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Lesson 4 of 7

Arithmetic Expressions · Lesson 4 of 7

Removing Brackets and Tinkering with Terms

“Understand why signs change when a whole quantity is subtracted, then use the reasoning for comparisons and mental subtraction.”

Learning Objectives

• Remove brackets after addition while preserving the inner signs. • Explain subtracting a sum as subtracting each part. • Explain subtracting a difference using subtraction and restoration. • Predict how changes in signed terms affect sums and differences. • Use compensation and bracket reasoning for mental subtraction.

Removing Brackets

Brackets tell us what is being treated as a whole, but a different expression can sometimes describe the same action without brackets. The aim is to preserve the value and meaning. In this lesson, we consider adding or subtracting a bracketed sum or difference; multiplying a bracket needs the distributive reasoning of the next lesson.

Example — Subtract a combined amount

Problem
Rewrite 200 − (40 + 3) without brackets and explain the result.

  1. 1.The bracketed amount is 43. Removing 43 can be done by removing 40 and then another 3.
  2. 2.Thus 200 − (40 + 3) = 200 − 40 − 3.
  3. 3.Evaluate either route: 200 − 43 = 157, or 160 − 3 = 157. The two subtractions account for all of the amount removed.
Example — Two purchases, two subtractions

Problem
Rewrite Irfan’s change, 100 − (15 + 56), without brackets.

  1. 1.The total cost contains both ₹15 and ₹56. Both must be removed from ₹100.
  2. 2.Write 100 − 15 − 56. First subtract 15 to get 85; then subtract 56 to get 29.
  3. 3.This is the same ₹29 change as 100 − 71. The expression 100 − 15 + 56 would restore ₹56 instead of removing it.

Subtracting a Difference

Now suppose the quantity to be removed has already been reduced. Subtracting its larger first part alone removes too much. Add back the reduction to reach the intended result; this gives a reason for the change of sign rather than a rule to memorise without meaning.

Example — Subtract and restore

Problem
Explain 500 − (250 − 100) without brackets.

  1. 1.The bracket has value 150. The intended action is therefore to remove 150 from 500.
  2. 2.If you first remove 250, you have removed 100 too much. Restore that 100 by adding it back.
  3. 3.So 500 − (250 − 100) = 500 − 250 + 100 = 350. The direct calculation 500 − 150 also gives 350.
Subtract too much, then restore the excessRemove the reduced amount 250 − 100 = 150.Start: 500Left after −250250 removed250+100= 350 left
Subtract too much, then restore the excess— The restored 100 corrects the temporary subtraction of 250. The actual amount removed is only 150.

Adding a Bracketed Quantity

When a quantity is added, its internal gains and losses are carried into the total unchanged. You can perform those gains and losses after adding its starting amount. This explains why adding a bracket leaves its inner signs unchanged.

Example — Hira’s collection

Problem
Hira has 28 rare coins in one bag and 35 in another. She gives away 10 coins from the second bag. Rewrite her total 28 + (35 − 10).

  1. 1.The second bag has 35 − 10 = 25 coins left.
  2. 2.Add that remaining bag to the first: 28 + 25 = 53 coins.
  3. 3.Alternatively, combine both original bags and then remove the gift: 28 + 35 − 10 = 53. Thus 28 + (35 − 10) = 28 + 35 − 10.

Use the following relationships after connecting them to these actions. The letters a, b and c stand for numbers. A plus outside the bracket preserves its signed terms; a minus outside adds their opposites, so every inner sign reverses. This includes negative terms: the opposite of −12 is +12.

Add a sum or differenceLaTeX
The inner plus or minus remains unchanged because the grouped quantity is being added.
Subtract a sum or differenceLaTeX
Every signed term inside is replaced by its additive inverse because the whole grouped quantity is subtracted.
ExpressionEquivalent expression without bracketsValue
14 + (12 + 10)14 + 12 + 1036
14 − (12 + 10)14 − 12 − 10−8
14 + (12 − 10)14 + 12 − 1016
14 − (12 − 10)14 − 12 + 1012
−14 + (12 − 10)−14 + 12 − 10−12
14 − (−12 − 10)14 + 12 + 1036
Subtraction does not permit arbitrary regrouping

16 − (8 − 3) is 11, but (16 − 8) − 3 is 5. Changing brackets around subtractions can change which quantities are removed. Rewrite as signed addition or reason about the whole removed quantity before changing the grouping.

Tinker the Terms I

You can often predict a new value from an old one by looking at how individual signed terms change. A term becomes greater when it moves toward the positive direction, even if it is still negative. Thus −15 is one greater than −16; this observation is central to comparing expressions containing negatives.

Use 53 + (−16) = 37 as a starting point. Increasing 53 to 54 raises the sum to 38. Increasing the second term from −16 to −15 also raises the sum to 38. For a difference such as 53 − 16, decreasing 53 lowers the result, while increasing the amount subtracted also lowers it. Think about the term actually added, not only the unsigned number printed after a subtraction sign.

FamilyExpressionValue and reasoning
Sum53 + (−16)37: starting value
Sum54 + (−16)38: first term rises by 1
Sum53 + (−15)38: second signed term rises by 1
Difference53 − 1637: starting value
Difference52 − 1636: starting quantity falls by 1
Difference53 − 1736: one more is removed
Negative terms−87 − 16−103: starting value
Negative terms−88 − 15−103: first term −1, second signed term +1
Negative terms−86 − 18−104: first term +1, second signed term −2
Negative terms−97 − 26−123: both signed terms fall by 10

Compensation and Mental Subtraction

A sum stays unchanged if one term increases by an amount while another decreases by the same amount. A difference stays unchanged if both its starting quantity and removed quantity change by the same amount. These compensation patterns let you replace awkward numbers with convenient ones and then account for the replacement.

Example — Subtract nine efficiently

Problem
Jasoda calculates 36 − 9 by first subtracting 10. Explain how to finish and adapt the method to subtract 8.

  1. 1.Subtracting 10 gives 26, but it removes one more than the required 9. Restore 1: 36 − 9 = 36 − 10 + 1 = 27.
  2. 2.To subtract 8, removing 10 removes 2 too many. Restore 2: 36 − 8 = 36 − 10 + 2 = 28.
  3. 3.The correction is determined by the extra amount removed. For 62 − 19, subtract 20 and restore 1, giving 43.

Jasoda’s strategy works for every starting number, not only 36. The number 9 is always 10 − 1, so subtracting 9 is always subtracting that difference: a − (10 − 1) = a − 10 + 1. The same reasoning works even when the result is negative. For 6 − 9, subtract 10 to get −4, then restore 1 to obtain −3.

Bracket placement is also a way to express compensation clearly. In 73 − (14 − 1), the amount removed is 13, so this is 73 − 14 + 1 = 60. In 73 − (14 + 1), the amount removed is 15, so this is 73 − 14 − 1 = 58. The two expressions differ because the bracketed quantities differ by 2.

Check Your Understanding

Try each question before opening the explanations. A useful answer includes the reason for your choice, not just its letter.

Quiz

Quick check

Which is equivalent to 80 − (23 + 7)?

Quick check

What is 90 − (40 − 6)?

Quick check

Which loses its brackets without changing the inner minus sign?

Quick check

Starting from −30 − 8 = −38, what is −31 − 7?

Quick check

Which method correctly calculates 54 − 19?

The combined 30 is removed, so both parts must be subtracted.

Work these problems on paper and explain any rewrite you use. Open the matching solution after you have made a first attempt; more than one valid expression or method may be possible.

Practice Problems

Practice Problems
  1. Rewrite without brackets, supplying the correct signs: 24 + (6 − 4); 38 + (9 − 4); 24 − (6 + 4); 24 − 6 − 4; 27 − (8 + 3); 27 − (8 − 3).
  2. Without brackets, rewrite and evaluate all six expressions in the bracket-removal table above.
  3. Compare each pair: (6 + 10) − 2 and 6 + (10 − 2); 16 − (8 − 3) and (16 − 8) − 3; 27 − (18 + 4) and 27 + (−18 − 4).
  4. Which of 319 + 537, 319 − 537, −537 + 319 and 537 − 319 have the same value?
  5. Group equal expressions: 87 + 46 − 109 (listed three times); 87 − 46 + 109; (87 − 46) + 109; 87 − (46 + 109).
  6. Insert brackets to make: 34 − 9 + 12 = 13; 56 − 14 − 8 = 34; −22 − 12 + 10 + 22 = −22.
  7. Find one pair of numbers for 423 + □ = 419 + □. Then complete 207 − 68 = 210 − □.
  8. Using 2, 3 and 5 once each, only +, − and brackets, make five different values.
  9. Use Jasoda’s method to find 74 − 9, 74 − 8 and 74 − 19.
  10. Match 73 − 14 + 1 and 73 − 14 − 1 with 73 − (14 − 1), 73 − (14 + 1), 73 + (−14 + 1) and 73 + (−14 − 1).

In order: 24 + 6 − 4 = 26; 38 + 9 − 4 = 43; 24 − 6 − 4 = 14; 24 − 6 − 4 = 14; 27 − 8 − 3 = 16; 27 − 8 + 3 = 22. In the already unbracketed fourth expression, both minus signs remain.

Key Takeaways

Key Takeaways

• Adding a bracketed quantity preserves its inner signed terms. • Subtracting a bracketed quantity adds the opposite of each inner term. • Subtracting a difference can be understood as removing too much and restoring the excess. • Subtraction cannot be regrouped freely like addition. • Compensating changes can preserve a sum or difference. • Subtract a convenient nearby amount and restore the excess for efficient mental subtraction.

Previous · Lesson 3

Swapping, Grouping, and Evaluating Expressions

Next · Lesson 5

The Distributive Property and Smarter Multiplication