Work from the highest known values toward the lower blanks. - Verify the finished wall by adding from bottom to top.

Solution

1. Find the left brick in the third row: \(800 - 450 = 350\). 2. Find the left and right bricks in the second row: \(350 - 200 = 150\) and \(450 - 200 = 250\). 3. Complete the bottom row: \(150 - 60 = 90\), \(200 - 90 = 110\), and \(250 - 110 = 140\). 4. Check by adding upward: \(60 + 90 = 150\), \(90 + 110 = 200\), and \(110 + 140 = 250\).

Answer

Bottom row: \(60\), \(90\), \(110\), \(140\) Second row: \(150\), \(200\), \(250\) Third row: \(350\), \(450\) Top: \(800\)

5353693

This five-row number wall is almost empty. Use the given values and the symmetry of the wall to complete it.

Figure for problem 535369

Hints

- Use each outside \(75\) with the adjacent \(25\) to find the two outer bottom bricks. - Once the bottom row is known, fill the middle values by adding upward. - Check that the top brick is \(900\).

Solution

1. The two outside bottom bricks are each \(75 - 25 = 50\), so the bottom row is \(50, 25, 100, 25, 50\). 2. The second row is \(50 + 25 = 75\), \(25 + 100 = 125\), \(100 + 25 = 125\), and \(25 + 50 = 75\). 3. The third row is \(75 + 125 = 200\), \(125 + 125 = 250\), and \(125 + 75 = 200\). 4. The fourth row is \(200 + 250 = 450\) and \(250 + 200 = 450\). 5. The top is \(450 + 450 = 900\).

Answer

Bottom row: \(50\), \(25\), \(100\), \(25\), \(50\) Second row: \(75\), \(125\), \(125\), \(75\) Third row: \(200\), \(250\), \(200\) Fourth row: \(450\), \(450\) Top: \(900\)

5354353

Complete all missing values in this four-row number wall.

Figure for problem 535435

Hints

- Begin at the top and subtract downward. - Verify the completed wall by adding upward from the bottom.

Solution

1. The right brick in the third row is \(800 - 380 = 420\). 2. The middle brick in the second row is \(380 - 180 = 200\). 3. The first bottom brick is \(180 - 100 = 80\). 4. The third bottom brick is \(200 - 100 = 100\). 5. The right brick in the second row is \(420 - 200 = 220\). 6. The last bottom brick is \(220 - 100 = 120\).

Answer

Bottom row: \(80\), \(100\), \(100\), \(120\) Second row: \(180\), \(200\), \(220\) Third row: \(380\), \(420\) Top: \(800\)

5352853

The top brick is given. Complete the number wall all the way down. Each brick is the sum of the two bricks directly below it.

Figure for problem 535285

Hints

- Work downward from the top one row at a time. - Check the bottom row by adding upward to reproduce every given brick.

Solution

1. Find the left brick in the third row: \(999 - 555 = 444\). 2. Find the left and right bricks in the second row: \(444 - 244 = 200\) and \(555 - 244 = 311\). 3. Complete the bottom row: \(200 - 80 = 120\), \(244 - 120 = 124\), and \(311 - 124 = 187\).

Answer

Bottom row: \(80\), \(120\), \(124\), \(187\) Second row: \(200\), \(244\), \(311\) Third row: \(444\), \(555\) Top: \(999\)

5354003

Challenge: Complete this number wall even though the blanks appear on several different levels.

Figure for problem 535400

Hints

- Begin with a connected group in which only one value is missing. - Some steps require subtracting downward, while others require adding upward.

Solution

1. The right brick in the third row is \(950 - 450 = 500\). 2. The left and right bricks in the second row are \(450 - 240 = 210\) and \(500 - 240 = 260\). 3. Complete the bottom row: \(210 - 100 = 110\), \(240 - 110 = 130\), and \(260 - 130 = 130\).

Answer

Bottom row: \(100\), \(110\), \(130\), \(130\) Second row: \(210\), \(240\), \(260\) Third row: \(450\), \(500\) Top: \(950\)

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