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Level 8Grade 8skill: integer_ops· 9 min read

Integer Operations: Negatives

In short: Rewrite every subtraction as adding the opposite, do × and ÷ before + and − (a leading minus waits its turn), count negative factors to fix the sign, and always take a percentage of the amount actually in front of you.

How to evaluate integer expressions without dropping a sign

Integer questions at this level rarely test whether you can compute 3 × 4. They test whether a minus sign survives a five-step expression intact. Almost every mark you lose here is a sign that wandered off or an operation done in the wrong order, not an arithmetic slip.

So do the boring, reliable thing first: turn every subtraction into an addition of the opposite. Once a − b is written as a + (−b), and a − (−b) as a + b, there are no more double negatives to trip over — just a list of signed numbers to combine in order, taking every × and ÷ before any + and −.

  1. Clear brackets first, working from the innermost pair outward.
  2. Rewrite each subtraction as adding the opposite: a − (−b) becomes a + b, a − b becomes a + (−b).
  3. Do every × and ÷ next, strictly left to right — they bind tighter than + and −, so a leading minus has to wait its turn.
  4. In each product or quotient, count the negative factors: an even count gives a positive result, an odd count gives a negative one.
  5. Finish with + and − from left to right.
  6. For any percentage, turn it into a multiplier and apply it to the amount actually in front of you at that step — never assume the original amount is the base.

The one idea that kills the traps: a minus sign is an instruction, not glue Why it works

a − (−b)
Two negatives in a row collapse to a plus. Subtracting a debt makes you richer.

7 − (−2) = 9, not 5.

− before ×
A leading minus is a sign, not a subtraction that fires first. × and ÷ always run before + and −.

−3 − 4 × 2 = −3 − 8 = −11, not −3 − 4 = −7 then −7 × 2 = −14.

count the signs
In a product or quotient, an even number of negatives is positive; an odd number is negative.

(−1)(−2)(−3) = −6 — three negatives, so negative.

+p% then −p%
Successive percentages multiply, they never add. Up then down by the same rate always ends below where you started.

×1.2 then ×0.8 = ×0.96 — a 4% net fall, not break-even.

right base
Take each percentage of the amount showing at that step, not the original price.

5% sales tax after a 20% discount is 5% of the discounted price.

Here is the insight the strong questions are built around: the minus sign that opens an expression is the lowest-priority operation in the room. Students read left to right and feel that the minus "happens first" because it comes first. It doesn't. Multiplication and division jump the queue, and a same-rate rise-then-fall never returns you to the start, because the fall is taken on the bigger number. Master those two facts and the double-negative and precedence traps stop being traps.

Worked examples

Evaluate −3 − (−4) × (−2). = −11. (Reading left to right gives the tempting −2 — that ignores that × comes first.)
  1. Multiplication first: (−4) × (−2) = 8 (two negatives → positive).
  2. Now −3 − 8.
  3. Add the opposite: −3 + (−8) = −11.
Evaluate 7 − (5 − 8) − (−2). = 12. (Miss the double negatives and you get 2.)
  1. Innermost bracket: 5 − 8 = −3.
  2. Rewrite: 7 − (−3) − (−2) = 7 + 3 + 2.
  3. Add: 12.
Evaluate −20 + 48 ÷ (−6) × 2. = −36.
  1. ÷ and × left to right, not × before ÷: 48 ÷ (−6) = −8.
  2. Then −8 × 2 = −16.
  3. Finally −20 + (−16) = −36.
A jacket is priced $2000. The shop takes 20% off, then adds 5% sales tax on the discounted price. What do you pay? = $1680. (Charging sales tax on the original 2000 gives $1700; forgetting sales tax leaves $1600.)
  1. Discount: 20% of 2000 = 400, so the price drops to 1600.
  2. sales tax is charged on the amount now showing: 5% of 1600 = 80.
  3. Pay 1600 + 80 = 1680.
A share is worth $500. It rises 20% one day and falls 20% the next. What is it worth now? = $480 — not $500. Equal-rate up-then-down always loses.
  1. Rise: 500 × 1.2 = 600.
  2. Fall: 600 × 0.8 = 480.
  3. Net multiplier is 1.2 × 0.8 = 0.96, a 4% loss.

Six mistakes that cost real marks

Keeping the minus on a product of two negatives

✗ (−6) × (−5) = −30

You multiplied the digits correctly but carried a negative sign that should have cancelled. Two negatives multiplied give a positive.

Fix: Count the negative factors before you write the sign. Even count → positive, odd → negative. Two negatives is even, so (−6) × (−5) = 30.

sign_error

Adding two negatives when the expression says multiply

✗ (−4) × (−2) = −6

You treated the × as a +, adding −4 and −2 to get −6. The operation, not just the sign, was wrong.

Fix: Multiply first, then decide the sign: 4 × 2 = 8, two negatives → positive, so (−4) × (−2) = 8.

wrong_operation

Stopping after the discount and forgetting the tax

✗ $1600

You found the discounted price of the $2000 jacket and stopped. That is a correct intermediate value, but the question asked for the final price including sales tax.

Fix: Read to the end. After the $1600 discounted price, add 5% sales tax ($80) to reach $1680.

partial_computation

Charging the tax on the original price

✗ $1700

You took 5% sales tax of the original $2000 ($100) instead of the discounted $1600. The percentage was applied to the wrong base.

Fix: Each percentage acts on the amount currently showing. After discounting to $1600, sales tax is 5% of 1600 = $80, giving $1680.

base_confusion

Using the percent number as if it were dollars

✗ $30 off a $50 book (pay $20)

For a 30% discount you subtracted 30 directly instead of computing 30% of 50. The percent number is not the amount.

Fix: Convert first: 30% of 50 = 0.30 × 50 = $15 off, so you pay $35.

percent_as_whole

Working left to right through a mixed expression

✗ −3 − (−4) × (−2) = −2

You computed −3 − (−4) = 1 first, then × (−2) = −2, evaluating in reading order and ignoring that × outranks −.

Fix: Do × and ÷ before + and −: (−4) × (−2) = 8, then −3 − 8 = −11.

order_of_ops_lr

Practice

Try a mixed set of signed-number and percentage problems built to spring the double-negative, precedence, and wrong-base traps. Work each one to the final value, then check which distractor you would have picked in a hurry.

Related skills

First, make sure of

Frequently asked questions

Why does −3 − 4 × 2 equal −11 and not −14?

Multiplication runs before subtraction, so 4 × 2 = 8 first, then −3 − 8 = −11. You only get −14 if you wrongly do −3 − 4 = −7 first and then −7 × 2, evaluating left to right and ignoring that × outranks −.

Do a 20% rise and a 20% fall cancel out?

No. Percentages multiply, not add: ×1.2 then ×0.8 = ×0.96, a 4% net fall. The fall is taken on the larger amount, so $500 becomes $600 then $480, never back to $500.

When a shop gives a discount and then charges sales tax, what is the tax based on?

The discounted price. Apply the discount first, then take sales tax of the amount that remains. On a $2000 jacket with 20% off then 5% sales tax, sales tax is 5% of $1600 = $80, giving $1680 — not 5% of the original $2000.

How do I stop losing signs in long expressions?

Turn every subtraction into adding the opposite, so a − (−b) becomes a + b. Then in each product count the negative factors: an even count is positive, an odd count is negative. That removes both double-negative and sign-of-product mistakes.

Is 48 ÷ (−6) × 2 the same as 48 ÷ (−12)?

No. Division and multiplication have equal priority and run left to right, so 48 ÷ (−6) = −8 first, then −8 × 2 = −16. Doing (−6) × 2 first is a precedence error and gives the wrong value.