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Evaluate expressions with grouping symbols

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Difficulty: 1 – 5

5100355

Evaluate the expression: \(672 - [(291 - 144) - (123 - 95)]\)

Hints

- Work from the innermost grouping symbols outward. - Keep track of each subtraction operation. - Evaluate one grouped expression at a time.

Solution

1. Evaluate the inner differences: \(291 - 144 = 147\) and \(123 - 95 = 28\). 2. Evaluate the bracketed expression: \(147 - 28 = 119\). 3. Complete the subtraction: \(672 - 119 = 553\).

Answer

\(553\)

5162255

Complete the table mentally. In each column, evaluate \((n \times 4) \div 2\), where \(n\) is the starting number. Record the value after multiplying by \(4\), and then record the final value after dividing by \(2\). <table> <tr> <td><strong>Starting number</strong></td> <td>\(50\)</td> <td>\(80\)</td> <td>\(120\)</td> </tr> <tr> <td><strong>Result after \(\times 4\)</strong></td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td><strong>Then \(\div 2\)</strong></td> <td> </td> <td> </td> <td> </td> </tr> </table>

Hints

- Work one column at a time. - Use the result in the second row as the starting value for the third row. - What single operation has the same effect as multiplying by \(4\) and then dividing by \(2\)?

Solution

1. For the first column, \(50 \times 4 = 200\), and then \(200 \div 2 = 100\). 2. For the second column, \(80 \times 4 = 320\), and then \(320 \div 2 = 160\). 3. For the third column, \(120 \times 4 = 480\), and then \(480 \div 2 = 240\).

Answer

<table> <tr> <td><strong>Starting number</strong></td> <td>\(50\)</td> <td>\(80\)</td> <td>\(120\)</td> </tr> <tr> <td><strong>Result after \(\times 4\)</strong></td> <td>\(200\)</td> <td>\(320\)</td> <td>\(480\)</td> </tr> <tr> <td><strong>Then \(\div 2\)</strong></td> <td>\(100\)</td> <td>\(160\)</td> <td>\(240\)</td> </tr> </table>

5163155

Which expression has the least value? A: \((560 \div 7)\) B: \((9 \times 9)\) C: \((400 \div 5)\) D: \((6 \times 13)\)

Hints

- Evaluate all four expressions before comparing them. - Use basic facts and place-value patterns for the divisions. - Break \(13\) into \(10 + 3\) for expression D. - Compare the four values carefully.

Solution

1. Evaluate A: \(560 \div 7 = 80\). 2. Evaluate B: \(9 \times 9 = 81\). 3. Evaluate C: \(400 \div 5 = 80\). 4. Evaluate D: \(6 \times 13 = 6 \times 10 + 6 \times 3 = 60 + 18 = 78\). 5. Since \(78 < 80 < 81\), expression D has the least value.

Answer

D: \((6 \times 13)\)

5178975

Evaluate the expression step by step: \(248 - (56 + 79) + 42\)

Hints

- Evaluate inside the grouping symbols first. - Rewrite the entire expression after each step. - When only addition and subtraction remain, work from left to right.

Solution

1. Evaluate inside the grouping symbols: \(56 + 79 = 135\). 2. Substitute: \(248 - 135 + 42\). 3. Work from left to right: \(113 + 42 = 155\).

Answer

\(155\)

5179395

Evaluate each expression. 1. \((16 \div 4) + (24 \div 4)\) 2. \((32 \div 4) - (12 \div 4)\) 3. \((40 \div 4) + (8 \div 4) - (20 \div 4)\) 4. \((28 \div 4) + (4 \div 4) + (12 \div 4)\)

Hints

- Which operations are inside parentheses? - Evaluate each division fact first. - Then follow the addition and subtraction signs carefully. - Use related multiplication facts to check the quotients.

Solution

1. Evaluate the divisions first: \(16 \div 4 = 4\) and \(24 \div 4 = 6\). Then \(4 + 6 = 10\). 2. Evaluate the divisions first: \(32 \div 4 = 8\) and \(12 \div 4 = 3\). Then \(8 - 3 = 5\). 3. Evaluate the divisions first: \(40 \div 4 = 10\), \(8 \div 4 = 2\), and \(20 \div 4 = 5\). Then \(10 + 2 - 5 = 7\). 4. Evaluate the divisions first: \(28 \div 4 = 7\), \(4 \div 4 = 1\), and \(12 \div 4 = 3\). Then \(7 + 1 + 3 = 11\).

Answer

1. \(10\) 2. \(5\) 3. \(7\) 4. \(11\)

5179435

Evaluate the expression step by step: \(1245 - (367 + 233)\)

Hints

- Evaluate inside the grouping symbols first. - Rewrite the expression using the grouped value. - Complete the remaining subtraction.

Solution

1. Evaluate inside the grouping symbols: \(367 + 233 = 600\). 2. Subtract: \(1245 - 600 = 645\).

Answer

\(645\)

5179865

Which comparison symbol makes each statement true? Write \(>\), \(<\), or \(=\). a) \((32 \div 4) + (8 \div 4) \; \square \; 40 \div 4\) b) \((18 \div 6) + (18 \div 9) \; \square \; 18 \div 3\) c) \((50 \div 5) - (15 \div 5) \; \square \; 30 \div 5\)

Hints

- Evaluate the left side and the right side separately. - Compare the two resulting numbers. - For the same dividend, how does a larger divisor affect the quotient?

Solution

1. For part a, the left side is \(8 + 2 = 10\), and the right side is \(10\). Therefore, the values are equal. 2. For part b, the left side is \(3 + 2 = 5\), and the right side is \(6\). Therefore, \(5 < 6\). 3. For part c, the left side is \(10 - 3 = 7\), and the right side is \(6\). Therefore, \(7 > 6\).

Answer

a) \(=\) b) \(<\) c) \(>\)

5185275

Evaluate each numerical expression. a) \((24 \div 3) \times (12 \div 3)\) b) \((42 \div 6) \times (30 \div 6)\) c) \((64 \div 8) \times (40 \div 8)\) d) \((27 \div 9) \times (81 \div 9)\)

Hints

- Evaluate each expression inside parentheses first. - Use multiplication and division facts to find the intermediate values. - Multiply the two intermediate values last.

Solution

1. Evaluate the parentheses: \(24 \div 3 = 8\) and \(12 \div 3 = 4\). Then \(8 \times 4 = 32\). 2. Evaluate the parentheses: \(42 \div 6 = 7\) and \(30 \div 6 = 5\). Then \(7 \times 5 = 35\). 3. Evaluate the parentheses: \(64 \div 8 = 8\) and \(40 \div 8 = 5\). Then \(8 \times 5 = 40\). 4. Evaluate the parentheses: \(27 \div 9 = 3\) and \(81 \div 9 = 9\). Then \(3 \times 9 = 27\).

Answer

a) \(32\) b) \(35\) c) \(40\) d) \(27\)

5191525

Determine which expressions are defined. Evaluate each defined expression and briefly explain any undefined expression. a) \(0 \div 452\) b) \(452 \div 0\) c) \((12 - 12) \div 12\) d) \(12 \div (12 - 12)\)

Hints

- Check whether zero is the dividend or the divisor. - Evaluate expressions inside parentheses first. - Division by a nonzero number is allowed; division by \(0\) is undefined.

Solution

1. \(0 \div 452 = 0\). Zero divided by a nonzero number is \(0\). 2. \(452 \div 0\) is undefined because division by zero is not defined. 3. \((12 - 12) \div 12 = 0 \div 12 = 0\). 4. Since \(12 - 12 = 0\), the expression becomes \(12 \div 0\), which is undefined.

Answer

a) Defined; \(0\) b) Undefined; division by \(0\) c) Defined; \(0\) d) Undefined; the divisor equals \(0\)

5191845

Evaluate the four expressions and determine which has the greatest value. 1) \(60 \times 40\) 2) \(12 \times 5 \times 4 \times 10\) 3) \(20{,}000 \div 8\) 4) \(15 \times 150\)

Hints

- Evaluate each expression separately. - Use place value when multiplying numbers with trailing zeros. - Compare the four results after calculating them.

Solution

1. The values are \(60 \times 40 = 2400\), \(12 \times 5 \times 4 \times 10 = 2400\), \(20{,}000 \div 8 = 2500\), and \(15 \times 150 = 2250\). 2. Since \(2500\) is greatest, expression 3 has the greatest value.

Answer

Expression 3, with a value of \(2500\)

5191855

Which expression has the least value? 1) \(130 \times 4\) 2) \(3500 \div 7\) 3) \(12 \times 5 \times 8\) 4) \(240 + 180 + 90\)

Hints

- Evaluate each expression exactly. - Use convenient grouping in the product with three factors. - Compare the four values and identify the least one.

Solution

1. The values are \(520\), \(500\), \(480\), and \(510\), respectively. 2. The least value is \(480\), from expression 3.

Answer

Expression 3, with a value of \(480\)

5192405

Evaluate the expression efficiently: \(54{,}321 \times (25 \times 4 - 100)\)

Hints

- Evaluate inside the grouping symbols first. - Decide whether the large first factor must actually be multiplied. - Recall the zero property of multiplication.

Solution

1. Evaluate inside the grouping symbols: \(25 \times 4 - 100 = 100 - 100 = 0\). 2. Any number multiplied by \(0\) equals \(0\), so the expression has value \(0\).

Answer

\(0\)

5194535

Evaluate expressions \(A\) and \(B\). Then find the positive difference between their values. \(A = 15 + 5 \times 4\) \(B = (15 + 5) \times 4\)

Hints

- Apply the order of operations to \(A\). - Notice how the grouping symbols change \(B\). - Subtract the lesser value from the greater value.

Solution

1. For \(A\), multiply before adding: \(15 + 5 \times 4 = 15 + 20 = 35\). 2. For \(B\), evaluate the grouping symbols first: \((15 + 5) \times 4 = 20 \times 4 = 80\). 3. The positive difference is \(80 - 35 = 45\).

Answer

\(A = 35\), \(B = 80\), and the positive difference is \(45\).

5196555

Evaluate each expression efficiently. Decide whether it is easier to evaluate inside the grouping symbols first or to use the distributive property. a) \((15 + 5) \times 24\) b) \(73 \times (18 - 8)\) c) \((125 - 25) \times 9\)

Hints

- Evaluate the numbers inside each set of grouping symbols. - Check whether that result is a convenient multiple of \(10\) or \(100\). - Compare this method with distributing before choosing a strategy.

Solution

1. In each expression, evaluating inside the grouping symbols first produces a convenient multiple of \(10\) or \(100\). 2. For a), \((15 + 5) \times 24 = 20 \times 24 = 480\). 3. For b), \(73 \times (18 - 8) = 73 \times 10 = 730\). 4. For c), \((125 - 25) \times 9 = 100 \times 9 = 900\).

Answer

a) Evaluate inside the grouping symbols first: \(20 \times 24 = 480\). b) Evaluate inside the grouping symbols first: \(73 \times 10 = 730\). c) Evaluate inside the grouping symbols first: \(100 \times 9 = 900\).

5213975

Evaluate each expression using the order of operations. a) \(0 \times 13 + 7\) b) \(25 - 0 \times 4\) c) \(6 \times 0 + 8 \times 1\) d) \(1 \times 9 - 9 \times 0\) e) \(0 \times (15 + 25)\)

Hints

- Evaluate multiplication before addition or subtraction. - Use the multiplication properties of \(0\) and \(1\). - In part e, notice the factor outside the grouping symbols.

Solution

1. For a), \(0 \times 13 + 7 = 0 + 7 = 7\). 2. For b), \(25 - 0 \times 4 = 25 - 0 = 25\). 3. For c), \(6 \times 0 + 8 \times 1 = 0 + 8 = 8\). 4. For d), \(1 \times 9 - 9 \times 0 = 9 - 0 = 9\). 5. For e), any product with factor \(0\) is \(0\), so the value is \(0\).

Answer

a) \(7\) b) \(25\) c) \(8\) d) \(9\) e) \(0\)

5352005

Write the numerical expression represented by the expression tree, and then evaluate it step by step.

Figure for problem 535200

Hints

- Identify which numbers are combined in the upper branch. - Use grouping symbols so that addition occurs before subtraction. - The root operation is performed last.

Solution

1. The expression is \(150 - (45 + 35)\). 2. Evaluate the grouped sum: \(45 + 35 = 80\). 3. Subtract: \(150 - 80 = 70\).

Answer

Expression: \(150 - (45 + 35)\) Value: \(70\)

5352075

Fill in the empty boxes in the expression tree. Then write and evaluate the corresponding numerical expression.

Figure for problem 535207

Hints

- Follow the expression tree from the top toward the root. - The sum is used as one factor in the next step. - Use grouping symbols because the addition occurs first.

Solution

1. Add: \(18 + 7 = 25\). 2. Multiply: \(25 \times 4 = 100\). 3. The expression is \((18 + 7) \times 4\).

Answer

Intermediate value: \(25\) Expression: \((18 + 7) \times 4\) Final value: \(100\)

5352085

Find the intermediate and final values in the expression tree. Then write the corresponding numerical expression.

Figure for problem 535208

Hints

- Follow one level of the tree at a time. - The subtraction must be completed before division. - Use grouping symbols to show that order.

Solution

1. Subtract: \(32 - 14 = 18\). 2. Divide: \(18 \div 3 = 6\). 3. The expression is \((32 - 14) \div 3\).

Answer

Intermediate value: \(18\) Expression: \((32 - 14) \div 3\) Final value: \(6\)

5352095

Complete the expression tree. Then write and evaluate the corresponding numerical expression.

Figure for problem 535209

Hints

- Identify the multiplication branch first. - Add \(26\) to that intermediate value. - Use the tree to check the order of operations.

Solution

1. Multiply: \(9 \times 6 = 54\). 2. Add: \(54 + 26 = 80\). 3. The expression is \(9 \times 6 + 26\).

Answer

Intermediate value: \(54\) Expression: \(9 \times 6 + 26\) Final value: \(80\)

5352115

Find the missing values in the expression tree. Then write and evaluate the corresponding numerical expression.

Figure for problem 535211

Hints

- Evaluate the product on the right branch first. - Subtract that product from \(60\). - The order of operations already matches the tree.

Solution

1. Multiply: \(8 \times 4 = 32\). 2. Subtract: \(60 - 32 = 28\). 3. The expression is \(60 - 8 \times 4\).

Answer

Intermediate value: \(32\) Expression: \(60 - 8 \times 4\) Final value: \(28\)

5352125

Find the final value of the expression tree. Then write the corresponding numerical expression.

Figure for problem 535212

Hints

- Identify the complete divisor in the tree. - Evaluate the difference before dividing. - Use grouping symbols around the divisor.

Solution

1. Evaluate the difference: \(25 - 5 = 20\). 2. Divide: \(100 \div 20 = 5\). 3. The expression is \(100 \div (25 - 5)\).

Answer

Expression: \(100 \div (25 - 5)\) Final value: \(5\)

5352175

Complete the expression tree. Then write and evaluate the corresponding numerical expression.

Figure for problem 535217

Hints

- Follow the tree from the upper operation toward the root. - Divide before subtracting.

Solution

1. Divide: \(150 \div 5 = 30\). 2. Subtract: \(30 - 10 = 20\). 3. The expression is \(150 \div 5 - 10\).

Answer

Expression: \(150 \div 5 - 10\) Final value: \(20\)

5352185

Write the numerical expression represented by the expression tree, and then evaluate it.

Figure for problem 535218

Hints

- The sum on the left branch must be found first. - Use grouping symbols around that sum.

Solution

1. Add: \(25 + 35 = 60\). 2. Divide: \(60 \div 5 = 12\). 3. The expression is \((25 + 35) \div 5\).

Answer

Expression: \((25 + 35) \div 5\) Final value: \(12\)

5352425

Write the numerical expression represented by the expression tree, and then evaluate it. Use grouping symbols only if they are needed.

Figure for problem 535242

Hints

- Identify the operation on the left branch. - Decide whether the usual order of operations already matches the tree.

Solution

1. Multiply: \(12 \times 8 = 96\). 2. Subtract: \(96 - 46 = 50\). 3. The expression is \(12 \times 8 - 46\). Grouping symbols around the product are optional because multiplication is performed before subtraction.

Answer

Expression: \(12 \times 8 - 46\) Final value: \(50\)

5353035

For a class party, Mr. Weber buys \(4\) cases of juice for \(\$14\) each and \(3\) large bags of pretzels for \(\$8\) each. He also spends \(\$6\) on decorations. The calculation tree shows how to find the total cost. a) Find the values for the empty boxes in the calculation tree. b) What is the total cost?

Figure for problem 535303

Hints

- Identify the operation at each branch of the tree. - Work from the starting numbers toward the final result. - Match each number in the word problem with its position in the tree.

Solution

1. The juice costs \(4\times \$14=\$56\). 2. The pretzels cost \(3\times \$8=\$24\). 3. The pretzels and decorations cost \(\$24+\$6=\$30\). 4. The total cost is \(\$56+\$30=\$86\).

Answer

a) The intermediate values are \(56\), \(24\), and \(30\). b) The total cost is \(\$86\).

5353365

Complete the missing values in the calculation tree. What final value do you obtain?

Figure for problem 535336

Hints

- Begin with the operation whose two input values are already shown. - Carry that result into the next operation to find the final value.

Solution

1. Evaluate the multiplication branch first: \(25 \times 3 = 75\). 2. Add the left branch value: \(75 + 75 = 150\).

Answer

The missing values are \(75\) in the multiplication box and \(150\) in the bottom box.

5353435

Use the expression tree to evaluate the numerical expression.

Figure for problem 535343

Hints

- Begin with the subtraction in the inner branch. - Use that result in the final addition.

Solution

1. Evaluate the first operation: \(1250 - 750 = 500\). 2. Add \(400\): \(500 + 400 = 900\).

Answer

The value in the final box is \(900\).

5162175

Evaluate \(((6 \times 70) \div 3) \times 5\). Work through the grouping symbols in order. What is the final value?

Hints

- Use basic facts and place-value patterns for the larger numbers. - Break hundreds or tens into parts when helpful. - Work in order and record each intermediate result.

Solution

1. Multiply the starting number by \(70\): \(6 \times 70 = 420\). 2. Divide by \(3\): \(420 \div 3 = 140\). 3. Multiply by \(5\): \(140 \times 5 = 700\).

Answer

The final number is \(700\).

5170305

Evaluate each expression using the grouping symbols, and compare the two values in each pair. Pair 1: \(12\times(9+1)+13\times(10+1)\) \(12\times(10+1)+13\times(9+1)\) Pair 2: \(20\times(29+1)+21\times(30+1)\) \(20\times(30+1)+21\times(29+1)\) What do you notice within each pair?

Hints

- Evaluate every expression inside parentheses first. - Perform each multiplication before adding. - Keep the two expressions in each pair separate. - Compare the final values within each pair.

Solution

1. In Pair 1, evaluate the grouping symbols first: \(9+1=10\) and \(10+1=11\). Then \(12\times 10+13\times 11=120+143=263\). 2. The other expression is \(12\times 11+13\times 10=132+130=262\). 3. In Pair 2, \(29+1=30\) and \(30+1=31\). Then \(20\times 30+21\times 31=600+651=1251\). 4. The other expression is \(20\times 31+21\times 30=620+630=1250\). 5. In each pair, the first value is exactly \(1\) greater than the second value.

Answer

Pair 1: \(263\) and \(262\) Pair 2: \(1251\) and \(1250\) In each pair, the first value is \(1\) greater than the second value.

5174735

Evaluate each expression. Compare parts a and b, and compare parts c and d. What do you notice? a) \((12 + 8) \times 3\) b) \(12 \times 3 + 8 \times 3\) c) \((24 - 12) \div 4\) d) \(24 \div 4 - 12 \div 4\)

Hints

- Evaluate the parentheses first in parts a and c. - In parts b and d, perform the multiplication or division before addition or subtraction. - Compare the values in each pair.

Solution

1. Part a: \((12 + 8) \times 3 = 20 \times 3 = 60\). 2. Part b: \(12 \times 3 + 8 \times 3 = 36 + 24 = 60\). 3. Part c: \((24 - 12) \div 4 = 12 \div 4 = 3\). 4. Part d: \(24 \div 4 - 12 \div 4 = 6 - 3 = 3\). 5. Parts a and b have the same value, and parts c and d have the same value. In these expressions, applying the operation to the grouped terms or to each term separately gives equivalent results.

Answer

a) \(60\) b) \(60\) c) \(3\) d) \(3\) The expressions in each pair have equal values.

5177835

Use properties of operations and grouping symbols to calculate efficiently. a) \(438 + 275 + 162\) b) \(5600 - 1200 - 400\) c) \(125 + 999 + 875\) d) \(10{,}000 - 2500 - 3500\)

Hints

- Look for two addends that make a multiple of \(100\) or \(1000\). - You may reorder and regroup addends without changing their sum. - For consecutive subtractions, consider combining the amounts being subtracted.

Solution

1. For a), regroup the addends: \((438 + 162) + 275 = 600 + 275 = 875\). 2. For b), combine the amounts being subtracted: \(5600 - (1200 + 400) = 5600 - 1600 = 4000\). 3. For c), regroup the addends: \((125 + 875) + 999 = 1000 + 999 = 1999\). 4. For d), combine the amounts being subtracted: \(10{,}000 - (2500 + 3500) = 10{,}000 - 6000 = 4000\).

Answer

a) \(875\) b) \(4000\) c) \(1999\) d) \(4000\)

5178985

Evaluate the expression step by step: \(1500 - (340 + 160) - (480 - 120)\)

Hints

- Evaluate both grouped expressions first. - Keep the subtraction signs outside the grouping symbols. - Complete the remaining subtractions from left to right.

Solution

1. Evaluate the grouped expressions: \(340 + 160 = 500\) and \(480 - 120 = 360\). 2. Substitute: \(1500 - 500 - 360\). 3. Work from left to right: \(1000 - 360 = 640\).

Answer

\(640\)

5178995

Evaluate the expression using the order of operations: \(980 - (120 + (340 - 160)) + 45\)

Hints

- Work from the innermost grouping symbols outward. - Rewrite the expression after each grouped calculation. - Complete the final addition and subtraction from left to right.

Solution

1. Evaluate the innermost grouping: \(340 - 160 = 180\). 2. Evaluate the outer grouping: \(120 + 180 = 300\). 3. Substitute and calculate: \(980 - 300 + 45 = 680 + 45 = 725\).

Answer

\(725\)

5179135

First estimate by rounding each number to the nearest hundred. Then evaluate the expression exactly and compare the two results. \(4873 - [(1214 + 786) + (654 - 354)]\)

Hints

- Round every number to the nearest hundred for the estimate. - Evaluate the exact expression from the innermost grouping outward. - Use the estimate to check whether the exact result is reasonable.

Solution

1. Estimate: \(4900 - [(1200 + 800) + (700 - 400)] = 4900 - 2300 = 2600\). 2. Evaluate exactly: \(1214 + 786 = 2000\) and \(654 - 354 = 300\). 3. Then \(4873 - (2000 + 300) = 4873 - 2300 = 2573\). 4. The exact value \(2573\) is close to the estimate \(2600\).

Answer

Estimate: \(2600\) Exact value: \(2573\)

5179145

Choose a reasonable rounding place to estimate the value. Then evaluate the expression exactly. \(1245 + 3782 - (567 + 1233)\)

Hints

- Round to values that make the addition and subtraction easy. - Evaluate inside the grouping symbols first for the exact value. - Compare the exact result with the estimate.

Solution

1. One reasonable estimate using hundreds is \(1200 + 3800 - (600 + 1200) = 5000 - 1800 = 3200\). 2. Evaluate exactly: \(567 + 1233 = 1800\), and \(1245 + 3782 = 5027\). 3. Therefore, \(5027 - 1800 = 3227\).

Answer

One estimate: \(3200\) Exact value: \(3227\)

5179155

Estimate the value by rounding to the nearest ten. Then evaluate the expression exactly. \(872 - [245 - (134 - 89)]\)

Hints

- Round each number to the nearest ten for the estimate. - For the exact value, work from the innermost grouping outward. - Check that the exact result is reasonably close to the estimate.

Solution

1. Estimate: \(870 - [250 - (130 - 90)] = 870 - 210 = 660\). 2. Evaluate exactly from the inside outward: \(134 - 89 = 45\), then \(245 - 45 = 200\). 3. Therefore, \(872 - 200 = 672\).

Answer

Estimate: \(660\) Exact value: \(672\)

5179285

Evaluate the expression using the order of operations: \(7500 - [(1200 + 800) - (450 + 150)]\)

Hints

- Work from the innermost grouping symbols outward. - Record the value of each inner sum. - Use those values to evaluate the bracketed difference.

Solution

1. Evaluate the inner sums: \(1200 + 800 = 2000\) and \(450 + 150 = 600\). 2. Evaluate the bracketed difference: \(2000 - 600 = 1400\). 3. Complete the final subtraction: \(7500 - 1400 = 6100\).

Answer

\(6100\)

5179445

Evaluate the expression step by step: \(456 + (321 - 105 - 66)\)

Hints

- Evaluate the grouped expression first. - With only addition and subtraction, work from left to right. - Keep the number outside the grouping symbols unchanged until the grouped value is known.

Solution

1. Inside the grouping symbols, work from left to right: \(321 - 105 = 216\). 2. Continue: \(216 - 66 = 150\). 3. Add: \(456 + 150 = 606\).

Answer

\(606\)

5179455

Evaluate the expression step by step: \(2100 - (850 + 450 - 300)\)

Hints

- Evaluate the grouped expression first. - Complete addition and subtraction from left to right. - Rewrite the outer subtraction after finding the grouped value.

Solution

1. Inside the grouping symbols, work from left to right: \(850 + 450 = 1300\). 2. Continue: \(1300 - 300 = 1000\). 3. Subtract: \(2100 - 1000 = 1100\).

Answer

\(1100\)

5179605

First estimate by rounding to the nearest hundred. Then evaluate the expression exactly. \((765 + 235) - (482 - 118)\)

Hints

- Round each number to the nearest hundred for the estimate. - Evaluate both grouped expressions exactly. - Compare the exact value with the estimate.

Solution

1. Estimate: \((800 + 200) - (500 - 100) = 1000 - 400 = 600\). 2. Evaluate exactly: \(765 + 235 = 1000\) and \(482 - 118 = 364\). 3. Therefore, \(1000 - 364 = 636\).

Answer

Estimate: \(600\) Exact value: \(636\)

5179615

First estimate the value, then evaluate the expression exactly: \(25{,}000 - [(8420 - 3120) + (1550 + 450)]\)

Hints

- Round to values that make the estimate easy to calculate. - Evaluate the inner grouped expressions before the bracketed sum. - Compare the exact result with the estimate.

Solution

1. One estimate is \(25{,}000 - [(8000 - 3000) + 2000] = 25{,}000 - 7000 = 18{,}000\). 2. Evaluate exactly: \(8420 - 3120 = 5300\) and \(1550 + 450 = 2000\). 3. The bracketed value is \(7300\), so the exact value is \(25{,}000 - 7300 = 17{,}700\).

Answer

Estimate: \(18{,}000\) Exact value: \(17{,}700\)

5179835

Insert \(+\) or \(-\) in each circle so that every equation is true. Evaluate the grouping symbols first. a) \(850 \bigcirc (420 \bigcirc 130) = 560\) b) \(1500 \bigcirc (600 \bigcirc 250) = 1150\) c) \(2100 \bigcirc (900 \bigcirc 400 \bigcirc 150) = 1750\)

Hints

- Evaluate the grouping symbols before the outside operation. - Determine what grouped value is needed to reach the target. - A subtraction sign before grouping symbols changes how the inner signs affect the total.

Solution

1. a) The grouped value must be \(290\): \(420 - 130 = 290\). Then \(850 - 290 = 560\), so \(850 - (420 - 130) = 560\). 2. b) \(600 - 250 = 350\), and \(1500 - 350 = 1150\). Therefore, \(1500 - (600 - 250) = 1150\). 3. c) \(900 - 400 - 150 = 350\), and \(2100 - 350 = 1750\). Therefore, \(2100 - (900 - 400 - 150) = 1750\).

Answer

a) \(850 - (420 - 130) = 560\) b) \(1500 - (600 - 250) = 1150\) c) \(2100 - (900 - 400 - 150) = 1750\)

5191355

Evaluate each expression mentally using the order of operations. a) \((15 + 25) \times (12 - 8)\) b) \(100 \div (2 \times 10 \div 4)\) c) \(7 \times 8 + 4 \times 11\)

Hints

- Evaluate grouping symbols first. - Within one level, multiplication and division are completed from left to right. - Complete multiplication before addition.

Solution

1. For a), evaluate both grouped expressions: \(40 \times 4 = 160\). 2. For b), work from left to right inside the grouping symbols: \(2 \times 10 \div 4 = 20 \div 4 = 5\). Then \(100 \div 5 = 20\). 3. For c), multiply before adding: \(56 + 44 = 100\).

Answer

a) \(160\) b) \(20\) c) \(100\)

5191535

Evaluate each expression when possible. If an expression cannot be evaluated, write “undefined” and explain why. a) \(750 \div (25 \times 2 - 50)\) b) \((25 \times 2 - 50) \div 750\) c) \(120 \div [30 \div (15 - 10)]\) d) \(120 \div [(30 - 10) \div 4 - 5]\)

Hints

- Work from the innermost grouping symbols outward. - Use multiplication and division before addition and subtraction within the same grouping. - Check whether a divisor becomes \(0\) at any step.

Solution

1. The parentheses equal \(25 \times 2 - 50 = 0\), so the expression is \(750 \div 0\). It is undefined. 2. The parentheses equal \(0\), so \(0 \div 750 = 0\). 3. Evaluate from the inside out: \(15 - 10 = 5\), \(30 \div 5 = 6\), and \(120 \div 6 = 20\). 4. Evaluate the bracket: \((30 - 10) \div 4 - 5 = 20 \div 4 - 5 = 0\). The expression becomes \(120 \div 0\), so it is undefined.

Answer

a) Undefined; division by \(0\) b) \(0\) c) \(20\) d) Undefined; division by \(0\)

5191545

A student claims, “Whenever \(0\) appears anywhere in a division expression, the value is either \(0\) or undefined.” Test the claim by evaluating each expression when possible. 1. \((15 + 0) \div 3\) 2. \(0 \div (24 \div 6)\) 3. \(48 \div (12 - 3 \times 4)\)

Hints

- Evaluate each expression normally before judging the claim. - Adding \(0\) does not change a number. - Only division by \(0\) is undefined.

Solution

1. \((15 + 0) \div 3 = 15 \div 3 = 5\). This value is neither \(0\) nor undefined, so it disproves the claim. 2. \(0 \div (24 \div 6) = 0 \div 4 = 0\). 3. The divisor is \(12 - 3 \times 4 = 0\), so \(48 \div 0\) is undefined.

Answer

The claim is false. 1. \(5\) 2. \(0\) 3. Undefined because the divisor is \(0\)

5191865

Which expression has a value closest to \(250\)? 1) \(16 \times 15\) 2) \(1024 \div 4\) 3) \(7 \times 6 \times 6\) 4) \(500 - 243\)

Hints

- Evaluate all four expressions. - Find the absolute difference between each value and \(250\). - The smallest difference identifies the closest value.

Solution

1. The expression values are \(240\), \(256\), \(252\), and \(257\). 2. Their distances from \(250\) are \(10\), \(6\), \(2\), and \(7\). 3. Expression 3 is closest because its value, \(252\), is only \(2\) away from \(250\).

Answer

Expression 3, with a value of \(252\)

5192265

Evaluate expressions \(A\), \(B\), \(C\), and \(D\). Then order their values from least to greatest. \(A = 36 + 12 \div 4\) \(B = (36 + 12) \div 4\) \(C = 36 \times 4 - 12\) \(D = 36 \times (12 - 4)\)

Hints

- Apply the order of operations to each expression separately. - Notice how grouping symbols change the order of calculation. - Compare the four values after evaluating them.

Solution

1. \(A = 36 + 3 = 39\). 2. \(B = 48 \div 4 = 12\). 3. \(C = 144 - 12 = 132\). 4. \(D = 36 \times 8 = 288\). 5. Therefore, \(12 < 39 < 132 < 288\), so the order is \(B, A, C, D\).

Answer

\(B = 12\), \(A = 39\), \(C = 132\), \(D = 288\) Order: \(B < A < C < D\)

5192315

Solve the expression chain to find a word. Begin with card [B]. The result of each card is the first number on the next card. The final result returns to the first number on card [B]. - \((5 \times 8) + 12\) [R] - \((100 - 40) \div 12\) [B] - \(40 \div 4 + 90\) [K] - \((52 - 4) \div 8\) [E] - \(6 \times 15 - 50\) [A]

Hints

- Start by evaluating card [B]. - Find the next card whose expression begins with your result. - Record each card letter in order.

Solution

1. Card [B]: \((100 - 40) \div 12 = 5\). 2. Card [R] begins with \(5\): \((5 \times 8) + 12 = 52\). 3. Card [E] begins with \(52\): \((52 - 4) \div 8 = 6\). 4. Card [A] begins with \(6\): \(6 \times 15 - 50 = 40\). 5. Card [K] begins with \(40\): \(40 \div 4 + 90 = 100\), returning to card [B]. The letters spell BREAK.

Answer

BREAK

5192415

Evaluate the expression efficiently: \((17{,}400 + 2600) \div (125 \div 5 \div 25)\)

Hints

- Evaluate each set of grouping symbols separately. - In the second set, perform division from left to right. - Recall what happens when a number is divided by \(1\).

Solution

1. Evaluate the first set of grouping symbols: \(17{,}400 + 2600 = 20{,}000\). 2. Evaluate the second set from left to right: \(125 \div 5 = 25\), and then \(25 \div 25 = 1\). 3. Divide by \(1\): \(20{,}000 \div 1 = 20{,}000\).

Answer

\(20{,}000\)

5192425

Evaluate the expression. First look for a part that makes the calculation simpler: \(45{,}678 - [(123 + 456) \times (10 - 2 \times 5)]\)

Hints

- Start with the innermost grouping symbols on the right. - Think about how that result affects the entire product in brackets. - Decide whether you actually need to calculate \(123 + 456\).

Solution

1. Evaluate the innermost grouping symbols using the order of operations: \(10 - 2 \times 5 = 10 - 10 = 0\). 2. The product in brackets is \((123 + 456) \times 0 = 0\). The sum \(123 + 456\) does not need to be calculated because any number multiplied by \(0\) equals \(0\). 3. Therefore, \(45{,}678 - 0 = 45{,}678\).

Answer

\(45{,}678\)

5193275

Evaluate each expression using the order of operations. a) \(1200 - 25 \times 30\) b) \((540 + 180) \div 12\) c) \(14 \times (100 - 3)\) d) \(640 \div (16 \times 4)\)

Hints

- Identify which operation must be performed first in each expression. - Evaluate grouping symbols before operations outside them. - Write one intermediate result at a time.

Solution

1. a) Multiply before subtracting: \(1200 - 25 \times 30 = 1200 - 750 = 450\). 2. b) Evaluate the grouping symbols first: \((540 + 180) \div 12 = 720 \div 12 = 60\). 3. c) Evaluate the grouping symbols first: \(14 \times (100 - 3) = 14 \times 97 = 1358\). 4. d) Multiply inside the grouping symbols first: \(640 \div (16 \times 4) = 640 \div 64 = 10\).

Answer

a) \(450\) b) \(60\) c) \(1358\) d) \(10\)

5193815

Evaluate the expression \((5712 \div 6) - (145 \times 4)\). What difference do you obtain?

Hints

- Evaluate each grouped operation separately. - A quotient is the result of division, and a product is the result of multiplication. - Subtract the smaller evaluated result from the larger one.

Solution

1. Evaluate the first grouped expression: \(5712 \div 6 = 952\). 2. Evaluate the second grouped expression: \(145 \times 4 = 580\). 3. Subtract the second result from the first: \(952 - 580 = 372\).

Answer

The difference is \(372\).

5194515

Evaluate the four expressions. Which expression has the greatest value, and which has the least value? 1. \(40 + 24 \div 4 - 2\) 2. \((40 + 24) \div 4 - 2\) 3. \(40 + 24 \div (4 - 2)\) 4. \((40 + 24) \div (4 - 2)\)

Hints

- Evaluate grouping symbols first. - Apply multiplication and division before addition and subtraction. - Compare the four values after evaluating them.

Solution

1. Expression 1: \(40 + 24 \div 4 - 2 = 40 + 6 - 2 = 44\). 2. Expression 2: \((40 + 24) \div 4 - 2 = 64 \div 4 - 2 = 16 - 2 = 14\). 3. Expression 3: \(40 + 24 \div (4 - 2) = 40 + 24 \div 2 = 52\). 4. Expression 4: \((40 + 24) \div (4 - 2) = 64 \div 2 = 32\). 5. The greatest value is \(52\), from expression 3. The least value is \(14\), from expression 2.

Answer

Greatest value: \(52\), expression 3 Least value: \(14\), expression 2

5195145

Evaluate the expression: \(8 \times [15 - (12 - 3 \times 2)]\)

Hints

- Work from the innermost grouping symbols outward. - Apply the order of operations inside the parentheses. - Multiply only after the brackets have been evaluated.

Solution

1. Evaluate the innermost grouping symbols: \(12 - 3 \times 2 = 12 - 6 = 6\). 2. Evaluate the brackets: \(15 - 6 = 9\). 3. Multiply: \(8 \times 9 = 72\).

Answer

\(72\)

5195155

Evaluate the expression: \([42 + (48 \div 8 + 2)] \div 5 + 17\)

Hints

- Work from the innermost grouping symbols outward. - Apply division before addition inside the parentheses. - After the brackets, continue using the order of operations.

Solution

1. Evaluate the parentheses: \(48 \div 8 + 2 = 6 + 2 = 8\). 2. Evaluate the brackets: \(42 + 8 = 50\). 3. Divide: \(50 \div 5 = 10\). 4. Add: \(10 + 17 = 27\).

Answer

\(27\)

5195165

Evaluate the expression step by step: \(140 - 3 \times [(7 + 13) \times 4 - 55]\)

Hints

- Begin with the innermost grouping symbols. - Finish evaluating the brackets before multiplying by \(3\). - The subtraction from \(140\) is the final step.

Solution

1. Evaluate the parentheses: \(7 + 13 = 20\). 2. Continue inside the brackets: \(20 \times 4 = 80\), and \(80 - 55 = 25\). 3. Multiply: \(3 \times 25 = 75\). 4. Subtract: \(140 - 75 = 65\).

Answer

\(65\)

5195275

Evaluate the expression using the order of operations: \((14 \times 5 - 62) \times (33 \div 3 + 4) - 80\)

Hints

- Evaluate each grouped quantity separately. - Apply multiplication or division before addition or subtraction inside the grouping symbols. - Multiply the two grouped values before subtracting \(80\).

Solution

1. Evaluate the first grouped quantity: \(14 \times 5 - 62 = 70 - 62 = 8\). 2. Evaluate the second grouped quantity: \(33 \div 3 + 4 = 11 + 4 = 15\). 3. Multiply: \(8 \times 15 = 120\). 4. Subtract: \(120 - 80 = 40\).

Answer

\(40\)

5195285

Evaluate the expression: \(200 - [4 \times (18 + 7) - 30] \div 5\)

Hints

- Work from the innermost grouping symbols outward. - Finish the brackets before dividing by \(5\). - Perform the division before subtracting from \(200\).

Solution

1. Evaluate the parentheses: \(18 + 7 = 25\). 2. Evaluate the brackets: \(4 \times 25 - 30 = 100 - 30 = 70\). 3. Divide: \(70 \div 5 = 14\). 4. Subtract: \(200 - 14 = 186\).

Answer

\(186\)

5203695

Use the numbers \(2\), \(4\), \(6\), and \(8\) in that order. Insert operation symbols and grouping symbols as needed to create an expression with each value. a) \(4\) b) \(10\) c) \(18\) d) \(20\)

Hints

- Keep the numbers in the given order. - Use multiplication when you need a larger value. - Apply the order of operations when checking each expression. - More than one answer may be possible.

Solution

Possible expressions are: 1. a) \(2 + 4 + 6 - 8 = 4\). 2. b) \(2 \times 4 - 6 + 8 = 10\). 3. c) \(2 + 4 \times 6 - 8 = 18\). 4. d) \(2 + 4 + 6 + 8 = 20\).

Answer

Possible answers are: a) \(2 + 4 + 6 - 8 = 4\) b) \(2 \times 4 - 6 + 8 = 10\) c) \(2 + 4 \times 6 - 8 = 18\) d) \(2 + 4 + 6 + 8 = 20\)

5211615

Evaluate each expression. Write each final answer as a decimal. a) \(2.5\,\text{h}+(1\,\text{h}\ 45\,\text{min}+45\,\text{min})\) b) \(4.8\,\text{kg}\div 8+325\,\text{g}\times 4\)

Hints

- Remember that one hour has \(60\) minutes, not \(100\). - Express \(30\) minutes as part of an hour. - Use the order of operations. - Convert the masses to the same unit before adding.

Solution

1. In a), \(1\,\text{h}\ 45\,\text{min}+45\,\text{min}=2\,\text{h}\ 30\,\text{min}=2.5\,\text{h}\). 2. Then \(2.5\,\text{h}+2.5\,\text{h}=5.0\,\text{h}\). 3. In b), \(4.8\,\text{kg}\div 8=0.6\,\text{kg}\). 4. Also, \(325\,\text{g}\times 4=1300\,\text{g}=1.3\,\text{kg}\). 5. Add the two masses: \(0.6\,\text{kg}+1.3\,\text{kg}=1.9\,\text{kg}\).

Answer

a) \(5.0\,\text{h}\) b) \(1.9\,\text{kg}\)

5213205

Evaluate the expression, or explain why it is undefined. \((12\,\text{m}+8\,\text{m})\div(45\,\text{min}-2700\,\text{s})\)

Hints

- Evaluate each set of parentheses separately. - Convert the seconds to minutes. - Recall what happens when the divisor is zero.

Solution

1. Evaluate the first set of parentheses: \(12\,\text{m}+8\,\text{m}=20\,\text{m}\). 2. Convert \(2700\,\text{s}\) to minutes: \(2700\div 60=45\), so \(2700\,\text{s}=45\,\text{min}\). 3. Evaluate the second set of parentheses: \(45\,\text{min}-45\,\text{min}=0\,\text{min}\). 4. The expression would require division by zero. Division by zero is undefined, so the expression has no value.

Answer

The expression is undefined because its divisor equals zero.

5214895

Evaluate the expression step by step: \(5\,\text{yd}^2\ 2\,\text{ft}^2-(18\,\text{ft}^2\div 2+3\,\text{ft}^2)\times 2\).

Hints

- Evaluate the expression inside the grouping symbols first. - Convert the mixed area to square feet. - Then multiply and subtract in order.

Solution

1. Convert the mixed area: \(5\,\text{yd}^2\ 2\,\text{ft}^2=45\,\text{ft}^2+2\,\text{ft}^2=47\,\text{ft}^2\). 2. Evaluate inside the grouping symbols: \(18\,\text{ft}^2\div 2=9\,\text{ft}^2\), and \(9\,\text{ft}^2+3\,\text{ft}^2=12\,\text{ft}^2\). 3. Multiply: \(12\,\text{ft}^2\times 2=24\,\text{ft}^2\). 4. Subtract: \(47\,\text{ft}^2-24\,\text{ft}^2=23\,\text{ft}^2\).

Answer

\(23\,\text{ft}^2\), or \(2\,\text{yd}^2\ 5\,\text{ft}^2\)

5316955

The expression tree is shown. a) Write the numerical expression represented by the tree. Use grouping symbols only where they are needed. b) Find the values of the empty boxes and the final value of the expression.

Figure for problem 531695

Hints

- Read the expression tree from the top branches toward the final box. - Use grouping symbols so the addition occurs before multiplication. - Evaluate one node at a time.

Solution

1. The left branch first adds \(15\) and \(25\), and then multiplies the sum by \(12\). 2. The root subtracts \(340\), so the expression is \((15 + 25) \times 12 - 340\). 3. Evaluate the tree: \(15 + 25 = 40\), \(40 \times 12 = 480\), and \(480 - 340 = 140\).

Answer

a) \((15 + 25) \times 12 - 340\) b) The intermediate values are \(40\) and \(480\). The final value is \(140\).

5316965

Use the calculation tree shown. a) Write a numerical expression that matches the tree. Use grouping symbols so the expression preserves the tree structure. b) Find the intermediate values and the final value.

Figure for problem 531696

Hints

- Read the two branches of the calculation tree separately before combining them. - Use grouping symbols to show which operations happen before the final operation. - Keep decimal place values aligned when evaluating the branch operations. - Evaluate the branch operations before finding the value at the bottom of the tree.

Solution

1. The left branch represents \(4.5+1.5\), and the right branch represents \(12.8-7.3\). The tree then multiplies those two results, so the expression is \((4.5+1.5)\times(12.8-7.3)\). 2. Evaluate the left branch: \(4.5+1.5=6\). 3. Evaluate the right branch: \(12.8-7.3=5.5\). 4. Multiply the branch results: \(6\times5.5=33\).

Answer

a) \((4.5+1.5)\times(12.8-7.3)\) b) The intermediate values are \(6\) and \(5.5\), and the final value is \(33\).

5316975

The expression tree is shown. a) Write the numerical expression represented by the tree. Use grouping symbols only where they are needed. b) Evaluate the expression. Show the calculation steps clearly.

Figure for problem 531697

Hints

- Read each branch from its top values toward the root. - Use grouping symbols where a sum or difference must be evaluated before multiplication or division. - Evaluate each branch before adding the results.

Solution

1. The left branch is the product of \(24\) and the difference between \(15\) and \(8\): \(24 \times (15 - 8)\). 2. The right branch divides the sum of \(42\) and \(18\) by \(5\): \((42 + 18) \div 5\). 3. The branches are added, so the expression is \(24 \times (15 - 8) + (42 + 18) \div 5\). 4. Evaluate: \(15 - 8 = 7\), \(42 + 18 = 60\), \(24 \times 7 = 168\), and \(60 \div 5 = 12\). 5. Add: \(168 + 12 = 180\).

Answer

a) \(24 \times (15 - 8) + (42 + 18) \div 5\) b) \(180\)

5316995

The expression tree is shown. Write the numerical expression represented by the tree, and then find its final value.

Figure for problem 531699

Hints

- Read the tree from the top values toward the final box. - Use grouping symbols for the addition and subtraction branches. - Evaluate the two branches before multiplying their values.

Solution

1. The left branch adds \(28\) and \(14\), then divides the sum by \(7\): \((28 + 14) \div 7\). 2. The right branch subtracts \(9\) from \(15\): \((15 - 9)\). 3. The root multiplies the branch values, so the expression is \((28 + 14) \div 7 \times (15 - 9)\). 4. Evaluate: \(28 + 14 = 42\), \(42 \div 7 = 6\), and \(15 - 9 = 6\). 5. Multiply: \(6 \times 6 = 36\).

Answer

Expression: \((28 + 14) \div 7 \times (15 - 9)\) Final value: \(36\)

5317025

Two expression trees are shown. a) Write the numerical expression represented by the left tree, and evaluate it. b) Write the numerical expression represented by the right tree, and evaluate it.

Figure for problem 531702

Hints

- Read each tree from the top values toward the root. - Use grouping symbols when they make a branch structure clear. - Evaluate each branch before combining the results.

Solution

1. a) The left tree adds the product \(15 \times 8\) and the difference \(130 - 45\). The expression is \(15 \times 8 + (130 - 45)\). Its value is \(120 + 85 = 205\). 2. b) The right tree subtracts the quotient \(80 \div 5\) from \(240\). The expression is \(240 - 80 \div 5\). Its value is \(240 - 16 = 224\).

Answer

a) \(15 \times 8 + (130 - 45) = 205\) b) \(240 - 80 \div 5 = 224\)

5317115

The expression tree is shown. a) Write the numerical expression represented by the tree. Use grouping symbols to show its structure exactly. b) Find the values of the empty boxes and the final value.

Figure for problem 531711

Hints

- Identify the two complete quantities used in the final division. - Use grouping symbols around the entire dividend and divisor. - Evaluate the tree one node at a time.

Solution

1. The left side of the final division is the difference between \(25 \times 6\) and \(30\): \((25 \times 6 - 30)\). 2. The right side is the product \(4 \times 5\). Therefore, the expression is \((25 \times 6 - 30) \div (4 \times 5)\). 3. Evaluate the nodes: \(25 \times 6 = 150\), \(150 - 30 = 120\), and \(4 \times 5 = 20\). 4. Divide: \(120 \div 20 = 6\).

Answer

a) \((25 \times 6 - 30) \div (4 \times 5)\) b) The intermediate values are \(150\), \(120\), and \(20\). The final value is \(6\).

5317335

For each expression tree, find all missing intermediate values and the final value.

Figure for problem 531733

Hints

- Follow the branches toward the root of each tree. - Evaluate the highest operation nodes first. - Check each intermediate value before finding the final value.

Solution

1. a) Evaluate the upper branches: \(120 - 45 = 75\) and \(14 + 6 = 20\). Then \(75 \times 20 = 1500\). 2. b) Evaluate the upper branches: \(250 \div 5 = 50\) and \(18 \times 4 = 72\). Then \(50 + 72 = 122\). 3. c) First \(12 \times 8 = 96\). Then \(96 + 34 = 130\). Finally, \(400 - 130 = 270\).

Answer

a) Intermediate values: \(75\) and \(20\); final value: \(1500\) b) Intermediate values: \(50\) and \(72\); final value: \(122\) c) Intermediate values: \(96\) and \(130\); final value: \(270\)

5351565

Write the numerical expression represented by the expression tree, and then evaluate it. Use grouping symbols correctly.

Figure for problem 535156

Hints

- Begin with the innermost branches of the tree. - Each operation node becomes one part of the expression. - Use nested grouping symbols to preserve the tree structure.

Solution

1. Evaluate the innermost branch: \(320 + 450 = 770\). 2. Evaluate the difference on the right: \(980 - 770 = 210\). 3. Add: \(540 + 210 = 750\). 4. The expression is \(540 + [980 - (320 + 450)]\).

Answer

Expression: \(540 + [980 - (320 + 450)]\) Value: \(750\)

5351575

Write the numerical expression represented by the expression tree, and then evaluate it.

Figure for problem 535157

Hints

- Identify the final operation at the root of the tree. - Work from the innermost branch outward. - Use grouping symbols to preserve the tree structure.

Solution

1. Evaluate the innermost sum: \(80 + 40 = 120\). 2. Subtract: \(200 - 120 = 80\). 3. Multiply: \(15 \times 80 = 1200\). 4. The expression is \(15 \times [200 - (80 + 40)]\).

Answer

Expression: \(15 \times [200 - (80 + 40)]\) Value: \(1200\)

5351585

What numerical expression matches the calculation tree? Write the expression and evaluate it.

Figure for problem 535158

Hints

- Identify the numbers joined directly by each operation. - Each branch of the tree becomes a grouped part of the expression.

Solution

1. Evaluate the innermost sum: \(30 + 20 = 50\). 2. Evaluate the difference: \(100 - 50 = 50\). 3. Add the left branch: \(45 + 50 = 95\). 4. The expression is \(45 + (100 - (30 + 20))\).

Answer

The expression is \(45 + (100 - (30 + 20))\), and its value is \(95\).

5351605

Write the numerical expression represented by the expression tree, and then evaluate it.

Figure for problem 535160

Hints

- Evaluate the two upper branches first. - Treat each branch value as one complete quantity. - Subtract the right branch value from the left branch value.

Solution

1. Evaluate the left sum: \(800 + 200 = 1000\). 2. Evaluate the right sum: \(350 + 150 = 500\). 3. Subtract: \(1000 - 500 = 500\). 4. The expression is \((800 + 200) - (350 + 150)\).

Answer

Expression: \((800 + 200) - (350 + 150)\) Value: \(500\)

5351615

Write the numerical expression represented by the expression tree, and then evaluate it.

Figure for problem 535161

Hints

- Read the operation symbols at each node. - Evaluate the two upper branches before the final subtraction. - Use the order of operations to check the expression.

Solution

1. Evaluate the product on the left: \(7 \times 9 = 63\). 2. Evaluate the quotient on the right: \(100 \div 5 = 20\). 3. Subtract: \(63 - 20 = 43\). 4. The expression is \(7 \times 9 - 100 \div 5\).

Answer

Expression: \(7 \times 9 - 100 \div 5\) Value: \(43\)

5351675

Write the numerical expression shown by the expression tree, then evaluate it.

Figure for problem 535167

Hints

- Follow the branches from the numbers toward the final operation. - Evaluate each inner operation before the operation that contains it.

Solution

1. Evaluate the sum in the inner branch: \(600 + 400 = 1000\). 2. Evaluate the right branch: \(1000 - 500 = 500\). 3. Evaluate the entire expression: \(2000 - 500 = 1500\). The expression is \(2000 - ((600 + 400) - 500)\).

Answer

The expression is \(2000 - ((600 + 400) - 500)\), and its value is \(1500\).

5351785

Write the numerical expression represented by the expression tree, and then evaluate it.

Figure for problem 535178

Hints

- Read the tree from the upper values toward the root. - Identify which operations belong together in one branch. - The root operation is performed last.

Solution

1. The expression is \(140 + (25 \times 4 - 60)\). 2. Evaluate the product: \(25 \times 4 = 100\). 3. Evaluate the difference: \(100 - 60 = 40\). 4. Add: \(140 + 40 = 180\).

Answer

Expression: \(140 + (25 \times 4 - 60)\) Value: \(180\)

5351815

Use the expression tree to evaluate the expression step by step. Then write the complete numerical expression with all necessary grouping symbols.

Figure for problem 535181

Hints

- Evaluate the upper branches before moving toward the root. - Find the complete dividend and divisor. - Use grouping symbols to preserve the right branch structure.

Solution

1. Evaluate the left product: \(12 \times 5 = 60\). 2. Evaluate the inner product on the right: \(2 \times 30 = 60\). 3. Evaluate the right difference: \(80 - 60 = 20\). 4. Divide: \(60 \div 20 = 3\). 5. The expression is \((12 \times 5) \div [80 - (2 \times 30)]\).

Answer

Expression: \((12 \times 5) \div [80 - (2 \times 30)]\) Value: \(3\)

5351835

Write the numerical expression represented by the expression tree, and then evaluate it step by step.

Figure for problem 535183

Hints

- Evaluate the left and right main branches separately. - Use nested grouping symbols on the right. - Check each intermediate value before the final subtraction.

Solution

1. The expression is \((200 + 300) - [600 - (150 + 250)]\). 2. Evaluate the left sum: \(200 + 300 = 500\). 3. Evaluate the inner right sum: \(150 + 250 = 400\). 4. Evaluate the right difference: \(600 - 400 = 200\). 5. Subtract: \(500 - 200 = 300\).

Answer

Expression: \((200 + 300) - [600 - (150 + 250)]\) Value: \(300\)

5352205

Fill in all empty boxes in the expression tree. Then write and evaluate the corresponding numerical expression.

Figure for problem 535220

Hints

- The tree adds two products. - Evaluate each product before adding.

Solution

1. Evaluate the left product: \(8 \times 5 = 40\). 2. Evaluate the right product: \(6 \times 7 = 42\). 3. Add: \(40 + 42 = 82\). 4. The expression is \(8 \times 5 + 6 \times 7\).

Answer

Intermediate values: \(40\) and \(42\) Expression: \(8 \times 5 + 6 \times 7\) Final value: \(82\)

5352215

Find the final value of the expression tree. Then write the corresponding numerical expression.

Figure for problem 535221

Hints

- Find the complete dividend and divisor. - Evaluate both branch values before dividing.

Solution

1. Evaluate the left difference: \(200 - 80 = 120\). 2. Evaluate the right sum: \(4 + 6 = 10\). 3. Divide: \(120 \div 10 = 12\). 4. The expression is \((200 - 80) \div (4 + 6)\).

Answer

Expression: \((200 - 80) \div (4 + 6)\) Final value: \(12\)

5352225

Complete the nested expression tree step by step. Then write and evaluate the full numerical expression.

Figure for problem 535222

Hints

- Begin with the deepest branch, \(45 + 35\). - Use nested grouping symbols to preserve the tree structure.

Solution

1. Evaluate the innermost sum: \(45 + 35 = 80\). 2. Subtract: \(120 - 80 = 40\). 3. Multiply: \(40 \times 2 = 80\). 4. The expression is \([120 - (45 + 35)] \times 2\).

Answer

Expression: \([120 - (45 + 35)] \times 2\) Final value: \(80\)

5352235

Complete the expression tree. Then write and evaluate the corresponding numerical expression.

Figure for problem 535223

Hints

- Begin with the multiplication branch. - Subtract the product from \(90\), then add \(40\).

Solution

1. Multiply: \(12 \times 5 = 60\). 2. Subtract: \(90 - 60 = 30\). 3. Add: \(30 + 40 = 70\). 4. The expression is \(90 - 12 \times 5 + 40\).

Answer

Expression: \(90 - 12 \times 5 + 40\) Final value: \(70\)

5352245

Write the structured calculation as a numerical expression, and then evaluate it.

Figure for problem 535224

Hints

- Begin with the deepest branch. - Use nested grouping symbols so the subtraction occurs before division.

Solution

1. Evaluate the innermost sum: \(15 + 45 = 60\). 2. Subtract: \(100 - 60 = 40\). 3. Divide: \(40 \div 4 = 10\). 4. The expression is \([100 - (15 + 45)] \div 4\).

Answer

Expression: \([100 - (15 + 45)] \div 4\) Final value: \(10\)

5352435

Evaluate the expression tree step by step. Then write the complete numerical expression.

Figure for problem 535243

Hints

- Evaluate the two upper branches first. - Use grouping symbols around the complete dividend and divisor.

Solution

1. Evaluate the left sum: \(24 + 36 = 60\). 2. Evaluate the right difference: \(15 - 5 = 10\). 3. Divide: \(60 \div 10 = 6\). 4. The expression is \((24 + 36) \div (15 - 5)\).

Answer

Expression: \((24 + 36) \div (15 - 5)\) Final value: \(6\)

5352875

Use the expression tree to find each missing intermediate value from top to bottom. Then give the final value.

Figure for problem 535287

Hints

- Follow the branches toward the root. - Each box contains the result of the operation directly above it. - Combine the two smaller branch values before multiplying by \(2\).

Solution

1. Evaluate the product on the left: \(25 \times 4 = 100\). 2. Evaluate the quotient on the right: \(150 \div 3 = 50\). 3. Add the branch values: \(100 + 50 = 150\). 4. Multiply: \(150 \times 2 = 300\).

Answer

Intermediate values: \(100\), \(50\), and \(150\) Final value: \(300\)

5353355

Complete the calculation tree. Work from the top operations toward the final result.

Figure for problem 535335

Hints

- Evaluate the multiplication branch first. - Then add that result to the number in the other branch.

Solution

1. Evaluate the multiplication branch: \(45 \times 4 = 180\). 2. Add the other branch: \(180 + 120 = 300\).

Answer

The missing numbers are \(180\) and \(300\).

5353385

Evaluate the two upper branches of the expression tree, and then find the final value.

Figure for problem 535338

Hints

- Evaluate the two upper branches independently. - The root multiplies the two intermediate values.

Solution

1. Evaluate the left sum: \(60 + 40 = 100\). 2. Evaluate the right difference: \(25 - 15 = 10\). 3. Multiply the branch values: \(100 \times 10 = 1000\).

Answer

Intermediate values: \(100\) and \(10\) Final value: \(1000\)

5353395

Fill in all empty boxes in the calculation tree.

Figure for problem 535339

Hints

- Evaluate the two upper branches first. - Add the two branch results to complete the bottom box.

Solution

1. Evaluate the left branch: \(120 \div 3 = 40\). 2. Evaluate the right branch: \(8 \times 5 = 40\). 3. Add the two branch results: \(40 + 40 = 80\).

Answer

The missing values are \(40\), \(40\), and \(80\).

5353415

Complete the calculation tree. You may use a separate calculation for the addition.

Figure for problem 535341

Hints

- Add the hundreds, tens, and ones carefully. - For the division, break \(615\) into \(600 + 15\).

Solution

1. Evaluate the grouped sum: \(368 + 247 = 615\). 2. Divide the result by \(5\): \(615 \div 5 = 123\).

Answer

The intermediate box contains \(615\), and the final box contains \(123\).

5353425

Complete the calculation tree step by step from top to bottom.

Figure for problem 535342

Hints

- Check the subtraction carefully. - Then double the difference to find the final value.

Solution

1. Find the difference: \(912 - 456 = 456\). 2. Multiply that result by \(2\): \(456 \times 2 = 912\).

Answer

The missing values are \(456\) and \(912\).

5353465

Use the expression tree to find the total length.

Figure for problem 535346

Hints

- Follow the operations from the inner branch to the final operation. - Include meters in the final answer.

Solution

1. Divide: \(500\,\text{m} \div 5 = 100\,\text{m}\). 2. Multiply the result by \(3\): \(100\,\text{m} \times 3 = 300\,\text{m}\).

Answer

The total length is \(300\,\text{m}\).

5353485

Complete the calculation tree. First find the two intermediate results, and then find the value in the final box.

Figure for problem 535348

Hints

- Work through the tree one level at a time. - Pay close attention to each operation symbol. - Break the larger numbers into hundreds and tens if helpful.

Solution

1. Find the left intermediate result: \(460 + 270 = 730\). 2. Find the right intermediate result: \(530 - 380 = 150\). 3. Subtract the intermediate results: \(730 - 150 = 580\).

Answer

The left intermediate box contains \(730\), the right intermediate box contains \(150\), and the final box contains \(580\).

5192495

Use the numbers \(2\), \(3\), \(4\), and \(5\) exactly once in each part. Insert operation symbols and grouping symbols as needed to create an expression with each value. a) \(1\) b) \(12\) c) \(50\)

Hints

- Use all four numbers exactly once in each part. - For a small value such as \(1\), try creating two equal quantities to divide. - For \(50\), try making \(10\) and then multiplying by \(5\). - Use grouping symbols to control the order of operations.

Solution

1. For a value of \(1\), make two equal quantities and divide: \((5 + 3) \div (4 \times 2) = 8 \div 8 = 1\). 2. For a value of \(12\), first make \(6\), then multiply by \(2\): \((5 + 4 - 3) \times 2 = 6 \times 2 = 12\). 3. For a value of \(50\), first make \(10\), then multiply by \(5\): \(5 \times (3 \times 4 - 2) = 5 \times (12 - 2) = 5 \times 10 = 50\).

Answer

Possible answers are: a) \((5 + 3) \div (4 \times 2) = 1\) b) \((5 + 4 - 3) \times 2 = 12\) c) \(5 \times (3 \times 4 - 2) = 50\)

5203705

Use the numbers \(1\), \(2\), \(3\), \(4\), and \(5\) in that order. Insert operation symbols and grouping symbols as needed to create an expression with each value. a) \(1\) b) \(15\) c) \(36\) d) \(50\)

Hints

- Keep the numbers in the given order. - For a small target such as \(1\), consider making a quantity that can be divided evenly. - For larger targets, try multiplying grouped sums. - Check each expression using the order of operations.

Solution

Possible expressions are: 1. a) \(((1 + 2) \times 3 - 4) \div 5 = (9 - 4) \div 5 = 1\). 2. b) \(1 + 2 + 3 + 4 + 5 = 15\). 3. c) \((1 + 2) \times (3 + 4 + 5) = 3 \times 12 = 36\). 4. d) \((1 \times 2 \times 3 + 4) \times 5 = (6 + 4) \times 5 = 50\).

Answer

Possible answers are: a) \(((1 + 2) \times 3 - 4) \div 5 = 1\) b) \(1 + 2 + 3 + 4 + 5 = 15\) c) \((1 + 2) \times (3 + 4 + 5) = 36\) d) \((1 \times 2 \times 3 + 4) \times 5 = 50\)

5203715

Use the numbers \(10\), \(5\), \(2\), and \(1\) in that order. Insert operation symbols and grouping symbols as needed to create an expression with each value. a) \(1\) b) \(4\) c) \(7\) d) \(13\) e) \(31\)

Hints

- Keep the numbers in the given order. - Remember that multiplication and division are performed before addition and subtraction. - For \(31\), try making a factor close to \(3\). - Use grouping symbols when you need to change the usual order.

Solution

Possible expressions are: 1. a) \(10 - 5 \times 2 + 1 = 1\). 2. b) \(10 \div 5 + 2 \times 1 = 4\). 3. c) \(10 - 5 + 2 \times 1 = 7\). 4. d) \(10 + 5 - 2 \times 1 = 13\). 5. e) \(10 \times (5 - 2) + 1 = 31\).

Answer

Possible answers are: a) \(10 - 5 \times 2 + 1 = 1\) b) \(10 \div 5 + 2 \times 1 = 4\) c) \(10 - 5 + 2 \times 1 = 7\) d) \(10 + 5 - 2 \times 1 = 13\) e) \(10 \times (5 - 2) + 1 = 31\)

5211485

Evaluate each expression. Use a helpful property when possible. a) \(\$18.45\times 25+\$11.55\times 25\) b) \((10\,\text{m}-45\,\text{cm}\times 8)\div 20\,\text{cm}\) c) \((2\,\text{kg}-1250\,\text{g})\div 25\,\text{g}\)

Hints

- In a), both products have the same factor. - Follow the order of operations inside each set of parentheses. - Decide whether units cancel in each division.

Solution

1. Use the distributive property: \((\$18.45+\$11.55)\times 25=\$30\times 25=\$750\). 2. First, \(45\,\text{cm}\times 8=360\,\text{cm}\). Also, \(10\,\text{m}=1000\,\text{cm}\). Then \((1000-360)\,\text{cm}\div 20\,\text{cm}=640\div 20=32\). 3. \(2\,\text{kg}=2000\,\text{g}\). Then \((2000-1250)\,\text{g}\div 25\,\text{g}=750\div 25=30\).

Answer

a) \(\$750\) b) \(32\) c) \(30\)

5351625

Write the numerical expression represented by the expression tree, and then evaluate it.

Figure for problem 535162

Hints

- Begin with the deepest branches of the tree. - Evaluate the left and right main branches separately. - Use nested grouping symbols to preserve the structure.

Solution

1. On the left, evaluate \(250 - 50 = 200\), and then \(600 - 200 = 400\). 2. On the right, evaluate \(80 - 30 = 50\), and then \(120 + 50 = 170\). 3. Subtract: \(400 - 170 = 230\). 4. The expression is \([600 - (250 - 50)] - [120 + (80 - 30)]\).

Answer

Expression: \([600 - (250 - 50)] - [120 + (80 - 30)]\) Value: \(230\)

5351635

Use the calculation tree with decimals. Write the numerical expression represented by the tree, then find its value.

Figure for problem 535163

Hints

- Read the calculation tree from the inner branch operations toward the final operation. - Keep decimal place values aligned as you calculate each branch.

Solution

1. On the left side, \(4.2-1.2=3.0\), then \(10.5-3.0=7.5\). 2. On the right side, \(2.8-0.8=2.0\), then \(1.5+2.0=3.5\). 3. Subtract the two branch results: \(7.5-3.5=4.0\). 4. The expression represented by the tree is \((10.5-(4.2-1.2))-(1.5+(2.8-0.8))\).

Answer

The expression is \((10.5-(4.2-1.2))-(1.5+(2.8-0.8))\), and its value is \(4.0\).

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