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Compare multi-digit numbers

Learning goals

  • Compares multi-digit whole numbers from the highest place value downward.
  • Uses <, > and = to compare whole numbers to 1,000,000.
  • Explains comparisons using place-value structure.
  • Orders several multi-digit numbers.

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Compare multi-digit numbers 1
Use the number line. What values do markers \(A\) and \(B\) represent? Then write a comparison of those two values using \(<\) or \(>\).
Figure for problem 554460

Hints

- Use the labeled ticks to determine the value of each unlabeled tick. - Give the marker values before comparing them.

Solution

1. The tick spacing is \(10{,}000\). 2. Marker \(A\) is on \(140{,}000\), and marker \(B\) is on \(170{,}000\). 3. Therefore, \(140{,}000<170{,}000\).

Answer

\(A=140{,}000\) \(B=170{,}000\) \(140{,}000<170{,}000\)
Compare multi-digit numbers 2
Compare \(72{,}504\) and \(72{,}540\). Which number is greater, and which place decides?

Hints

- Start comparing at the leftmost digit. - Keep moving right until the digits first differ.

Solution

1. Both numbers have the same ten-thousands, thousands, and hundreds digits. 2. The tens digits are \(0\) and \(4\), so \(72{,}540\) is greater.

Answer

\(72{,}540>72{,}504\); the tens place decides.
Compare multi-digit numbers 3
Which is greater, \(99{,}800\) or \(100{,}200\)? Explain using place value, not subtraction.

Hints

- Compare the numbers starting at the left. - If you wrote both numbers in a place-value chart, which column would decide their order?

Solution

1. \(99{,}800\) has no hundred-thousands, while \(100{,}200\) has \(1\) hundred-thousand. 2. The hundred-thousands place decides the comparison, so \(100{,}200\) is greater.

Answer

\(100{,}200>99{,}800\); the hundred-thousands place decides.
Compare multi-digit numbers 4
Read points A and B on the number line. Which point shows the greater number? Explain using their positions.
Figure for problem 559100

Hints

- Work out how much each tick is worth. - Read the two markers from the labeled points. - Numbers grow as you move right on a number line.

Solution

1. The ticks count by \(1000\). Point A is \(31{,}000\) and point B is \(35{,}000\). 2. Point B lies farther to the right, so \(35{,}000>31{,}000\).

Answer

A is \(31{,}000\), B is \(35{,}000\), and B is greater because it lies farther to the right on the number line.
Compare multi-digit numbers 5
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) \(89{,}800\)
Compare multi-digit numbers 6
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) \(89{,}679\)
Compare multi-digit numbers 7
Four zoos counted their visitors last year: Lakeside Zoo: \(506{,}780\) visitors Pine Ridge Zoo: \(560{,}870\) visitors Riverbend Zoo: \(506{,}870\) visitors Meadow Park Zoo: \(560{,}780\) visitors a) Compare the visitor counts for Lakeside Zoo and Riverbend Zoo. Insert \(<\) or \(>\). b) Compare the visitor counts for Pine Ridge Zoo and Meadow Park Zoo. 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. - For the final order, first group numbers with the same hundred-thousands and ten-thousands digits.

Solution

1. Lakeside Zoo and Riverbend Zoo match through the thousands place. Since the hundreds digit \(7\) is less than \(8\), \(506{,}780 < 506{,}870\). 2. Pine Ridge Zoo and Meadow Park Zoo 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) \(506{,}780 < 506{,}870 < 560{,}780 < 560{,}870\)
Compare multi-digit numbers 8
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

\(243{,}650 > 243{,}560 > 234{,}650 > 234{,}560\)
Compare multi-digit numbers 9
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
Compare multi-digit numbers 10
Compare each pair of whole numbers without subtracting. Write \(<\), \(=\), or \(>\), and name the first place value that decides the comparison. If every digit matches, write “all places match.” a) \(240{,}080\ \square\ 240{,}800\) b) \(609{,}999\ \square\ 610{,}001\) c) \(74{,}305\ \square\ 74{,}305\) d) \(502{,}410\ \square\ 502{,}401\) e) \(999{,}999\ \square\ 1{,}000{,}000\)

Hints

- Compare the digits from left to right, starting with the greatest place value. - Stop when corresponding place values first differ. - If one number has another place on the left, include it in the comparison.

Solution

1. In a), the first differing place is hundreds: \(0<8\), so \(240{,}080<240{,}800\). 2. In b), the first differing place is ten-thousands: \(0<1\), so \(609{,}999<610{,}001\). 3. In c), all corresponding digits match, so the numbers are equal. 4. In d), the first differing place is tens: \(1>0\), so \(502{,}410>502{,}401\). 5. In e), the millions place decides: \(999{,}999\) has no millions, so it is less than \(1{,}000{,}000\).

Answer

a) \(240{,}080<240{,}800\); hundreds place. b) \(609{,}999<610{,}001\); ten-thousands place. c) \(74{,}305=74{,}305\); all places match. d) \(502{,}410>502{,}401\); tens place. e) \(999{,}999<1{,}000{,}000\); millions place.
Compare multi-digit numbers 11
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) \(450{,}000>45{,}000\)
Compare multi-digit numbers 12
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) \(100<120<150\)
Compare multi-digit numbers 13
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

\(89<98<507<567<576<1092<1209<1290<10{,}045<10{,}450\)
Compare multi-digit numbers 14
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

\(45{,}890\,\text{mi}>45{,}809\,\text{mi}>45{,}098\,\text{mi}>45{,}089\,\text{mi}>40{,}859\,\text{mi}>40{,}589\,\text{mi}\)
Compare multi-digit numbers 15
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

\(7007<7070<7077<7700<7707<7770\)
Compare multi-digit numbers 16
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) \(234{,}567>234{,}557\)
Compare multi-digit numbers 17
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

\(34{,}051<34{,}501<34{,}510<35{,}041<35{,}401\)
Compare multi-digit numbers 18
Complete each comparison with \(<\), \(>\), or \(=\). a) \(49\ \square\ 51\) b) \(700\ \square\ 600\) c) \(900\ \square\ 1100\) d) \(1040\ \square\ 1004\)

Hints

- Compare digits from the greatest place value to the least. - If one place matches, move one place to the right until you find a difference.

Solution

1. a) \(49<51\). 2. b) \(700>600\). 3. c) \(900<1100\). 4. d) The thousands digits match. At the hundreds place, \(0=0\); at the tens place, \(4>0\), so \(1040>1004\).

Answer

a) \(49<51\) b) \(700>600\) c) \(900<1100\) d) \(1040>1004\)
Compare multi-digit numbers 19
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 (\(1886\))
Compare multi-digit numbers 20
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.
Compare multi-digit numbers 21
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 \(36{,}000\,\text{lb}\)
Compare multi-digit numbers 22
The number line shows points \(P\), \(Q\), and \(R\). Write their values in order from least to greatest.
Figure for problem 554461

Hints

- Determine the value of one tick interval. - Read the marker positions from left to right. - Check that your ordered values increase each time.

Solution

1. The ticks increase by \(1000\). 2. The markers are \(P=453{,}000\), \(R=455{,}000\), and \(Q=457{,}000\). 3. Reading from left to right gives \(453{,}000<455{,}000<457{,}000\).

Answer

\(453{,}000,\ 455{,}000,\ 457{,}000\)
Compare multi-digit numbers 23
Find every digit that can replace the box to make \(52\square{,}071<526{,}050\) true. Explain why your list is complete.

Hints

- Compare the two numbers from the greatest place to the least. - Which place contains the box, and what digit is in that place in the other number? - Check the equal-digit case separately; later digits may then decide the comparison.

Solution

1. The hundred-thousands and ten-thousands digits match in both numbers. 2. For the first number to be smaller, its thousands digit must be less than \(6\), giving \(0,1,2,3,4,5\). 3. If the thousands digit were \(6\), the hundreds would match too, but the tens digits would be \(7>5\), making the first number greater. Digits above \(6\) also fail.

Answer

\(0,1,2,3,4,5\). Larger digits fail: \(7,8,9\) make the thousands digit too large, while \(6\) makes the first number \(526{,}071>526{,}050\).
Compare multi-digit numbers 24
Put \(384{,}506\), \(384{,}650\), and \(384{,}560\) in order from least to greatest. Which place decides the comparison between the first two numbers in your final order?

Hints

- Compare digits beginning with the leftmost place. - Numbers with the same larger places may differ first in the hundreds or tens. - Use the first differing place to order each pair.

Solution

1. The numbers agree through the thousands place. 2. \(384{,}506\) and \(384{,}560\) both have \(5\) hundreds, while \(384{,}650\) has \(6\) hundreds, so the last number is greatest. 3. Between the two numbers with \(5\) hundreds, compare the tens: \(0<6\). Thus \(384{,}506<384{,}560<384{,}650\).

Answer

\(384{,}506<384{,}560<384{,}650\). The tens place decides the comparison between the first two.
Compare multi-digit numbers 25
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) \(450{,}003\)
Compare multi-digit numbers 26
Compare each pair of whole numbers. Write \(<\), \(>\), or \(=\), and name the first place value that decides the comparison. If the numbers are equal, write “all places match.” a) \(340{,}000\ \square\ 345{,}000\) b) \(4800\ \square\ 5000\) c) \(90{,}900\ \square\ 90{,}900\) d) \(999{,}998\ \square\ 1{,}000{,}000\)

Hints

- Compare from the largest place value to the smallest. - If two numbers have different numbers of digits, think about the extra place on the left.

Solution

1. In a), the first different digits are in the thousands place: \(0<5\), so \(340{,}000<345{,}000\). 2. In b), the thousands digits are \(4\) and \(5\), so \(4800<5000\). 3. In c), all corresponding digits match, so the numbers are equal. 4. In d), \(999{,}998\) has no millions, while \(1{,}000{,}000\) has \(1\) million. The millions place decides the comparison.

Answer

a) \(340{,}000<345{,}000\); thousands place. b) \(4800<5000\); thousands place. c) \(90{,}900=90{,}900\); all places match. d) \(999{,}998<1{,}000{,}000\); millions place.
Compare multi-digit numbers 27
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) \(0\)
Compare multi-digit numbers 28
Compare each pair of whole numbers. Write \(<\), \(>\), or \(=\), and name the first place value that decides the comparison. a) \(408{,}050\ \square\ 408{,}500\) b) \(79{,}999\ \square\ 80{,}001\) c) \(620{,}400\ \square\ 620{,}040\) d) \(1{,}000{,}000\ \square\ 999{,}999\)

Hints

- Start at the leftmost place value. - Find the first place where the values differ, including any place missing from the shorter number.

Solution

1. In a), the hundreds digits are the first that differ: \(0<5\). 2. In b), the ten-thousands digits differ: \(7<8\). 3. In c), the hundreds digits differ: \(4>0\). 4. In d), \(1{,}000{,}000\) has \(1\) million, while \(999{,}999\) has no millions. The millions place decides the comparison.

Answer

a) \(408{,}050<408{,}500\); hundreds place. b) \(79{,}999<80{,}001\); ten-thousands place. c) \(620{,}400>620{,}040\); hundreds place. d) \(1{,}000{,}000>999{,}999\); millions place.
Compare multi-digit numbers 29
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
Compare multi-digit numbers 30
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 \(345{,}900 > 342{,}100 > 339{,}850\).
Compare multi-digit numbers 31
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
Compare multi-digit numbers 32
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
Compare multi-digit numbers 33
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) \(\{0,1\}\)
Compare multi-digit numbers 34
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: \(900{,}900<909{,}090<990{,}009\)
Compare multi-digit numbers 35
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, \(C<D<B<A\).
Compare multi-digit numbers 36
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) \(200{,}600 < 206{,}000 < 260{,}000\)
Compare multi-digit numbers 37
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.
Compare multi-digit numbers 38
A list shows the populations of three cities. Check each number carefully. Northgate: four hundred five thousand — \(405{,}000\) Pinecrest: four hundred fifty thousand — \(400{,}050\) Westhaven: 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. Northgate is correct because four hundred five thousand is \(405{,}000\). 2. Pinecrest is incorrect. Four hundred fifty thousand is \(450{,}000\), not \(400{,}050\). 3. Westhaven 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) Pinecrest is incorrect. The correct number is \(450{,}000\). b) \(40{,}500 < 405{,}000 < 450{,}000\)
Compare multi-digit numbers 39
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) \(304{,}500 > 304{,}050\)
Compare multi-digit numbers 40
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. \(0\)
Compare multi-digit numbers 41
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.
Compare multi-digit numbers 42
Two cargo ships have total weights of two hundred five thousand ninety tons and two hundred five thousand nine hundred tons. a) Write both weights in standard form. b) Ethan says, “The number with a \(9\) in the hundreds place is greater than the number with a \(9\) in the tens place.” Is Ethan correct? Explain using place value.

Hints

- Use zeros as placeholders when writing both numbers. - Find the first place where the two numbers differ. - Explain why that place decides which number is greater.

Solution

1. Two hundred five thousand ninety is \(205{,}090\), so the first weight is \(205{,}090\,\text{tons}\). 2. Two hundred five thousand nine hundred is \(205{,}900\), so the second weight is \(205{,}900\,\text{tons}\). 3. Both numbers match through the thousands place. Their first different digits are in the hundreds place, where \(9>0\). So Ethan is correct: \(205{,}900>205{,}090\).

Answer

a) First ship: \(205{,}090\,\text{tons}\); second ship: \(205{,}900\,\text{tons}\). b) Yes. The thousands and larger places match, and \(9\) hundreds are greater than \(0\) hundreds. Therefore, \(205{,}900>205{,}090\).
Compare multi-digit numbers 43
Point \(X\) is on a tick of the number line. a) What value does \(X\) represent? b) Compare \(X\) with \(365{,}000\) using \(<\) or \(>\).
Figure for problem 554462

Hints

- Use the labeled ticks to determine the value of each interval. - Count equal intervals from a labeled benchmark to \(X\). - After finding \(X\), compare the numbers place by place.

Solution

1. The ticks are spaced by \(10{,}000\). Starting at \(350{,}000\), two more ticks give \(370{,}000\), so \(X=370{,}000\). 2. Since \(370{,}000\) is greater than \(365{,}000\), \(X>365{,}000\).

Answer

a) \(370{,}000\) b) \(370{,}000>365{,}000\)
Compare multi-digit numbers 44
Find two six-digit whole numbers where the smaller number has a greater digit sum than the larger number. Write the comparison and both digit sums. Explain why this is possible.

Hints

- Does having a larger digit sum always mean a number is greater? - How do you compare two six-digit numbers when their digits differ? - Check both the order of your example numbers and their digit sums.

Solution

1. One example is \(199{,}999<200{,}000\). 2. The digit sum of \(199{,}999\) is \(1+9+9+9+9+9=46\), but the digit sum of \(200{,}000\) is \(2\). 3. The number \(200{,}000\) is greater because its hundred-thousands digit is larger. The digit sum does not decide which whole number is greater.

Answer

Example: \(199{,}999<200{,}000\), with digit sums \(46\) and \(2\). The first differing place determines the greater number, not the digit sum.
Compare multi-digit numbers 45
Oak School and Pine School are counting steps in a walking challenge. Oak's total is between \(89{,}980\) and \(90{,}020\). Pine's total is between \(90{,}000\) and \(90{,}040\). Both ends of each range are allowed. Give three possible results: Oak has more steps, Pine has more steps, and a tie. For each result, give a possible total for both schools. Explain why neither school is sure to win.

Hints

- Write down some allowed step totals for each school. - Do the two ranges have any possible totals in common? - For each result, check that your two chosen totals are allowed and compare them.

Solution

1. Oak can win: Oak \(90{,}020\) steps and Pine \(90{,}000\) steps. Both totals are in their allowed ranges, and \(90{,}020>90{,}000\). 2. Pine can win: Oak \(89{,}980\) steps and Pine \(90{,}040\) steps. Both are allowed, and \(89{,}980<90{,}040\). 3. They can tie: both totals can be \(90{,}010\), which is in both ranges. 4. All three outcomes are possible, so neither school is sure to win.

Answer

Oak wins: Oak \(90{,}020\); Pine \(90{,}000\). Pine wins: Oak \(89{,}980\); Pine \(90{,}040\). Tie: Oak \(90{,}010\); Pine \(90{,}010\). Neither school is certain to win. Other correct totals are possible.
Compare multi-digit numbers 46
Kai and Ada wrote six-digit numbers. Kai: \(4\square7{,}900\) Ada: \(4\square8{,}100\) The two boxes can be different digits. Tell exactly when Kai's number is greater and when Ada's is greater. Could their numbers ever be equal? Explain why your rule works for any digits in the boxes.

Hints

- First find the leftmost place where the two numbers might differ. - Think about what happens if the two box digits happen to match. - Your rule must work for every digit from \(0\) through \(9\), not just one example.

Solution

1. Both numbers begin with \(4\) in the hundred-thousands place. The next digit is in the ten-thousands place. 2. If Kai's box is larger than Ada's box, Kai's number is greater. The difference in the ten-thousands place decides the order. 3. If Kai's box is smaller than Ada's, Ada's number is greater. 4. If the box digits match, compare the thousands places: Ada has \(8\) and Kai has \(7\), so Ada is greater. 5. The numbers can never be equal because either the box digits differ, or the thousands digits decide the order.

Answer

Kai is greater only if his box digit is greater than Ada's. Ada is greater if her box digit is greater than or equal to Kai's. They cannot be equal. The ten-thousands digits decide when they differ. If they match, Ada's \(8\) thousands are greater than Kai's \(7\) thousands.

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