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Level 10Grade 10skill: percent_multi_step· 9 min read

Multi-Step Percentage Problems

In short: Turn every percentage step into a multiplier (+p% is ×(1+p/100), −p% is ×(1−p/100)). Chain successive changes by MULTIPLYING the multipliers, never by adding the percentages. To find an original value from a changed one, DIVIDE by the multiplier instead of multiplying.

Turn every percentage into a multiplier

Multi-step percentage problems fall apart the moment you try to track them as running totals of "plus this much, minus that much." They become almost mechanical once you stop thinking in changes and start thinking in multipliers.

A rise of p% is the same as multiplying by (1 + p/100). A fall of p% is multiplying by (1 − p/100). So +25% is ×1.25, a 20% discount is ×0.80, adding 8% tax is ×1.08. Once each step is a single number you multiply by, a chain of changes is just a chain of multiplications you can resolve in one line.

This buys you the two things that trip strong students up. Successive changes: multiply the multipliers together (never add the percentages). Reverse problems — where you're told the value after a change and asked for the original — become a division: if new = old × multiplier, then old = new ÷ multiplier.

  1. Write each percentage step as a multiplier: a p% rise is ×(1 + p/100); a p% fall is ×(1 − p/100).
  2. To apply changes in sequence, multiply the multipliers together to get one overall multiplier.
  3. Multiply the starting amount by that overall multiplier to get the final amount.
  4. To go backwards (find the original from a changed value), DIVIDE the known value by the multiplier instead of multiplying.
  5. Read the overall multiplier itself: 1.05 means +5% net, 0.77 means a 23% net fall.

Why successive percentages never just add up Why it works

Rise then fall
A p% rise then a p% fall does NOT return you to the start — you land at ×(1+p/100)(1−p/100), which is always less than 1.

Up 20% then down 20%: 1.20 × 0.80 = 0.96, so you end 4% below where you began — not back to even.

Two discounts
"20% off, then an extra 10% off" is ×0.80 × 0.90 = ×0.72 — a 28% discount, not 30%. The second cut is taken on the already-reduced price.

$150 → 150 × 0.72 = $108. Adding the percentages (30% off) would wrongly give $105.

Order doesn't matter
The ORDER of two percentage changes never changes the final amount: 1.20 × 0.90 = 0.90 × 1.20.

Mark up 20% then discount 10%, or discount 10% then mark up 20% — both give ×1.08, the same final figure.

The one idea to carry away: a percentage always acts on whatever base it currently sees, and after the first step that base has moved. That single fact explains why you multiply rather than add, why "down 20% then up 20%" loses money, and why reversing a change is a division. Track the base, and the traps disappear.

Worked examples

A share is worth $50. In week 1 it rises 10%; in week 2 it falls 30%. What is it worth after the two weeks? = $38.50 (a 23% overall fall, not 20%).
  1. Week 1 multiplier: +10% → ×1.10, so 50 × 1.10 = 55.
  2. Week 2 multiplier: −30% → ×0.70, so 55 × 0.70 = 38.50.
  3. Check with the overall multiplier: 1.10 × 0.70 = 0.77, a 23% net fall. 50 × 0.77 = 38.50.
  4. Note the temptation: +10 − 30 = −20% would give $40 — but the 30% fall acts on $55, not $50.
After a 15% discount, a bag sells for $102. What was the original price? = $120.
  1. A 15% discount is a multiplier of ×0.85, so: original × 0.85 = 102.
  2. This is a reverse problem — divide, don't take a percentage of $102: original = 102 ÷ 0.85.
  3. 102 ÷ 0.85 = 120.
  4. Check: 120 × 0.85 = 102. ✓
A shop buys a phone for $200, marks it up 40%, then in a sale takes 25% off the marked price. What is the sale price, and what is the shop's profit percentage? = Sale price $210; profit 5%.
  1. Marked price: 200 × 1.40 = 280.
  2. Sale price: 280 × 0.75 = 210.
  3. Profit = 210 − 200 = 10, on a cost of 200, so 10 ÷ 200 = 5%.
  4. Shortcut: overall multiplier on cost is 1.40 × 0.75 = 1.05 → a 5% profit, straight away.
A town's population grew 20% one year, then fell 10% the next, and now stands at 5,400. What was the population before those two years? = 5,000 people.
  1. Overall multiplier: 1.20 × 0.90 = 1.08.
  2. Reverse problem: original × 1.08 = 5,400, so original = 5,400 ÷ 1.08.
  3. 5,400 ÷ 1.08 = 5,000.
  4. Check: 5,000 × 1.20 = 6,000; 6,000 × 0.90 = 5,400. ✓
A store advertises "20% off everything, then an extra 10% off at the checkout." What single discount is this equivalent to, and what does a $150 coat cost? = Equivalent to 28% off; the coat costs $108.
  1. First cut ×0.80, second cut ×0.90; combine: 0.80 × 0.90 = 0.72.
  2. A multiplier of 0.72 means you pay 72%, i.e. a 28% discount overall — not 30%.
  3. Coat: 150 × 0.72 = 108.
  4. Adding the percentages (30% off → $105) undercharges by $3, because the 10% is taken on the already-reduced $120.

Four traps that catch strong students

Adding successive percentages instead of chaining them

✗ For +10% then −30%, writing 10 − 30 = −20% and giving 50 × 0.80 = $40.

The second percentage acts on the new amount ($55), not the original $50, so the changes don't add. The real overall multiplier is 1.10 × 0.70 = 0.77.

Fix: Convert each step to a multiplier and MULTIPLY them: 1.10 × 0.70 = 0.77, giving 50 × 0.77 = $38.50.

wrong_operation

Stopping after the first step

✗ For the phone marked up 40% then 25% off, giving the marked price $280 as the answer and forgetting the sale.

You did one percentage step correctly but the question asked for the value after both. The 25% sale still has to be applied to $280.

Fix: Carry the chain to the end: 280 × 0.75 = $210, then read off profit = 10 ÷ 200 = 5%.

partial_computation

Working forward when the problem is a reverse

✗ For "after 15% off it costs $102, find the original," taking 15% of 102 (= 15.30) and adding it to get $117.30.

$102 is the reduced price, so 15% of it isn't the discount that was actually applied — the discount was 15% of the larger original. You must undo the ×0.85, not build on $102.

Fix: Divide by the multiplier: 102 ÷ 0.85 = $120. Check forward: 120 × 0.85 = 102. ✓

reversed_operands

Treating the percent number as a raw amount

✗ Reading "adds 8% sales tax" on a $250 item as "add $8," giving $258.

8% is a proportion of the price, not a fixed $8. On $250 the tax is 250 × 0.08 = $20, so the total is $270.

Fix: Always compute the percentage of the correct base: 250 × 1.08 = $270. The percent number is never the money until you multiply.

percent_as_whole

Practice: multi-step percentages

Work these the multiplier way — chain the steps, and divide to reverse. Watch for the questions where the obvious "add the percentages" answer is exactly the distractor.

Before and after this skill

First, make sure of

Frequently asked questions

Why can't I just add successive percentage changes together?

Because each percentage acts on the amount in front of it, and that amount moves after the first step. A 10% rise then a 30% fall isn't −20%: the 30% is taken on the already-raised value, so the true overall multiplier is 1.10 × 0.70 = 0.77, a 23% fall. Multiply the multipliers; don't add the percentages.

How do I find the original price after a discount or tax has been applied?

Treat it as a reverse problem and divide. If a 15% discount left the price at $102, then original × 0.85 = 102, so original = 102 ÷ 0.85 = $120. The classic mistake is taking 15% of $102 — but that percentage was applied to the larger original, not the reduced price.

If something falls 20% and then rises 20%, am I back where I started?

No. Down 20% then up 20% is 0.80 × 1.20 = 0.96, so you end 4% below the start. Equal-sized rises and falls never cancel, because the rise acts on a smaller base than the fall did.

Does the order of two percentage changes affect the final amount?

No — multiplication can be done in any order, so a 20% markup followed by a 10% discount gives the same final figure as the discount first: 1.20 × 0.90 = 0.90 × 1.20 = 1.08. The order only matters if you need the value at an intermediate stage.

Is "20% off then an extra 10% off" the same as 30% off?

No, it's a 28% discount. The two cuts multiply: 0.80 × 0.90 = 0.72, meaning you pay 72% of the price. On a $150 item that's $108, not the $105 you'd get from a single 30% cut.