# GMAT Reverse Percentages and Successive Changes | topin

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## Translate the sentence into a multiplier equation

GMAC describes current Quant content as arithmetic and elementary algebra. The worked questions below are original teaching examples, not official questions or a prediction of their frequency. They develop one narrow skill within [Problem Solving](/gmat/quant/problem-solving): identifying which amount the percentage uses as its base.

Write final = original × multiplier. A 20% increase uses 1.20; a 20% decrease uses 0.80\. The multiplier is not 20 or 0.20 because the original amount remains part of the final amount. The equation also tells you the direction: recovering the original means dividing, not subtracting an arbitrary percentage of the final value.

Original question: after a 20% discount, a bag costs £96\. What was its original price? Use 96 = 0.8P, so P = 96/0.8 = £120\. A tempting £115.20 comes from adding 20% of £96, but that uses the discounted amount as the base and therefore does not reverse the discount.

Verify by applying the original change: 20% of £120 is £24, and £120 − £24 = £96\. This check is short and tests the actual relationship. If you instead check that your answer is merely greater than £96, you confirm only the direction, not the required percentage.

## Combine changes by multiplying, not adding

Original question: a subscription’s price increases by 25% and then decreases by 20%. What is the overall percentage change? The combined factor is 1.25 × 0.80 = 1.00\. There is no net change. Adding 25 − 20 = 5 produces the wrong result because the second change uses the increased price as its base.

Use a convenient starting value when the original amount is unspecified. Starting at 100 gives 125 after the increase and 100 after the decrease. That is a teaching shortcut, not an assumption that the real price equals £100\. Any positive starting amount produces the same combined percentage change when those percentage factors apply.

Original question: revenue falls by 10% and then rises by 10%. The factor is 0.90 × 1.10 = 0.99, so the final revenue is 1% below the original. The equal numerical percentages do not cancel. A rise needs a larger percentage than the preceding fall to recover the initial value.

__Original multiplier calculations__
| Changes         | Combined factor    | Net result   |
| --------------- | ------------------ | ------------ |
| +25%, then −20% | 1.25 × 0.80 = 1    | No change    |
| −10%, then +10% | 0.90 × 1.10 = 0.99 | 1% decrease  |
| +20%, then +25% | 1.20 × 1.25 = 1.50 | 50% increase |
| −20%, then −25% | 0.80 × 0.75 = 0.60 | 40% decrease |

## Find the recovery percentage after a fall

Suppose a fund’s value falls from 200 to 150 in an invented problem. The fall is 50/200 = 25%. To return from 150 to 200, the required increase is 50/150 = one third, or 33⅓%. The lost amount is the same, but the recovery uses a smaller base.

In multiplier form, a fall of fraction d leaves 1 − d of the original. The recovery multiplier is 1/(1 − d). The required increase as a fraction is d/(1 − d). Use the formula when helpful, but derive it from final = original × factor if memorised symbols make you lose track of the base.

Original question: sales after a 40% fall are 360 units. How many units were sold before the fall, and what percentage rise restores that volume? Original = 360/0.60 = 600\. Recovery = 240/360 = two thirds, or 66⅔%. Check that 360 × 5/3 = 600.

This calculation describes a hypothetical volume change, not investment advice or a forecast. In exam practice, keep the unit attached to each quantity. If the question asks only for the original units, stop at 600; if it asks for recovery growth, use the final reduced volume as the denominator.

## Distinguish markup from margin using the same profit amount

Original question: a shop buys an item for £80 and sells it for £100\. Profit is £20\. Markup on cost is 20/80 = 25%; profit margin on selling price is 20/100 = 20%. Both results are correct for their respective bases, so the wording determines which answer you need.

If the prompt defines margin differently, use its definition. Do not let a familiar business term override an explicit exam condition. Writing “profit/cost” or “profit/selling price” before arithmetic is more reliable than choosing an answer based on recognising a round percentage.

Original question: an item sells for £150 with a profit margin of 20% of selling price. Profit is £30, so cost is £120\. Its markup on cost is £30/£120 = 25%. An answer of £125 for cost would come from dividing by 1.20 as though the 20% were a markup; it is not.

For reverse problems involving a discount after markup, combine both relationships. A cost of C marked up by 50% and then discounted by 20% sells for 1.5 × 0.8C = 1.2C. If the final price is £96, cost is £80\. The price and cost have different roles, even though both appear as money.

Original combined-cost check: an item costs £50, is marked up 40%, and then receives a 10% discount from the marked price. The selling price is 50 × 1.4 × 0.9 = £63\. Profit is £13, giving a 26% markup relative to the original cost. Adding 40 − 10 would predict 30% and overlook the second base.

If the final selling price is given as £63 instead, recover cost with 63/(1.4 × 0.9) = £50\. This shows why the combined multiplier is useful in both directions. Before rounding, simplify the factors or fractions where possible; an early rounded intermediate can make nearby choices harder to distinguish.

## Separate percentage points, relative change and total volume

A conversion rate rising from 20% to 25% increases by five percentage points. Its relative increase is 5/20 = 25%. Those are different descriptions of the same change. If an answer asks for percentage growth, entering five without checking the requested measure can be wrong.

Original question: a campaign has 200 visitors and a 20% conversion rate, then 300 visitors and a 25% conversion rate. Purchases rise from 40 to 75, an increase of 35/40 = 87.5%. The conversion-rate change alone does not describe the total purchase change because the visitor count also changed.

Write the total as population × rate. The visitors rose by a factor of 1.5 and the conversion rate by a factor of 1.25, so purchases rose by 1.875\. This offers a second check on 87.5%. It also connects the method to [weighted-average and table questions](/articles/gmat-table-analysis-weighted-averages), where group sizes matter.

Do not infer total changes from a rate when the denominator is unknown. A rising success rate can coexist with fewer successes if fewer candidates were tested. In a calculation problem, use the given denominator; in a reasoning problem, recognise when the data do not determine it.

## Choose a method and verify it on fresh problems

Use multipliers when the task reverses a change, combines several changes or mixes an unknown base with a final amount. Use an easy starting value when only a percentage relationship is needed. Use fractions when cancellation is cleaner: 25% is one quarter, so a 25% rise is multiplication by 5/4.

Try this original mini-drill before reading the answers. A price after a 30% reduction is £84; find the original. A quantity rises 20% then falls 30%; find the net change. A 16% rate rises to 20%; find both percentage points and relative growth. The respective answers are £120, a 16% decrease, and four points with 25% relative growth.

For every wrong answer, record the base you used and the base required. That note is more useful than “percentage mistake”. Then change the figures and solve another example without looking at the method. Our [ten-week working-professional plan](/articles/gmat-study-plan-working-professionals) shows where these repair sessions fit between timed practice.

topin’s [free full-length GMAT mock](/gmat/practice-test), marked on the official scale, can later check whether the method transfers under timing. Do not claim readiness solely because these examples now look familiar. The useful evidence is choosing the correct base and operation on an unfamiliar problem while keeping the section moving.

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## FAQs

How do I reverse a percentage decrease?

Divide the final amount by one minus the decrease expressed as a fraction. After a 20% reduction, divide by 0.8.

Do a 20% increase and 20% decrease cancel?

No. Their combined factor is 1.2 × 0.8 = 0.96, a 4% decrease.

What increase reverses a 25% fall?

A 33⅓% increase on the reduced amount. The recovery uses the smaller final base.

Is a rise from 20% to 25% a 5% increase?

It is five percentage points and a 25% relative increase. Use the measure the question asks for.

How are profit margin and markup different?

In the examples here, markup divides profit by cost; margin divides profit by selling price. Follow any explicit definition in the prompt.

## Sources (checked 5 October 2026)

* [mba.com: arithmetic and algebra in current Quant (checked 5 October 2026)](https://www.mba.com/exams/gmat-exam/about/exam-content)
* [mba.com: calculator and note-taking policies (checked 5 October 2026)](https://support.mba.com/hc/en-us/articles/14076264362907-GMAT-Calculator-and-Scratch-Paper-Note-Taking-Policies)

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