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Math · Number Sense & Place Value

Place Value to 100,000: Digit Value, Expanded Form, Compare, Order

Read, write and compare numbers up to 100,000 by place value, use zeros as placeholders, and see what happens to digits when you multiply or divide by 10.

Grades 3–5 · 32 practice questions · Original material
Learning objectiveName the value of each digit in a whole number up to 100,000, write the number in standard, word and expanded form, compare and order numbers by place, and explain how multiplying or dividing by 10 moves every digit one place.

The idea

In the last few lessons you rounded, estimated and regrouped with big numbers. All of that depends on place value: where a digit sits tells you how much it is worth. Today you look at that idea closely so that large numbers stop being just strings of digits.

Each place is worth 10 times the place to its right. Going left from the ones: ones, tens, hundreds, thousands, ten-thousands. In 74,205 the 7 is in the ten-thousands place and is worth 70,000, the 4 is worth 4,000, the 2 is worth 200, the 0 is worth 0 and the 5 is worth 5. The digit and its value are not the same thing: the digit is 4, the value is 4,000.

A zero is a placeholder. It shows that a place is empty and keeps the other digits where they belong. 4,000 + 50 + 6 is 4,056, not 456, because there are no hundreds. Expanded form writes the value of every non-zero digit as an addition: 52,060 = 50,000 + 2,000 + 60. Word form says the same number in words: fifty-two thousand sixty. Do not say “and” when you read a whole number.

The commas in this lesson group the digits in threes to make them easier to read. Some books and Canadian style guides use a space instead (52 060). The groups are read the same way either way, and a four-digit number such as 4,056 may be written without a comma or a space.

To compare two numbers, first count digits. A number with more digits is greater, so 10,234 is greater than 9,876 even though 9 is bigger than 1. If the digit counts match, compare place by place from the left and stop at the first place where the digits are different. 48,950 is greater than 48,905 because 5 tens is greater than 0 tens.

Multiplying by 10 makes every digit worth 10 times as much, so every digit moves one place to the left, and a zero fills the empty ones place: 3,400 × 10 = 34,000. Dividing by 10 does the reverse: every digit moves one place to the right. The saying “add a zero” gives the right answer for whole numbers, but it hides the reason, which is the move of the digits.

Worked examples

A place-value chart, multiplying by 10 and comparing two numbersTop panel: a place-value chart for 74,205. The places from left to right are ten-thousands, thousands, hundreds, tens and ones, and each place is worth 10 times the place to its right. The digits are 7, 4, 2, 0 and 5, with values 70,000, 4,000, 200, 0 and 5. Bottom left: multiplying 3,400 by 10 moves every digit one place left, so 3 thousands and 4 hundreds become 3 ten-thousands and 4 thousands, and a zero fills the ones place, giving 34,000. Bottom right: comparing 48,905 and 48,950. The digits match until the tens place, where 0 is less than 5, so 48,950 is greater.Where a digit sits tells you what it is worthOriginal schematic, drawn to the labelled values.1 · Place-value chart for 74,205Ten-thousands7= 70,000Thousands4= 4,000Hundreds2= 200Tens0= 0Ones5= 5◀ × 10◀ × 10◀ × 10◀ × 10Each place is worth 10 times the place to its right.2 · Multiply by 10: digits move left3400340003,400 × 10 = 34,000. The zero holds the ones place.3 · Compare from the left48905489500 < 5Same until the tens: 48,950 is greater.
Original schematic, drawn to the labelled values. Each place is worth 10 times the place to its right, so a digit and its value are different: the 4 in 74,205 is worth 4,000. Multiplying by 10 moves every digit one place left. To compare, count digits first, then compare from the left and stop at the first difference.

Example 1 · Value of a digit

What is the value of the 3 in 63,408?

  1. Write the places under the digits: 6 ten-thousands, 3 thousands, 4 hundreds, 0 tens, 8 ones.
  2. The 3 is in the thousands place.
  3. Its value is 3 × 1,000 = 3,000.
  4. The digit is 3. The value is 3,000.

3,000

Example 2 · Expanded form

Write 52,060 in expanded form.

  1. The digits are 5 ten-thousands, 2 thousands, 0 hundreds, 6 tens, 0 ones.
  2. Skip the places that hold zero.
  3. Write the value of each other digit: 50,000, 2,000 and 60.
  4. Add them: 50,000 + 2,000 + 60 = 52,060.

50,000 + 2,000 + 60

Example 3 · Compare two numbers

Which is greater, 48,905 or 48,950?

  1. Both numbers have 5 digits.
  2. Compare from the left: 4 and 4, 8 and 8, 9 and 9 are the same.
  3. In the tens place, 0 is less than 5.
  4. So 48,950 is greater than 48,905.

48,950

Example 4 · Multiply by 10

What is 3,400 × 10? Which way do the digits move?

  1. Each digit becomes worth 10 times as much.
  2. The 3 moves from thousands to ten-thousands, and the 4 moves from hundreds to thousands.
  3. A zero fills the empty ones place so the digits stay in their places.
  4. 3,400 × 10 = 34,000. Check: 34,000 ÷ 10 = 3,400.

34,000

Example 5 · Catch the mistake

Sam says 9,876 is greater than 12,345 because 9 is greater than 1. Is he right?

  1. He compared only the first digits.
  2. 9,876 has 4 digits and 12,345 has 5 digits.
  3. The 1 in 12,345 is in the ten-thousands place and is worth 10,000.
  4. 12,345 is greater. Count the digits first.

No. 12,345 is greater.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Confusing a digit with its value.

Why this fails: In 74,205 the digit is 4 but its value is 4,000 because it is in the thousands place.

Dropping a placeholder zero.

Why this fails: 4,000 + 50 + 6 is 4,056. Writing 456 moves the 4 into the hundreds place and loses 3,600.

Comparing only the first digit when the lengths differ.

Why this fails: 9,876 < 10,234 because the number with more digits is greater. Count digits first, then compare place by place.

Reading digits from the right when comparing.

Why this fails: 48,905 and 48,950 differ in the tens place. Start at the left, where the places are worth the most, and stop at the first difference.

Treating “add a zero” as the reason for multiplying by 10.

Why this fails: Each digit becomes worth 10 times as much and moves one place left, and the zero holds the ones place. “Add a zero” is a shortcut that hides this reason, and it does not explain why 47 × 10 is 470 and not 4,070.

Foundation

Foundation practice

Read, write and name the value of digits in numbers up to 100,000, and compare numbers by place.

Value of a digit

1. What is the value of the digit 4 in 74,205?

  1. 4
  2. 4,000
  3. 40
  4. 40,000
Name the place

2. In 38,561, which digit is in the hundreds place?

  1. 3
  2. 5
  3. 6
  4. 8
Words to digits

3. Which number is twenty-six thousand forty?

  1. 26,004
  2. 2,640
  3. 26,400
  4. 26,040
Digits to words

4. Which is the correct word form of 30,507?

  1. thirty-five thousand seven
  2. thirty thousand five hundred seven
  3. thirty thousand fifty-seven
  4. three thousand five hundred seven
Expanded form to number

5. What number is 6,000 + 300 + 5?

  1. 635
  2. 60,305
  3. 6,350
  4. 6,305
Greatest number

6. Which number is the greatest?

  1. 10,245
  2. 9,870
  3. 9,999
  4. 8,999
How many?

7. How many hundreds are in 4,700?

  1. 7
  2. 47
  3. 4
  4. 470
Compare

8. Which comparison is correct?

  1. 36,480 > 36,048
  2. 36,480 = 36,048
  3. Cannot tell without adding
  4. 36,480 < 36,048

Practice

Practice practice

Use zeros as placeholders, order numbers, and multiply or divide by 10.

Zero as a placeholder

9. What number is 40,000 + 7,000 + 5?

  1. 4,075
  2. 47,005
  3. 4,705
  4. 40,750
Order the numbers

10. Which list shows the numbers in order from least to greatest?

  1. 2,999, 20,999, 21,450, 21,504
  2. 20,999, 21,450, 21,504, 2,999
  3. 21,504, 21,450, 20,999, 2,999
  4. 2,999, 20,999, 21,504, 21,450
Compare values

11. In 77,000, the first 7 is worth 70,000 and the second 7 is worth 7,000. How many times as great is the first 7?

  1. 7 times
  2. 100 times
  3. 1,000 times
  4. 10 times
Multiply by 10

12. What is 5,280 × 10?

  1. 528,000
  2. 5,290
  3. 52,800
  4. 52,080
Divide by 10

13. What is 60,000 ÷ 10?

  1. 600
  2. 600,000
  3. 6,000
  4. 59,990
Same digit, different value

14. The number 82,481 has two 8s. What are their values?

  1. 8,000 and 80
  2. 8,000 and 8
  3. 80,000 and 80
  4. 80,000 and 8
One thousand more

15. What is 1,000 more than 46,750?

  1. 46,850
  2. 56,750
  3. 46,760
  4. 47,750
Between two numbers

16. Which number is between 34,800 and 35,200?

  1. 35,020
  2. 34,790
  3. 33,990
  4. 35,210

Think

Think practice

Reason about digit values, missing digits and real-life comparisons.

Which is true?

17. Which comparison is true?

  1. 70,003 < 7,303
  2. 31,400 = 3,140
  3. 45,062 > 45,602
  4. 9,999 < 10,000
Make a number

18. Use the digits 7, 0, 3 and 5 once each. What is the least 4-digit number you can make?

  1. 3,075
  2. 0357
  3. 3,057
  4. 3,570
Missing digit

19. What is the smallest digit that can go in the box to make 4□,620 greater than 47,000?

  1. 7
  2. 0
  3. 8
  4. 6
Compare in context

20. An arena (invented example) sold 23,480 tickets on Friday and 23,408 on Saturday. Which statement is correct?

  1. Saturday sold more, because it ends in 8.
  2. Friday sold more, because 8 tens is greater than 0 tens.
  3. Friday sold more, because 480 is a longer number than 408.
  4. They sold the same, because they use the same digits.
Expanded form

21. Which is the expanded form of 80,090?

  1. 80,000 + 9
  2. 8,000 + 90
  3. 80,000 + 90
  4. 80,000 + 900
How many tens?

22. A prize (invented example) is worth $1,250. How many $10 bills equal $1,250?

  1. 25
  2. 12
  3. 1,250
  4. 125
Ten times as great

23. A number is 10 times as great as 4,300. Which digit is in its ten-thousands place?

  1. 1
  2. 0
  3. 3
  4. 4
Find the error

24. Priya wrote 4,000 + 50 + 6 as 456. What is the correct number?

  1. 456
  2. 4,056
  3. 4,506
  4. 4,560

Challenge

Challenge practice

Solve riddles, count possibilities and explain place-value rules in your own words.

Number riddle

25. Find the number: the ten-thousands digit is 3, the thousands digit is twice as big, the hundreds digit is 0, the tens digit is 5 and the ones digit is 9.

  1. 63,059
  2. 33,059
  3. 36,059
  4. 36,509
Value of a digit

26. In 16,205 the 6 is worth 6,000. In which number is the value of the 6 exactly 10 times as great?

  1. 16,205
  2. 61,205
  3. 26,105
  4. 2,605
Count the possibilities

27. How many different 3-digit numbers can be made with the digits 2, 5 and 8 (each used once) that are greater than 500?

  1. 4
  2. 3
  3. 2
  4. 6
Rank the data

28. Four towns (invented data) have populations: Maple Bay 12,807; Pine Ridge 12,780; Cedar Falls 12,870; Oak Point 9,999. Which town has the second-greatest population?

  1. Maple Bay
  2. Pine Ridge
  3. Cedar Falls
  4. Oak Point
Explain the rule

29. Which statement best explains why 47 × 10 = 470?

  1. Each digit becomes worth 10 times as much, so each moves one place left and a zero holds the ones place.
  2. A zero is added, and adding a zero makes any number bigger.
  3. 47 + 10 = 57, then a zero is added.
  4. 4 × 10 = 40 and 7 × 10 = 70, and the two answers are written side by side.
Compare two 5s

30. In 25,051, how many times as great is the value of the first 5 (from the left) as the value of the second 5?

  1. 10 times
  2. 1,000 times
  3. 100 times
  4. 5 times
Explain

31. Explain why 9,876 is less than 10,234 even though the first digit, 9, is greater than 1.

Write it three ways

32. Write “forty thousand six hundred two” in standard form and expanded form. What is the value of the 6?

Answer key

  1. 1. 4,000
    The 4 is in the thousands place (7 ten-thousands, 4 thousands, 2 hundreds, 0 tens, 5 ones), so its value is 4 × 1,000 = 4,000. 4 is the digit itself, 40 treats it as tens, and 40,000 treats it as ten-thousands.
  2. 2. 5
    Read from the right: 1 ones, 6 tens, 5 hundreds, 8 thousands, 3 ten-thousands. So the hundreds digit is 5. 8 is the thousands digit, 3 is the ten-thousands digit, and 6 is the tens digit.
  3. 3. 26,040
    Twenty-six thousand is 26,000 and forty is 40, so the number is 26,040. There are no hundreds, so a zero holds the hundreds place. 26,400 reads as four hundred, 26,004 reads as four, and 2,640 is only two thousand six hundred forty.
  4. 4. thirty thousand five hundred seven
    30,507 is 30 thousand, 5 hundreds, 0 tens and 7 ones: thirty thousand five hundred seven. Thirty-five thousand seven is 35,007, three thousand five hundred seven is 3,507, and thirty thousand fifty-seven is 30,057.
  5. 5. 6,305
    There are 6 thousands, 3 hundreds, no tens and 5 ones, so the number is 6,305. 6,350 puts the 5 in the tens place, 635 leaves out the thousands, and 60,305 puts the 6 in the ten-thousands place.
  6. 6. 10,245
    10,245 has 5 digits, and the other three have only 4 digits. A number with more digits (and no leading zero) is greater. 9,999 looks big because of its 9s, but it is 1 less than 10,000.
  7. 7. 47
    4,700 is 4 thousands and 7 hundreds. Each thousand is 10 hundreds, so 4,700 = 47 hundreds. 4 counts only the thousands, 7 only the hundreds, and 470 is the number of tens.
  8. 8. 36,480 > 36,048
    Both numbers have 5 digits and match in the ten-thousands, thousands and hundreds places. In the tens place 8 > 0, so 36,480 > 36,048. The digits are the same, but their places are different, so the numbers are not equal.
  9. 9. 47,005
    4 ten-thousands, 7 thousands, 0 hundreds, 0 tens and 5 ones make 47,005. The zeros must stay as placeholders. 4,705 loses a place, 40,750 puts the 7 in the hundreds place, and 4,075 loses a digit and moves the 7.
  10. 10. 2,999, 20,999, 21,450, 21,504
    2,999 has only 4 digits, so it is the least. Then compare the 5-digit numbers place by place: 20,999 < 21,450 < 21,504. The first wrong list is greatest to least, the second puts 2,999 last because of its 9s, and the third swaps 21,450 and 21,504 (4 hundreds is less than 5 hundreds).
  11. 11. 10 times
    70,000 ÷ 7,000 = 10. Each place is worth 10 times the place to its right. 100 times would skip a place, 7 times compares the digits and ignores their places, and 1,000 times skips two places.
  12. 12. 52,800
    Multiplying by 10 makes every digit worth 10 times as much, so each digit moves one place to the left: 5,280 × 10 = 52,800. 5,290 adds 10, 528,000 multiplies by 100, and 52,080 puts the zero in the middle instead of keeping the digits in order.
  13. 13. 6,000
    Dividing by 10 makes every digit worth one tenth as much, so the 6 moves one place to the right, from ten-thousands to thousands: 6,000. 600 moves two places, 59,990 subtracts 10, and 600,000 multiplies by 10 instead of dividing.
  14. 14. 80,000 and 80
    Reading from the left, the first 8 is in the ten-thousands place (80,000). The second 8 is in the tens place (80). 82,481 has 8 ten-thousands, 2 thousands, 4 hundreds, 8 tens and 1 one. The other pairs each misread at least one place.
  15. 15. 47,750
    Adding 1,000 changes the thousands digit by 1: 46,750 + 1,000 = 47,750. 46,850 adds 100, 56,750 adds 10,000, and 46,760 adds 10.
  16. 16. 35,020
    35,020 is greater than 34,800 and less than 35,200. 34,790 is just under 34,800 (79 tens < 80 tens), 35,210 is just over 35,200, and 33,990 is far below.
  17. 17. 9,999 < 10,000
    9,999 has 4 digits and 10,000 has 5, so 9,999 < 10,000 (it is exactly 1 less). 45,062 < 45,602 because 0 hundreds < 6 hundreds, 70,003 > 7,303 because it has more digits, and 31,400 is 10 times 3,140, not equal.
  18. 18. 3,057
    To make the least number, put the smallest digit that is not zero in the thousands place (3), then the zero, then the rest in order: 3,057. 0357 is not a 4-digit number because it starts with a zero, 3,075 has the last two digits in the wrong order, and 3,570 has the zero too far right.
  19. 19. 7
    With 7 in the box, 47,620 > 47,000 (620 > 0 in the last three digits). With 6, 46,620 is less than 47,000. 8 works too (48,620), but it is not the smallest. 0 gives 40,620, which is far less than 47,000.
  20. 20. Friday sold more, because 8 tens is greater than 0 tens.
    The numbers match in the first three places. The first difference is in the tens place: 8 tens (Friday) is greater than 0 tens (Saturday). Ending in 8 does not make a number greater, both 3-digit endings have the same length, and the same digits in different places make different numbers.
  21. 21. 80,000 + 90
    80,090 is 8 ten-thousands, 0 thousands, 0 hundreds, 9 tens and 0 ones: 80,000 + 90. 80,000 + 9 uses ones, 80,000 + 900 uses hundreds, and 8,000 + 90 gives 8,090.
  22. 22. 125
    Every 100 is 10 tens, and 1,250 is 12 hundreds and 5 tens, so it is 120 + 5 = 125 tens (1,250 ÷ 10 = 125). 12 counts only the hundreds, 1,250 counts ones, and 25 counts the last two digits.
  23. 23. 4
    4,300 × 10 = 43,000. In 43,000 the 4 is in the ten-thousands place, the 3 is in the thousands place, and the rest are zeros. 1 does not appear in the number at all.
  24. 24. 4,056
    There are no hundreds, so a zero must hold the hundreds place: 4,056. Writing 456 drops the placeholder zero, so the 4 is only worth 400. 4,560 and 4,506 put the zero in the wrong place and change the values of 5 or 6.
  25. 25. 36,059
    The thousands digit is 2 × 3 = 6, so the digits are 3, 6, 0, 5, 9: 36,059. 36,509 swaps the hundreds and tens, 33,059 repeats the 3 instead of doubling it, and 63,059 reverses the first two digits.
  26. 26. 61,205
    10 × 6,000 = 60,000, so the 6 must be in the ten-thousands place: 61,205. In 26,105 the 6 is worth 6,000, in 2,605 it is worth 600, and in 16,205 it is worth 6,000.
  27. 27. 4
    The six numbers are 258, 285, 528, 582, 825 and 852. The four that start with 5 or 8 are greater than 500: 528, 582, 825, 852. 2 counts only those starting with 8, 3 miscounts, and 6 counts every number including 258 and 285.
  28. 28. Maple Bay
    Oak Point has only 4 digits, so it is the least. Among the 5-digit numbers, compare hundreds: 12,870 and 12,807 and 12,780. The order from greatest is Cedar Falls (12,870), Maple Bay (12,807), Pine Ridge (12,780), Oak Point (9,999). Cedar Falls is greatest, not second.
  29. 29. Each digit becomes worth 10 times as much, so each moves one place left and a zero holds the ones place.
    Multiplying by 10 makes 4 tens into 4 hundreds and 7 ones into 7 tens, so the digits shift one place left and a zero is needed in the ones place. “Add a zero” is a shortcut that hides the reason, side-by-side products would give 4,070, and 47 + 10 is addition.
  30. 30. 100 times
    The first 5 is in the thousands place (5,000) and the second 5 is in the tens place (50). 5,000 ÷ 50 = 100. There are two places between thousands and tens (hundreds and tens), so the ratio is 10 × 10 = 100, not 10 or 1,000. 5 times compares the digits only.
  31. 31. 9,876 has only 4 digits and 10,234 has 5. The 1 in 10,234 is in the ten-thousands place, worth 10,000, which is more than anything 9,876 can reach (9,876 is less than 10,000). Compare the number of digits first, then compare place by place from the left.
    Self-review with this: did you compare the places (ten-thousands against thousands) and not just the first digits? Did you say that a number with more digits is greater? Check: 10,234 − 9,876 = 358.
  32. 32. 40,602; 40,000 + 600 + 2; the 6 is worth 600.
    Self-review: 40 thousand is 40,000, six hundred is 600, and two is 2, so the number is 40,602. There are no thousands and no tens, so zeros hold those places. The 6 is in the hundreds place, so its value is 6 × 100 = 600.

Related practice

Rounding to 10, 100 and 1000: then estimate and checkAdd and subtract with regrouping: trade, then checkDecimals: tenths and hundredths on a number lineMath hub