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Remove One Pair of Parentheses: Smart Arithmetic Without Traps

Learn when parentheses can be safely removed, when minus signs reverse terms, how multiplication distributes, and why division over a grouped sum is different. Includes 28 original Grade 4–5 questions with worked explanations.

Grades 4–5 · 28 practice questions · Original material
Learning objectiveRewrite one pair of parentheses correctly, justify each step with equivalent arithmetic, and detect invalid division or sign-changing shortcuts.

The idea

Parentheses show what must be calculated together. A good mental-math shortcut does not ignore them; it replaces an expression with another expression that has exactly the same value.

When a plus sign comes before parentheses, you can combine the inside terms without changing their signs: 48 + (27 − 9) = 48 + 27 − 9. When a minus sign comes before parentheses, subtracting a sum subtracts every part: 91 − (31 + 20) = 91 − 31 − 20. Subtracting a difference reverses the second subtraction: 83 − (28 − 7) = 83 − 28 + 7.

Multiplication can distribute across a sum or difference: 6 × (25 + 5) = 6 × 25 + 6 × 5. The multiplier must apply to both terms. If the entire sum is divided by the same number, splitting can also work: (42 + 18) ÷ 6 = 42 ÷ 6 + 18 ÷ 6.

Do not apply that last rule backwards. For example, 72 ÷ (6 + 3) equals 8, not 72 ÷ 6 + 72 ÷ 3. The grouped divisor must be evaluated first. When a shortcut looks too easy, compute both forms or use a simpler example to test whether they are truly equivalent.

Worked examples

Safe parentheses rewrite versus invalid division by a sum83 minus the difference of 28 and 7 equals 83 minus 28 plus 7, giving 62. But 72 divided by the sum of 6 and 3 is 8, not 36.Do both sides give the SAME answer?VALID: subtract a difference83 − (28 − 7) = 83 − 28 + 762 = 62 ✓INVALID: split division across a grouped divisor72 ÷ (6 + 3) ≠ 72 ÷ 6 + 72 ÷ 38 is NOT 36
Check that every proposed shortcut preserves the value. Subtraction can reverse a sign; division by a grouped sum cannot be split across its terms.

Example 1 · Plus before parentheses

Find 48 + (27 − 9).

  1. Because the outside sign is plus, the inside signs stay the same.
  2. 48 + 27 − 9 = 75 − 9.
  3. The result is 66.

48 + (27 − 9) = 66.

Example 2 · Minus before a sum

Find 91 − (31 + 20).

  1. The minus applies to the entire sum.
  2. Rewrite as 91 − 31 − 20.
  3. 60 − 20 = 40.

91 − (31 + 20) = 40.

Example 3 · Minus before a difference

Find 83 − (28 − 7).

  1. Subtracting 28 and then adding back 7 is equivalent to subtracting (28 − 7).
  2. Rewrite as 83 − 28 + 7.
  3. 55 + 7 = 62.

83 − (28 − 7) = 62.

Example 4 · Multiply every part

Find 9 × (40 − 6).

  1. Distribute 9 across both terms.
  2. 9 × 40 − 9 × 6 = 360 − 54.
  3. The result is 306.

9 × (40 − 6) = 306.

Example 5 · A valid division split

Find (42 + 18) ÷ 6.

  1. The sum is the dividend, so split it into (42 ÷ 6) + (18 ÷ 6).
  2. 7 + 3 = 10.
  3. Check directly: 60 ÷ 6 = 10.

(42 + 18) ÷ 6 = 10.

Example 6 · Reject a false division rule

Can 72 ÷ (6 + 3) become 72 ÷ 6 + 72 ÷ 3?

  1. First calculate the grouped divisor: 6 + 3 = 9.
  2. 72 ÷ 9 = 8.
  3. The proposed split gives 12 + 24 = 36, which is not 8.

No. Dividing by a sum is not the sum of separate divisions.

Common mistakes

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

Dropping the parentheses after a minus without changing signs.

Why this fails: In 83 − (28 − 7), removing parentheses requires +7, not −7.

Distributing multiplication to only the first term.

Why this fails: 9 × (40 − 6) must become 9 × 40 − 9 × 6.

Splitting division across a grouped divisor.

Why this fails: 72 ÷ (6 + 3) is 8; 72 ÷ 6 + 72 ÷ 3 is 36.

Mixing an exact answer with an unrelated convenient calculation.

Why this fails: A shortcut is valid only when the original and rewritten expressions are equal.

Treating every pair of parentheses in the same way.

Why this fails: The sign or operation immediately outside the parentheses determines which rewrite is safe.

Foundation

Foundation practice

One pair of parentheses. Keep signs or reverse signs for a justified reason.

Remove parentheses

1. 48 + (27 − 9)

Remove parentheses

2. 91 − (31 + 20)

Remove parentheses

3. 83 − (28 − 7)

Remove parentheses

4. 35 + (25 + 15)

Remove parentheses

5. 100 − (46 − 14)

Remove parentheses

6. 60 + (18 − 8)

Remove parentheses

7. 120 − (45 + 15)

Practice

Practice practice

Apply the multiplier to every term, or split a whole sum being divided.

Distribute

8. 6 × (25 + 5)

Distribute

9. 9 × (40 − 6)

Distribute

10. 12 × (30 + 2)

Divide a sum

11. (42 + 18) ÷ 6

Divide a difference

12. (81 − 27) ÷ 9

Distribute

13. 7 × (50 − 4)

Distribute

14. 8 × (125 − 25)

Think

Think practice

Decide whether a rewrite is actually equivalent before rushing into arithmetic.

Grouped divisor

15. 72 ÷ (6 + 3)

Grouped divisor

16. 96 ÷ (12 − 4)

Equivalent rewrite

17. Which expression equals 75 − (30 − 8)?

  1. 75 − 30 − 8
  2. 75 − 30 + 8
  3. 75 + 30 − 8
  4. 75 + 30 + 8
Find the error

18. Is 48 − (20 − 7) equal to 48 − 20 − 7?

Distributive property

19. Which expression equals 5 × (20 + 6)?

  1. 5 × 20 + 6
  2. 5 × 20 + 5 × 6
  3. 5 + 20 × 6
  4. 5 × 20 − 5 × 6
Spot a trap

20. A student writes 6 × (40 − 3) = 6 × 40 − 3. Is that correct?

Valid division split

21. (24 + 36) ÷ 12

Challenge

Challenge practice

Use one pair of parentheses with a missing number, larger factors or a tempting false shortcut.

Missing number

22. 100 − (x + 25) = 45. Find x.

Distribute

23. 15 × (20 − 6)

Grouped divisor

24. 90 ÷ (9 − 4)

Distribute

25. 4 × (50 + 7)

Minus before a difference

26. 250 − (75 − 25)

Distribute

27. 18 × (30 − 5)

Grouped divisor

28. 144 ÷ (12 + 6)

Answer key

  1. 1. 66
    48 + 27 − 9 = 66.
  2. 2. 40
    91 − 31 − 20 = 40.
  3. 3. 62
    83 − 28 + 7 = 62.
  4. 4. 75
    35 + 25 + 15 = 75.
  5. 5. 68
    100 − 46 + 14 = 68.
  6. 6. 70
    60 + 18 − 8 = 70.
  7. 7. 60
    120 − 45 − 15 = 60.
  8. 8. 180
    6 × 25 + 6 × 5 = 150 + 30 = 180.
  9. 9. 306
    9 × 40 − 9 × 6 = 360 − 54 = 306.
  10. 10. 384
    12 × 30 + 12 × 2 = 360 + 24 = 384.
  11. 11. 10
    42 ÷ 6 + 18 ÷ 6 = 7 + 3 = 10.
  12. 12. 6
    81 ÷ 9 − 27 ÷ 9 = 9 − 3 = 6.
  13. 13. 322
    7 × 50 − 7 × 4 = 350 − 28 = 322.
  14. 14. 800
    8 × 125 − 8 × 25 = 1,000 − 200 = 800.
  15. 15. 8
    The divisor is 9, so 72 ÷ 9 = 8. Do not divide by 6 and 3 separately.
  16. 16. 12
    The divisor is 8; 96 ÷ 8 = 12.
  17. 17. 75 − 30 + 8
    Subtracting a difference changes the second inside minus to plus; both expressions equal 53.
  18. 18. No
    The first expression is 48 − 13 = 35; the second is 21. Correct rewrite: 48 − 20 + 7.
  19. 19. 5 × 20 + 5 × 6
    The 5 multiplies both 20 and 6; the common value is 130.
  20. 20. No
    They forgot to multiply the 3 by 6. Correct: 240 − 18 = 222.
  21. 21. 5
    24 ÷ 12 + 36 ÷ 12 = 2 + 3 = 5. Here the entire numerator is divided by 12.
  22. 22. 30
    The grouped amount must be 55 because 100 − 55 = 45. Then x + 25 = 55, so x = 30.
  23. 23. 210
    15 × 20 − 15 × 6 = 300 − 90 = 210.
  24. 24. 18
    9 − 4 = 5 and 90 ÷ 5 = 18. Splitting the divisor would be incorrect.
  25. 25. 228
    4 × 50 + 4 × 7 = 200 + 28 = 228.
  26. 26. 200
    250 − 75 + 25 = 200.
  27. 27. 450
    18 × 30 − 18 × 5 = 540 − 90 = 450.
  28. 28. 8
    12 + 6 = 18, and 144 ÷ 18 = 8.

Related practice

Order of operations and parenthesesMultiplication distribution and trapsMath hub