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Division That Makes Sense: Quotients, Remainders, Estimate, Check

Build reliable Grade 4–5 division with inverse multiplication, partial quotients, sensible remainders, estimation and 40 original practice questions.

Grades 4–5 · 40 practice questions · Original material
Learning objectiveDivide whole numbers accurately, interpret remainders in context, estimate before calculating, and check every quotient with multiplication.

The idea

Division asks how many equal groups fit, or how large each equal group will be. Multiplication is the fastest way to check the answer because the two operations undo each other.

For a division with a remainder, use this structure: dividend = divisor × quotient + remainder. The remainder must be at least 0 and smaller than the divisor. For 347 ÷ 8 = 43 R3, the check is 8 × 43 + 3 = 347.

Estimate before or after you calculate. For 1,248 ÷ 31, 1,200 ÷ 30 is about 40, so an answer near 40 is sensible. In a word problem, the remainder may mean leftovers, an extra container, or a fraction of a group, depending on the question.

Worked examples

Example 1 · Use multiplication facts to divide

Find 156 ÷ 12.

  1. 12 × 10 = 120, leaving 36.
  2. 12 × 3 = 36.
  3. 10 + 3 = 13, so 156 ÷ 12 = 13.

156 ÷ 12 = 13.

Example 2 · Keep the remainder legal

Find 347 ÷ 8.

  1. 8 × 40 = 320, leaving 27.
  2. 8 × 3 = 24, leaving 3.
  3. The quotient is 43 and the remainder is 3. Check: 8 × 43 + 3 = 347, and 3 < 8.

347 ÷ 8 = 43 R3.

Example 3 · Estimate first

Estimate, then find 1,248 ÷ 31.

  1. Use compatible numbers: 1,200 ÷ 30 ≈ 40.
  2. 31 × 40 = 1,240.
  3. 1,248 − 1,240 = 8, so the exact result is 40 R8.

Estimate ≈ 40; exact answer = 40 R8.

Example 4 · Leftovers stay leftovers

965 stickers are packed in groups of 24. How many full packs can be made, and how many stickers remain?

  1. 24 × 40 = 960.
  2. 965 − 960 = 5.
  3. The five stickers do not make another full pack.

40 full packs with 5 stickers left.

Example 5 · Sometimes the remainder means one more group

173 students ride in vans that hold 8 students each. What is the minimum number of vans?

  1. 173 ÷ 8 = 21 R5 because 8 × 21 = 168.
  2. Twenty-one vans leave 5 students without seats.
  3. One more van is needed.

22 vans.

Example 6 · Check a proposed answer

A student says 182 ÷ 9 = 20 R2. Is the student correct?

  1. Multiply: 9 × 20 = 180.
  2. Add the remainder: 180 + 2 = 182.
  3. The remainder 2 is smaller than 9, so the division statement is valid.

Yes. 182 ÷ 9 = 20 R2.

Common mistakes

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

Writing a remainder that is as large as the divisor.

Why this fails: If the divisor is 7, a remainder of 7 or more contains another full group and the quotient is not finished.

Using the remainder the same way in every word problem.

Why this fails: Leftover cookies may stay as leftovers, but leftover passengers require another vehicle. The context decides what to do.

Skipping the multiplication check.

Why this fails: divisor × quotient + remainder should rebuild the original dividend. This catches many arithmetic errors quickly.

Treating division as if order does not matter.

Why this fails: 48 ÷ 6 = 8, but 6 ÷ 48 is not 8. Division is not commutative.

Trusting an exact-looking quotient without estimating.

Why this fails: Compatible-number estimates can reveal a misplaced digit or an impossible quotient before you accept it.

Foundation

Foundation practice

Use multiplication facts, compatible numbers and simple remainders. Check the relationship between dividend, divisor and quotient.

Calculate

1. 96 ÷ 8

Calculate

2. 144 ÷ 12

Calculate

3. 175 ÷ 7

Calculate

4. 252 ÷ 9

Calculate

5. 320 ÷ 16

Calculate

6. 225 ÷ 15

Calculate

7. 432 ÷ 12

Estimate

8. What is a sensible estimate for 598 ÷ 21?

Inverse multiplication

9. ? × 14 = 238. What number belongs in the blank?

Remainder

10. 365 ÷ 10

Core

Core practice

Divide larger numbers, keep remainders smaller than the divisor, and decide what a remainder means in context.

Remainder

11. 287 ÷ 6

Remainder

12. 514 ÷ 8

Remainder

13. 743 ÷ 12

Remainder

14. 1,005 ÷ 25

Estimate and calculate

15. 1,248 ÷ 31

Remainder

16. 2,004 ÷ 48

Calculate

17. 936 ÷ 24

Calculate

18. 1,056 ÷ 32

Word problem

19. 425 apples are packed 18 per crate. How many full crates can be filled, and how many apples remain?

Word problem

20. 173 students need vans that hold 8 students each. What is the minimum number of vans?

Think

Think practice

Test remainder rules, work backwards and decide whether a proposed quotient really rebuilds the dividend.

Remainder rule

21. Which number cannot be the remainder when dividing by 7?

  1. 0
  2. 3
  3. 6
  4. 7
Reverse problem

22. 23 × q + 5 = 350. Find q.

Reverse problem

23. 800 ÷ ? = 40. Find the missing divisor.

Compare

24. How much larger is 615 ÷ 15 than 600 ÷ 15?

Check the statement

25. Is 450 ÷ 20 = 22 R10 valid?

  1. Yes
  2. No
Check the statement

26. Is 325 ÷ 16 = 20 R5 valid?

  1. Yes
  2. No
Remainder rule

27. What is the largest possible remainder when the divisor is 12?

Full groups

28. How many full groups of 36 can be made from 1,000, and what remains?

Check the statement

29. Is 275 ÷ 13 = 21 R2 valid?

  1. Yes
  2. No
Explain

30. Explain why 5,030 ÷ 50 = 100 R30 is valid.

Challenge

Challenge practice

Use the division equation as a structure for reverse problems, estimation and decisions about rounding a quotient.

Reverse problem

31. A number divided by 9 gives 34 R7. What is the number?

Find the divisor

32. 487 = d × 24 + 7. Find d.

Remainder

33. 1,234 ÷ 29

Remainder

34. 2,500 ÷ 64

Word problem

35. 1,257 people will sit at tables that hold 12 people each. What is the minimum number of tables?

Word problem

36. A 940 cm ribbon is cut into 28 cm strips. How many full strips can be cut, and how much ribbon remains?

Reverse remainder

37. 17 × q + r = 500, with q = 29 and 0 ≤ r < 17. Find r.

Estimate and calculate

38. Estimate 2,047 ÷ 51, then give the exact result.

Range reasoning

39. For whole-number division by 23, what is the smallest and largest dividend that can have quotient 18?

Prealgebra preview

40. If n = 12q + r, q = 37 and r = 5, find n.

Answer key

  1. 1. 12
    8 × 12 = 96.
  2. 2. 12
    12 × 12 = 144.
  3. 3. 25
    7 × 25 = 175.
  4. 4. 28
    9 × 28 = 252.
  5. 5. 20
    16 × 20 = 320.
  6. 6. 15
    15 × 15 = 225.
  7. 7. 36
    12 × 36 = 432.
  8. 8. About 30
    Use 600 ÷ 20 = 30 as a quick compatible-number estimate.
  9. 9. 17
    238 ÷ 14 = 17 because 14 × 17 = 238.
  10. 10. 36 R5
    10 × 36 = 360, leaving 5.
  11. 11. 47 R5
    6 × 47 = 282; 287 − 282 = 5.
  12. 12. 64 R2
    8 × 64 = 512; 2 remains.
  13. 13. 61 R11
    12 × 61 = 732; 743 − 732 = 11, which is smaller than 12.
  14. 14. 40 R5
    25 × 40 = 1,000; 5 remains.
  15. 15. 40 R8
    31 × 40 = 1,240; 8 remains. The result matches the estimate of about 40.
  16. 16. 41 R36
    48 × 41 = 1,968; 2,004 − 1,968 = 36.
  17. 17. 39
    24 × 39 = 936.
  18. 18. 33
    32 × 33 = 1,056.
  19. 19. 23 full crates with 11 apples left
    18 × 23 = 414; 425 − 414 = 11.
  20. 20. 22 vans
    173 ÷ 8 = 21 R5. The five remaining students need one more van.
  21. 21. 7
    A remainder must be smaller than the divisor. A remainder of 7 would make one more full group.
  22. 22. 15
    350 − 5 = 345, and 345 ÷ 23 = 15.
  23. 23. 20
    40 × 20 = 800.
  24. 24. 1
    615 ÷ 15 = 41 and 600 ÷ 15 = 40.
  25. 25. Yes
    20 × 22 + 10 = 450, and 10 is smaller than 20.
  26. 26. Yes
    16 × 20 + 5 = 325, and 5 < 16.
  27. 27. 11
    Remainders can be 0 through 11. Twelve would make another full group.
  28. 28. 27 full groups, R28
    36 × 27 = 972 and 1,000 − 972 = 28.
  29. 29. Yes
    13 × 21 = 273; adding 2 gives 275.
  30. 30. 50 × 100 + 30 = 5,030, and 30 < 50.
    Both the rebuild check and the remainder rule are satisfied.
  31. 31. 313
    9 × 34 + 7 = 306 + 7 = 313.
  32. 32. 20
    487 − 7 = 480, and 480 ÷ 24 = 20.
  33. 33. 42 R16
    29 × 42 = 1,218; 1,234 − 1,218 = 16.
  34. 34. 39 R4
    64 × 39 = 2,496; 4 remains.
  35. 35. 105 tables
    12 × 104 = 1,248, leaving 9 people, so one more table is needed.
  36. 36. 33 full strips and 16 cm left
    28 × 33 = 924; 940 − 924 = 16.
  37. 37. 7
    17 × 29 = 493; 500 − 493 = 7.
  38. 38. Estimate ≈ 40; exact = 40 R7
    2,000 ÷ 50 ≈ 40. Then 51 × 40 = 2,040, leaving 7.
  39. 39. Smallest 414; largest 436
    23 × 18 = 414. The largest legal remainder is 22, so 414 + 22 = 436.
  40. 40. 449
    12 × 37 + 5 = 444 + 5 = 449.

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