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5100254
Which number is closest to \(90{,}000\)? a) \(89{,}000\) b) \(89{,}800\) c) \(90{,}300\) d) \(90{,}500\)

Hints

- Find how far each choice is from \(90{,}000\). - Compare the four differences. - “Closest” means the smallest difference.

Solution

1. The distance from \(89{,}000\) to \(90{,}000\) is \(1000\). 2. The distance from \(89{,}800\) to \(90{,}000\) is \(200\). 3. The distance from \(90{,}300\) to \(90{,}000\) is \(300\), and the distance from \(90{,}500\) is \(500\). 4. The smallest distance is \(200\), so \(89{,}800\) is closest.

Answer

b) \(89{,}800\)
5100284
Which of these numbers is the least? a) \(217{,}213\) b) \(341{,}231\) c) \(89{,}679\) d) \(91{,}312\)

Hints

- First compare the number of digits. - For numbers with the same number of digits, compare from left to right. - Stop at the first place where the digits differ.

Solution

1. The numbers \(89{,}679\) and \(91{,}312\) have five digits, while the other two numbers have six digits. 2. Every five-digit number is less than every six-digit number. 3. Compare the five-digit numbers. Since \(8\) is less than \(9\) in the ten-thousands place, \(89{,}679 < 91{,}312\). 4. Therefore, \(89{,}679\) is the least number.

Answer

c) \(89{,}679\)
5164914
Four zoos counted their visitors last year: Zoo A: \(506{,}780\) visitors Zoo B: \(560{,}870\) visitors Zoo C: \(506{,}870\) visitors Zoo D: \(560{,}780\) visitors a) Compare the visitor counts for Zoo A and Zoo C. Insert \(<\) or \(>\). b) Compare the visitor counts for Zoo B and Zoo D. Insert \(<\) or \(>\). c) Order all four visitor counts from least to greatest using \(<\).

Hints

- Compare each pair from left to right. - Find the first place where the digits differ. - Order the two numbers beginning with \(506\) before the two numbers beginning with \(560\).

Solution

1. Zoo A and Zoo C match through the thousands place. Since the hundreds digit \(7\) is less than \(8\), \(506{,}780 < 506{,}870\). 2. Zoo B and Zoo D also match through the thousands place. Since the hundreds digit \(8\) is greater than \(7\), \(560{,}870 > 560{,}780\). 3. Ordering all four counts gives \(506{,}780 < 506{,}870 < 560{,}780 < 560{,}870\).

Answer

a) \(506{,}780 < 506{,}870\) b) \(560{,}870 > 560{,}780\) c) \(506{,}780 < 506{,}870 < 560{,}780 < 560{,}870\)
5164924
A bicycle factory made these numbers of bicycles in four consecutive years: Year 1: \(243{,}650\) Year 2: \(234{,}560\) Year 3: \(243{,}560\) Year 4: \(234{,}650\) Order the production totals from greatest to least using \(>\).

Hints

- Compare the digits from left to right. - Group the numbers that begin with the same three digits. - Insert \(>\) between each pair in your final order.

Solution

1. The numbers beginning with \(243\) thousand are greater than the numbers beginning with \(234\) thousand. 2. Within the first pair, \(243{,}650 > 243{,}560\) because the hundreds digit \(6\) is greater than \(5\). 3. Within the second pair, \(234{,}650 > 234{,}560\). 4. Therefore, \(243{,}650 > 243{,}560 > 234{,}650 > 234{,}560\).

Answer

\(243{,}650 > 243{,}560 > 234{,}650 > 234{,}560\)
5166344
The table shows ticket sales for three amusement parks in two different years. <table> <tr> <th>Amusement park</th> <th>Year 1</th> <th>Year 2</th> </tr> <tr> <td>Bear Creek Park</td> <td>\(456{,}780\)</td> <td>\(461{,}200\)</td> </tr> <tr> <td>Water World</td> <td>\(890{,}120\)</td> <td>\(887{,}500\)</td> </tr> <tr> <td>Castle Park</td> <td>\(123{,}450\)</td> <td>\(125{,}000\)</td> </tr> </table> a) Which parks had greater ticket sales in Year 2 than in Year 1? b) Which park had lower ticket sales in Year 2 than in Year 1?

Hints

- Compare the two values in each row from left to right. - Greater sales means the Year 2 number is greater. - Lower sales means the Year 2 number is less.

Solution

1. Bear Creek Park increased because \(461{,}200 > 456{,}780\). 2. Water World decreased because \(887{,}500 < 890{,}120\). 3. Castle Park increased because \(125{,}000 > 123{,}450\).

Answer

a) Bear Creek Park and Castle Park b) Water World
5170134
Insert \(<\), \(=\), or \(>\) in each comparison. a) \(240{,}000 \times 0 \mathrel{\square} 240{,}000\) b) \(240{,}000 \div 1 \mathrel{\square} 240{,}000\) c) \(240{,}000 \times 1 \mathrel{\square} 240{,}000\) d) \(240{,}000 \div 4 \mathrel{\square} 240{,}000\) e) \(240{,}000 \times 4 \mathrel{\square} 240{,}000\)

Hints

- Recall what multiplying by \(0\) or \(1\) does. - Decide whether dividing by a number greater than \(1\) makes a positive whole number greater or less. - Decide whether multiplying by a number greater than \(1\) makes a positive whole number greater or less.

Solution

1. Multiplying by \(0\) gives \(0\), so \(240{,}000 \times 0 < 240{,}000\). 2. Dividing by \(1\) leaves the number unchanged, so \(240{,}000 \div 1 = 240{,}000\). 3. Multiplying by \(1\) leaves the number unchanged, so \(240{,}000 \times 1 = 240{,}000\). 4. Since \(240{,}000 \div 4 = 60{,}000\), the quotient is less than \(240{,}000\). 5. Since \(240{,}000 \times 4 = 960{,}000\), the product is greater than \(240{,}000\).

Answer

a) \(<\) b) \(=\) c) \(=\) d) \(<\) e) \(>\)
5171924
Write \(<\) or \(>\) in each blank to make a true comparison. a) \(67{,}802\) ___ \(67{,}208\) b) \(101{,}010\) ___ \(110{,}010\) c) \(450{,}000\) ___ \(45{,}000\)

Hints

- Compare the digits from left to right. - A positive whole number with more digits is greater. - Find the first place where the digits differ.

Solution

1. a) The ten-thousands and thousands digits match. In the hundreds place, \(8>2\), so \(67{,}802>67{,}208\). 2. b) The hundred-thousands digits match. In the ten-thousands place, \(0<1\), so \(101{,}010<110{,}010\). 3. c) \(450{,}000\) has six digits and \(45{,}000\) has five digits, so \(450{,}000>45{,}000\).

Answer

a) \(67{,}802>67{,}208\) b) \(101{,}010<110{,}010\) c) \(450{,}000>45{,}000\)
5171934
Write each statement using \(<\) or \(>\). a) Twelve thousand five hundred is less than twelve thousand five hundred five. b) Eight hundred thousand is greater than seven hundred ninety-nine thousand. c) The number \(120\) is between \(100\) and \(150\). Use a chain with two comparison signs.

Hints

- Write each number name as a numeral first. - The open side of a comparison sign faces the greater number. - For three numbers, place them in order from least to greatest.

Solution

1. a) Twelve thousand five hundred is \(12{,}500\), and twelve thousand five hundred five is \(12{,}505\). Therefore, \(12{,}500<12{,}505\). 2. b) Eight hundred thousand is \(800{,}000\), and seven hundred ninety-nine thousand is \(799{,}000\). Therefore, \(800{,}000>799{,}000\). 3. c) Put the least number on the left and the greatest on the right: \(100<120<150\).

Answer

a) \(12{,}500<12{,}505\) b) \(800{,}000>799{,}000\) c) \(100<120<150\)
5171954
Order the numbers from least to greatest: \(567\); \(1209\); \(89\); \(10{,}450\); \(576\); \(1290\); \(98\); \(10{,}045\); \(507\); \(1092\)

Hints

- First count the digits. A positive whole number with more digits is greater. - For numbers with the same number of digits, compare from left to right. - When leading digits match, compare the next place value.

Solution

1. Group the numbers by digit count: two-digit numbers \(89,98\); three-digit numbers \(507,567,576\); four-digit numbers \(1092,1209,1290\); and five-digit numbers \(10{,}045,10{,}450\). 2. Order each group by comparing digits from left to right. 3. Combine the groups from the fewest digits to the most digits.

Answer

\(89<98<507<567<576<1092<1209<1290<10{,}045<10{,}450\)
5171964
A car dealership recorded the odometer readings of several used cars. Order the readings from greatest to least: \(45{,}089\,\text{mi}\); \(45{,}809\,\text{mi}\); \(40{,}589\,\text{mi}\); \(45{,}098\,\text{mi}\); \(40{,}859\,\text{mi}\); \(45{,}890\,\text{mi}\)

Hints

- Greatest to least means begin with the greatest value. - Compare digits from left to right. - First sort the readings into groups based on the thousands digit.

Solution

1. Separate the readings into the \(45{,}000\)-mile group and the \(40{,}000\)-mile group. Every reading in the first group is greater. 2. Order the \(45{,}000\)-mile readings by comparing hundreds, tens, and ones: \(45{,}890>45{,}809>45{,}098>45{,}089\). 3. Order the \(40{,}000\)-mile readings: \(40{,}859>40{,}589\). 4. Combine the two groups.

Answer

\(45{,}890\,\text{mi}>45{,}809\,\text{mi}>45{,}098\,\text{mi}>45{,}089\,\text{mi}>40{,}859\,\text{mi}>40{,}589\,\text{mi}\)
5171974
Order the numbers from least to greatest: \(7077\); \(7707\); \(7007\); \(7770\); \(7070\); \(7700\)

Hints

- Align the digits by place value. - Compare the thousands digits first, then the hundreds digits. - Continue from left to right until the digits differ.

Solution

1. All the numbers have \(7\) in the thousands place. 2. Numbers with \(0\) in the hundreds place come before numbers with \(7\) in the hundreds place. 3. Order the first group: \(7007<7070<7077\). 4. Order the second group: \(7700<7707<7770\). 5. Combine the groups.

Answer

\(7007<7070<7077<7700<7707<7770\)
5171984
Compare each pair. Write \(<\) or \(>\) in the box. a) \(7382\;\square\;7832\) b) \(45{,}019\;\square\;45{,}109\) c) \(9999\;\square\;10{,}001\) d) \(234{,}567\;\square\;234{,}557\)

Hints

- First compare the number of digits. - If the numbers have the same number of digits, compare from left to right. - The first place where the digits differ determines which number is greater.

Solution

1. a) In the hundreds place, \(3<8\), so \(7382<7832\). 2. b) The ten-thousands and thousands digits match. In the hundreds place, \(0<1\), so \(45{,}019<45{,}109\). 3. c) \(9999\) has four digits and \(10{,}001\) has five digits, so \(9999<10{,}001\). 4. d) The digits match through the hundreds place. In the tens place, \(6>5\), so \(234{,}567>234{,}557\).

Answer

a) \(7382<7832\) b) \(45{,}019<45{,}109\) c) \(9999<10{,}001\) d) \(234{,}567>234{,}557\)
5171994
Order the numbers from least to greatest using \(<\). \(34{,}501\); \(35{,}401\); \(34{,}051\); \(35{,}041\); \(34{,}510\)

Hints

- Use the leftmost digits to sort the numbers into groups. - When leading digits match, compare the next place value. - Align the digits by place value.

Solution

1. Separate the numbers into those beginning with \(34\) thousand and those beginning with \(35\) thousand. 2. Order the \(34{,}000\) group: \(34{,}051<34{,}501<34{,}510\). 3. Order the \(35{,}000\) group: \(35{,}041<35{,}401\). 4. Combine the groups.

Answer

\(34{,}051<34{,}501<34{,}510<35{,}041<35{,}401\)
5176354
Complete each comparison with \(<\), \(>\), or \(=\). a) \(49\ \square\ 51\) b) \(700\ \square\ 600\) c) \(900\ \square\ 1100\) d) \(24+16\ \square\ 40\)

Hints

- Compare the greatest place value first. - For the last item, find the value of the expression before comparing. - Use \(=\) only when both sides have the same value.

Solution

1. a) Since \(49\) is less than \(51\), write \(<\). 2. b) Since \(700\) is greater than \(600\), write \(>\). 3. c) Since \(900\) is less than \(1100\), write \(<\). 4. d) Compute \(24+16=40\), so write \(=\).

Answer

a) \(49<51\) b) \(700>600\) c) \(900<1100\) d) \(24+16=40\)
5176564
The following inventions are associated with these years: - movable-type printing press: \(1450\) - early telephone demonstration: \(1861\) - long-lasting electric light bulb: \(1879\) - early gasoline-powered automobile: \(1886\) List the inventions in chronological order, beginning with the earliest.

Hints

- Compare the years from the greatest place value to the least. - Put the smallest year first. - Check that each later year is greater than the one before it.

Solution

1. Compare the thousands digits. The year \(1450\) comes before all three years in the \(1800\)s. 2. Among \(1861, 1879\), and \(1886\), compare the tens and ones digits: \(1861<1879<1886\). 3. Therefore, the order is printing press, telephone demonstration, electric light bulb, and automobile.

Answer

movable-type printing press (\(1450\)); early telephone demonstration (\(1861\)); long-lasting electric light bulb (\(1879\)); early gasoline-powered automobile (\(1886\))
5199854
A national park recorded the numbers of two kinds of trees. There are eight hundred thirty-two thousand four hundred three pine trees. There are eight hundred twenty-three thousand three hundred four fir trees. a) Write both numbers in standard form. b) Which kind of tree is more common in the park?

Hints

- Separate each number name at the word “thousand.” - Use a place-value chart if needed. - Compare the numbers beginning with the greatest place value on the left.

Solution

1. Eight hundred thirty-two thousand four hundred three is \(832{,}403\). 2. Eight hundred twenty-three thousand three hundred four is \(823{,}304\). 3. Compare from left to right. Both numbers have \(8\) in the hundred-thousands place, but \(832{,}403\) has \(3\) in the ten-thousands place while \(823{,}304\) has \(2\). Thus, \(832{,}403 > 823{,}304\), so pine trees are more common.

Answer

a) Pine trees: \(832{,}403\); fir trees: \(823{,}304\) b) Pine trees are more common.
5382214
A commercial orchard recorded this fruit harvest. <table><tr><th>Fruit</th><th>Weight</th></tr><tr><td>Apples</td><td>\(36{,}000\,\text{lb}\)</td></tr><tr><td>Pears</td><td>\(22{,}000\,\text{lb}\)</td></tr><tr><td>Plums</td><td>\(28{,}000\,\text{lb}\)</td></tr><tr><td>Peaches</td><td>\(14{,}000\,\text{lb}\)</td></tr></table> Order the fruits from least to greatest harvest. Include each weight.

Hints

- Compare the four weights. - Start with the least number, then match each number to its fruit.

Solution

1. Compare the four weights. 2. Order them from least to greatest: \(14{,}000 < 22{,}000 < 28{,}000 < 36{,}000\). 3. Match each weight to its fruit: Peaches, Pears, Plums, Apples.

Answer

Peaches \(14{,}000\,\text{lb}\), Pears \(22{,}000\,\text{lb}\), Plums \(28{,}000\,\text{lb}\), Apples \(36{,}000\,\text{lb}\)
5163794
Use these five numbers: \(450{,}300\), \(405{,}030\), \(450{,}003\), \(405{,}300\), \(540{,}003\) a) Order the numbers from least to greatest using \(<\). b) Which number is closest to \(450{,}000\)?

Hints

- Compare digits from left to right, beginning with the hundred-thousands place. - When two digits match, move to the next place. - For part b, find each number's difference from \(450{,}000\).

Solution

1. Compare the numbers from the greatest place value to the least. This gives \(405{,}030 < 405{,}300 < 450{,}003 < 450{,}300 < 540{,}003\). 2. The distances from \(450{,}000\) are \(44{,}970\), \(44{,}700\), \(3\), \(300\), and \(90{,}003\), respectively. 3. The smallest distance is \(3\), so \(450{,}003\) is closest to \(450{,}000\).

Answer

a) \(405{,}030 < 405{,}300 < 450{,}003 < 450{,}300 < 540{,}003\) b) \(450{,}003\)
5163804
Insert \(<\), \(>\), or \(=\). Calculate each expression before comparing. a) \(300{,}000 + 40{,}000 \mathrel{\square} 350{,}000 - 5000\) b) \(120{,}000 \times 4 \mathrel{\square} 1{,}000{,}000 \div 2\) c) \(720{,}000 \div 8 \mathrel{\square} 9000 \times 10\) d) \(999{,}998 + 2 \mathrel{\square} 1{,}000{,}000\)

Hints

- Evaluate the expression on each side of the box. - Keep track of place value when multiplying or dividing numbers with zeros. - Compare only after both values have been calculated.

Solution

1. For a), \(300{,}000 + 40{,}000 = 340{,}000\) and \(350{,}000 - 5000 = 345{,}000\), so \(340{,}000 < 345{,}000\). 2. For b), \(120{,}000 \times 4 = 480{,}000\) and \(1{,}000{,}000 \div 2 = 500{,}000\), so \(480{,}000 < 500{,}000\). 3. For c), \(720{,}000 \div 8 = 90{,}000\) and \(9000 \times 10 = 90{,}000\), so the values are equal. 4. For d), \(999{,}998 + 2 = 1{,}000{,}000\), so the values are equal.

Answer

a) \(<\) b) \(<\) c) \(=\) d) \(=\)
5163814
Insert one digit in each box to make the comparison true. a) \(345{,}\square20 < 345{,}100\) b) \(8\square9{,}000 > 889{,}000\) c) \(600{,}\square00 > 600{,}800\) d) \(490{,}\square99 < 490{,}199\)

Hints

- Identify the place value represented by each box. - Compare from left to right and focus on the first digit that can differ. - Check that each answer is the only digit that works.

Solution

1. For a), the hundreds digit must be less than \(1\). The only possible digit is \(0\), and \(345{,}020 < 345{,}100\). 2. For b), the ten-thousands digit must be greater than \(8\). The only possible digit is \(9\), and \(899{,}000 > 889{,}000\). 3. For c), the hundreds digit must be greater than \(8\). The only possible digit is \(9\), and \(600{,}900 > 600{,}800\). 4. For d), the hundreds digit must be less than \(1\). The only possible digit is \(0\), and \(490{,}099 < 490{,}199\).

Answer

a) \(0\) b) \(9\) c) \(9\) d) \(0\)
5164714
Insert \(<\), \(>\), or \(=\). Evaluate both sides before comparing. a) \(100{,}000 \div 2 \mathrel{\square} 200{,}000 \div 4\) b) \(100{,}000 \div 4 \mathrel{\square} 10{,}000 \times 2\) c) \(1{,}000{,}000 \div 8 \mathrel{\square} 100{,}000 + 25{,}000\) d) \(100{,}000 \div 10 \mathrel{\square} 10{,}000 \times 1\)

Hints

- Evaluate each side separately. - Write down both results before choosing a comparison symbol.

Solution

1. For a), \(100{,}000 \div 2 = 50{,}000\) and \(200{,}000 \div 4 = 50{,}000\), so the values are equal. 2. For b), \(100{,}000 \div 4 = 25{,}000\) and \(10{,}000 \times 2 = 20{,}000\), so the left side is greater. 3. For c), \(1{,}000{,}000 \div 8 = 125{,}000\) and \(100{,}000 + 25{,}000 = 125{,}000\), so the values are equal. 4. For d), \(100{,}000 \div 10 = 10{,}000\) and \(10{,}000 \times 1 = 10{,}000\), so the values are equal.

Answer

a) \(=\) b) \(>\) c) \(=\) d) \(=\)
5164904
The table lists six cities and their populations. <table> <tr> <td>Lakeview</td> <td>\(363{,}441\)</td> <td>Westfield</td> <td>\(354{,}572\)</td> </tr> <tr> <td>Riverton</td> <td>\(334{,}002\)</td> <td>Fairview</td> <td>\(331{,}285\)</td> </tr> <tr> <td>Pinecrest</td> <td>\(317{,}713\)</td> <td>Cedar Grove</td> <td>\(309{,}964\)</td> </tr> </table> a) Order the cities from greatest to least population. b) Find the difference between the greatest and least populations in the table.

Hints

- Compare the digits from left to right. - When digits match, move to the next place value. - For part b, subtract the least population from the greatest population.

Solution

1. Comparing the population values gives \(363{,}441 > 354{,}572 > 334{,}002 > 331{,}285 > 317{,}713 > 309{,}964\). 2. Therefore, the cities in order are Lakeview, Westfield, Riverton, Fairview, Pinecrest, and Cedar Grove. 3. Subtract the least population from the greatest: \(363{,}441 - 309{,}964 = 53{,}477\).

Answer

a) Lakeview, Westfield, Riverton, Fairview, Pinecrest, Cedar Grove b) \(53{,}477\) people
5166354
A wildlife group counted migrating birds in the same area for three years. Year 1: \(345{,}900\) birds Year 2: \(342{,}100\) birds Year 3: \(339{,}850\) birds Describe the trend. Are the counts increasing or decreasing? Support your answer by comparing the numbers.

Hints

- Compare Year 1 with Year 2. - Then compare Year 2 with Year 3. - Decide whether the same type of change occurs both times.

Solution

1. From Year 1 to Year 2, the count decreased because \(342{,}100 < 345{,}900\). 2. From Year 2 to Year 3, the count decreased again because \(339{,}850 < 342{,}100\). 3. Therefore, the counts decrease each year: \(345{,}900 > 342{,}100 > 339{,}850\).

Answer

The counts are decreasing because \(345{,}900 > 342{,}100 > 339{,}850\).
5166364
A region measured the amount of material recycled over three years. <table> <tr> <th>Material</th> <th>Year 1</th> <th>Year 2</th> <th>Year 3</th> </tr> <tr> <td>Paper</td> <td>\(789{,}400\,\text{tons}\)</td> <td>\(785{,}200\,\text{tons}\)</td> <td>\(780{,}100\,\text{tons}\)</td> </tr> <tr> <td>Plastic</td> <td>\(345{,}600\,\text{tons}\)</td> <td>\(350{,}000\,\text{tons}\)</td> <td>\(355{,}200\,\text{tons}\)</td> </tr> <tr> <td>Glass</td> <td>\(120{,}800\,\text{tons}\)</td> <td>\(125{,}400\,\text{tons}\)</td> <td>\(122{,}000\,\text{tons}\)</td> </tr> </table> a) Which material increases every year? b) Which material decreases every year? c) Which material increases and then decreases?

Hints

- Read one row at a time. - Compare Year 1 with Year 2, then Year 2 with Year 3. - Look for two increases, two decreases, or a change in direction.

Solution

1. Paper decreases because \(789{,}400 > 785{,}200 > 780{,}100\). 2. Plastic increases because \(345{,}600 < 350{,}000 < 355{,}200\). 3. Glass first increases because \(120{,}800 < 125{,}400\), then decreases because \(125{,}400 > 122{,}000\).

Answer

a) Plastic b) Paper c) Glass
5171944
Complete the comparisons. a) Order \(24{,}900\), \(25{,}100\), and \(25{,}000\) in a chain using \(<\). b) Compare \(306{,}040\) and \(306{,}400\) using \(>\). c) Which place value determines the comparison in part b)?

Hints

- In a chain using \(<\), begin with the least number. - In part b), compare the digits from left to right until they differ. - Name the place values: ones, tens, hundreds, thousands, and so on.

Solution

1. a) In increasing order, the numbers are \(24{,}900\), \(25{,}000\), and \(25{,}100\), so \(24{,}900<25{,}000<25{,}100\). 2. b) The digits match through the thousands place. In the hundreds place, \(4>0\), so \(306{,}400>306{,}040\). 3. c) The hundreds place is the first place from the left where the digits differ, so it determines the comparison.

Answer

a) \(24{,}900<25{,}000<25{,}100\) b) \(306{,}400>306{,}040\) c) the hundreds place
5172004
Which digits from \(\{0,1,2,3,4,5,6,7,8,9\}\) can replace the box to make each statement true? List every possible digit. a) \(56{,}342<56{,}\square42\) b) \(19{,}283>19{,}2\square3\) c) \(8\square{,}765<82{,}765\)

Hints

- Compare the digits that are already given. - When all other digits match, compare only the digit in the box with the digit in the same place. - Decide whether the missing digit must be greater or less than the given digit. - Remember that \(0\) may be a possible digit.

Solution

1. a) Only the hundreds digits differ. The box must contain a digit greater than \(3\): \(\{4,5,6,7,8,9\}\). 2. b) Only the tens digits differ. The box must contain a digit less than \(8\): \(\{0,1,2,3,4,5,6,7\}\). 3. c) Only the thousands digits differ. The box must contain a digit less than \(2\): \(\{0,1\}\).

Answer

a) \(\{4,5,6,7,8,9\}\) b) \(\{0,1,2,3,4,5,6,7\}\) c) \(\{0,1\}\)
5172454
Three numbers are written in words. Write each as a numeral, and then order them from least to greatest using \(<\). Number A: nine hundred nine thousand ninety Number B: nine hundred thousand nine hundred Number C: nine hundred ninety thousand nine

Hints

- First write all three numbers in the same form using numerals. - Compare digits from left to right, beginning with the greatest place value. - When two digits match, move one place to the right.

Solution

1. Write each number as a numeral: \(A=909{,}090\), \(B=900{,}900\), and \(C=990{,}009\). 2. Compare from the greatest place value. All three have \(9\) in the hundred-thousands place. Number B has \(0\) in the ten-thousands place, Number A also has \(0\), and Number C has \(9\), so C is greatest. 3. Compare A and B at the thousands place: A has \(9\), while B has \(0\). Therefore, \(B<A\). 4. The order is \(900{,}900<909{,}090<990{,}009\).

Answer

A: \(909{,}090\) B: \(900{,}900\) C: \(990{,}009\) Order: \(900{,}900<909{,}090<990{,}009\)
5172514
Order the numbers from least to greatest. Use the less-than sign \(<\). \(A\): “five hundred fifty thousand” \(B\): \(505{,}000\) \(C\): “five hundred thousand” \(D\): \(500{,}550\)

Hints

- First write every number in the same form using numerals. - Compare digits from the greatest place value to the least. - When the digits match, move one place to the right.

Solution

1. Write all numbers as numerals: \(A=550{,}000\), \(B=505{,}000\), \(C=500{,}000\), and \(D=500{,}550\). 2. Compare the digits from left to right. C and D both have \(500\) in the thousands period, but D has \(550\) in the ones period, so \(C<D\). 3. Number B is \(505{,}000\), which is greater than D but less than A. 4. Therefore, \(500{,}000<500{,}550<505{,}000<550{,}000\).

Answer

\(500{,}000<500{,}550<505{,}000<550{,}000\) Equivalently, \(C<D<B<A\).
5179554
Subtract \(2489\) from the least five-digit number whose digits are all different.

Hints

- The first digit of a five-digit number cannot be \(0\). - After choosing the first digit, place the remaining smallest available digits from left to right. - Check that no digit is repeated.

Solution

1. The first digit must be the smallest nonzero digit, \(1\). Use the remaining smallest digits in increasing order, including \(0\), to get \(10{,}234\). 2. Subtract: \(10{,}234 - 2489 = 7745\).

Answer

\(7745\)
5188934
Write each number name in standard form. a) two hundred thousand six hundred b) two hundred six thousand c) two hundred sixty thousand d) Order the three numbers from least to greatest. Use the symbol \(<\).

Hints

- Listen for the word “thousand” and place the remaining part in the final three places. - Use zeros for place values that are not named. - Compare the numbers digit by digit from left to right.

Solution

1. For a), two hundred thousand six hundred is \(200{,}600\). 2. For b), two hundred six thousand is \(206{,}000\). 3. For c), two hundred sixty thousand is \(260{,}000\). 4. For d), compare the digits from left to right. The first two numbers both have \(2\) in the hundred-thousands place and \(0\) in the ten-thousands place, so compare their thousands digits. This gives \(200{,}600 < 206{,}000\). The number \(260{,}000\) has \(6\) in the ten-thousands place, so it is greatest. 5. The order is \(200{,}600 < 206{,}000 < 260{,}000\).

Answer

a) \(200{,}600\) b) \(206{,}000\) c) \(260{,}000\) d) \(200{,}600 < 206{,}000 < 260{,}000\)
5189104
Sofia and Ethan are playing a number riddle game. Sofia says, “My secret number has \(4\) hundred-thousands, \(2\) thousands, \(8\) hundreds, and \(5\) ones.” Ethan says, “My number is four hundred twenty thousand eight hundred five.” a) Write Sofia’s number in standard form. b) Write Ethan’s number in standard form. c) Who chose the greater number?

Hints

- Use a place-value chart to put each digit in the correct place. - What digit belongs in a place value that Sofia did not name? - Compare the numbers digit by digit from left to right.

Solution

1. Sofia’s number has \(4\) in the hundred-thousands place, \(2\) in the thousands place, \(8\) in the hundreds place, and \(5\) in the ones place. The ten-thousands and tens digits are zero, so her number is \(402{,}805\). 2. Four hundred twenty thousand eight hundred five is \(420{,}805\). 3. Compare from left to right. Both numbers have \(4\) in the hundred-thousands place, but Ethan’s number has \(2\) in the ten-thousands place while Sofia’s has \(0\). Therefore, \(420{,}805 > 402{,}805\), so Ethan chose the greater number.

Answer

a) \(402{,}805\) b) \(420{,}805\) c) Ethan chose the greater number.
5199464
A list shows the populations of three cities. Check each number carefully. City A: four hundred five thousand — \(405{,}000\) City B: four hundred fifty thousand — \(400{,}050\) City C: forty thousand five hundred — \(40{,}500\) a) One city’s standard form does not match its number name. Which city is it? Write the correct number. b) Order the three correct populations from least to greatest. Use the symbol \(<\).

Hints

- Write each number name in standard form and compare it with the given numeral. - Check each place value from left to right. - When ordering the numbers, first compare the number of digits. - If two numbers have the same number of digits, compare their digits from left to right.

Solution

1. City A is correct because four hundred five thousand is \(405{,}000\). 2. City B is incorrect. Four hundred fifty thousand is \(450{,}000\), not \(400{,}050\). 3. City C is correct because forty thousand five hundred is \(40{,}500\). 4. The five-digit number \(40{,}500\) is less than both six-digit numbers. Between \(405{,}000\) and \(450{,}000\), compare the ten-thousands digits: \(0 < 5\). 5. The correct order is \(40{,}500 < 405{,}000 < 450{,}000\).

Answer

a) City B is incorrect. The correct number is \(450{,}000\). b) \(40{,}500 < 405{,}000 < 450{,}000\)
5199594
Write each number name in standard form. Then insert \(<\), \(>\), or \(=\). a) eight hundred thousand twelve \(\mathrel{\square}\) eight hundred twelve thousand b) three hundred four thousand five hundred \(\mathrel{\square}\) three hundred four thousand fifty

Hints

- Write each number name in a place-value chart or standard form. - Use zeros as placeholders for missing place values. - Compare the standard-form numbers from left to right.

Solution

1. “Eight hundred thousand twelve” is \(800{,}012\), and “eight hundred twelve thousand” is \(812{,}000\). 2. Therefore, \(800{,}012 < 812{,}000\). 3. “Three hundred four thousand five hundred” is \(304{,}500\), and “three hundred four thousand fifty” is \(304{,}050\). 4. Therefore, \(304{,}500 > 304{,}050\).

Answer

a) \(800{,}012 < 812{,}000\) b) \(304{,}500 > 304{,}050\)
5199614
Three numbers are written in word form. Number \(A\): four hundred thousand six hundred Number \(B\): four hundred six thousand Number \(C\): four hundred thousand six 1. Write Numbers \(A\), \(B\), and \(C\) in standard form. 2. Order the three numbers from least to greatest. Use the symbol \(<\). 3. What digit is in the ten-thousands place of Number \(B\)?

Hints

- Write each number in a place-value chart first. - Compare the numbers digit by digit from left to right. - In a six-digit number, locate the ten-thousands place by counting places from the right.

Solution

1. The numbers in standard form are \(A = 400{,}600\), \(B = 406{,}000\), and \(C = 400{,}006\). 2. Compare the digits from left to right. Numbers \(A\) and \(C\) both begin with \(400\) thousand, and \(6 < 600\), so \(400{,}006 < 400{,}600\). Number \(B\) has \(6\) in the thousands place, making it greatest. Thus, \(400{,}006 < 400{,}600 < 406{,}000\). 3. In \(406{,}000\), the digit in the ten-thousands place is \(0\).

Answer

1. \(A = 400{,}600\); \(B = 406{,}000\); \(C = 400{,}006\) 2. \(400{,}006 < 400{,}600 < 406{,}000\) 3. \(0\)
5199684
Consider these two numbers. Number A has \(5\) hundred-thousands, \(3\) ten-thousands, \(8\) hundreds, and \(1\) ten. Number B is five hundred thirty thousand eight hundred one. Write both numbers in standard form. Which number is greater?

Hints

- Write both numbers in a place-value chart. - Use zero in any place that is not named. - Compare the numbers digit by digit from left to right. - Pay close attention to the tens and ones digits.

Solution

1. Number A is \(500{,}000 + 30{,}000 + 800 + 10 = 530{,}810\). 2. Number B, five hundred thirty thousand eight hundred one, is \(530{,}801\). 3. The digits agree through the hundreds place. In the tens place, Number A has \(1\) and Number B has \(0\). Therefore, \(530{,}810 > 530{,}801\), so Number A is greater.

Answer

Number A is \(530{,}810\). Number B is \(530{,}801\). Number A is greater.

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