# SAT Reverse Percentages and Successive Changes | topin

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## Name the base amount before writing a percentage

College Board’s [Problem-Solving and Data Analysis page](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/problem-solving) includes percentages and proportional relationships. The key mathematical distinction is between a fraction of a base and a change relative to that base. For p% of B, the amount is (p/100)B. For a p% increase from B, the new amount is (1 + p/100)B. A decrease uses 1 − p/100.

Our suggested first annotation is “percentage of what?” Write that quantity next to the percentage. If a question says a price decreased by 20%, the old price is the change base. If it says the new price is 80% of the old price, the same base appears through another wording. Recognising the relationship avoids treating every percentage in a paragraph as a standalone number to add or subtract.

Choose a simple hypothetical base such as 100 to inspect wording when useful. If a quantity rises by 30%, 100 becomes 130\. If it rises to 30% of the original, 100 becomes 30\. The words “by” and “to” create different models. The base-of-100 check helps interpret the statement, but solve using the actual quantities when the question gives them.

## Original example: recover a price before a discount

A jacket costs £72 after a 20% discount. What was its original price? Let P be the original. The remaining fraction is 80%, so 0.80P = 72\. Dividing gives P = 72/0.80 = 90\. Check forward: 20% of £90 is £18, and £90 − £18 = £72\. The forward check uses the original base, exactly as the discount statement requires.

Adding 20% of £72 produces £86.40, which is too low. That operation applies the percentage to the new price rather than the old one. The issue is not arithmetic accuracy; it is choosing the wrong base. A reverse percentage reconstructs the base from the stated final relationship. It does not undo the change by applying the same percentage in the opposite direction.

If the original cost is £90 and the new cost £72, the increase required to return from £72 to £90 is £18/£72 = 25%. Thus a 20% decrease requires a 25% increase to reverse. The two percentages use different starting amounts. This original calculation explains why “up 20%, down 20%” is not a general cancellation rule and why opposite signs alone cannot determine the result.

## Original example: chain changes using multiplication

A subscription price rises by 15% and later falls by 10%. What percentage of its original price remains? The multipliers are 1.15 and 0.90, so the final multiplier is 1.035\. The price is 103.5% of the original, a net 3.5% increase. A hypothetical £200 price becomes £230, then £207, which confirms the calculation directly.

Adding +15% and −10% gives +5%, but it ignores that the second change uses the increased price. The product includes that changed base. Keep the multipliers visible until the final step, then interpret the result as a percentage of the original or a percentage change, depending on the question. Those two requested quantities differ by 100 percentage points in this example.

If the same final price £207 is given and the original is unknown, divide by 1.035 to recover £200\. If only the second step needs undoing, divide by 0.90 to recover £230\. Identify which stage the question asks about before calculating. A long story can contain several valid amounts, but only one is the target. Use a labelled timeline: original, after increase, after decrease.

## Original example: reverse a relative comparison

Quantity A is 40% greater than quantity B. By what percentage is B less than A? The first statement gives A = 1.40B. The difference is 0.40B, but the second question uses A as its base. Therefore the required fraction is 0.40B/1.40B = 2/7, about 28.57%. A hypothetical B = 50 gives A = 70 and a difference of 20; 20/70 confirms the fraction.

The answer is not 40%, because reversing the comparison changes the denominator. It is also not 60%, which would describe a different fraction-of-base relationship. Use the sentence to establish numerator and denominator: “B is less than A” means the difference divided by A. Keep the exact fraction until you know whether the question asks an exact value or a rounded percentage.

Now let A be 25% less than B. Then A = 0.75B, and B/A = 4/3\. B is therefore 33 1/3% greater than A. The two original examples share the same method: write the stated relationship, identify the new comparison base and calculate the difference relative to it. You do not need a separate memorised conversion formula for every percentage.

## Original example: percentage points are not percentage change

In a fictional survey, a preference rises from 30% to 42%. The difference is 12 percentage points. The relative percentage increase is (42 − 30)/30 = 0.40, or 40%. These quantities answer different questions. The first compares the reported percentages directly; the second measures the change relative to the original percentage value. State which one is being requested.

If the survey sizes differ, the number of respondents with that preference may not move in the same way. Thirty per cent of 200 is 60, while 42% of 100 is 42\. The preference percentage rises while the raw count falls. This original example shows why denominators and totals belong in a data interpretation. A larger percentage does not automatically mean more people when group sizes change.

For a probability or two-way-table question, identify the relevant group before calculating the fraction. The percentage of all students and the percentage of students in one category use different totals. Use the [word-problem and units guide](/articles/sat-math-word-problems-units) for modelling counts, and the [statistics study-design guide](/articles/sat-statistics-sampling-causation) when the question asks what a survey can establish rather than only its arithmetic.

## Original example: interpret repeated multipliers and exponents

A fictional population starts at 800 and grows by 5% each year in a model. After t years, the model is P = 800(1.05)^t. The 800 is the initial amount; 1.05 is the annual multiplier. At t = 2, the amount is 800 × 1.1025 = 882\. The gain is 82, which is 10.25% of the original rather than exactly 10%.

If the question asks when the model reaches a given amount, you may compare the exponential expression using an appropriate calculation or graph. But identifying the growth rate does not require solving for time: 1.05 means 5% growth each one-year step. If the exponent is t/3, the same multiplier applies every three units of t, not every one. Read the time unit and exponent together.

College Board’s [Advanced Math content](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/advanced) includes exponential functions. The original population is a mathematical model, not a real-world prediction or a claim that populations always grow by a fixed percentage. Keep model assumptions separate from interpretation. A word problem can ask what a coefficient means without asking whether its assumed growth law is true outside the stated scenario.

## Use a compact decision table and verify forwards

__Original percentage method map__
| Question wording               | Operation                                              | Useful check                         |
| ------------------------------ | ------------------------------------------------------ | ------------------------------------ |
| Original before p% discount    | Final / (1 − p/100)                                    | Apply discount to recovered original |
| Original before p% increase    | Final / (1 + p/100)                                    | Apply increase forwards              |
| Several successive changes     | Multiply the change factors                            | Use a simple starting value          |
| Relative comparison reversed   | Difference / new comparison base                       | Label the denominator                |
| Change in reported percentages | Subtract for points; divide by old for relative change | Name the requested measure           |
| Repeated fixed-rate growth     | Initial × multiplier^steps                             | Check exponent time unit             |

Practise by changing the requested stage or base while keeping the same story. For the jacket example, ask for discount amount, original price and reverse increase in turn. Explain why each uses a different expression. Then use fresh numbers so the answer cannot come from remembering £90\. Our goal is a stable interpretation habit before adding time pressure, not a large collection of copied solutions.

[topin’s free SAT mock](/sat/practice-test), marked on the official scale, can test this habit under mixed conditions. Use the [error-log guide](/articles/sat-error-log-score-plateau) to record wrong base, wrong multiplier, wrong stage or wrong requested measure. If the same category returns, repair that step directly. A final forward calculation is a quick way to detect many reverse-percentage errors without rebuilding the whole solution.

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## FAQs

How do I find the original before a 20% discount?

Divide the final amount by 0.80, then check by applying the discount forwards.

Why can I not add the discount percentage back?

The final amount is a different base. Adding the same percentage of it does not reconstruct the original.

Do successive percentage changes add?

Multiply their factors. Addition generally ignores that the base changes between steps.

What is the difference between percentage points and percentage increase?

Percentage points are the direct difference between percentages. Relative increase divides that difference by the starting percentage.

Does 1.05 in a growth model mean a 105% increase?

No. It means the new amount is 105% of the previous amount, a 5% increase per stated step.

## Sources (checked 5 October 2026)

* [College Board: Problem-Solving and Data Analysis content (checked 5 October 2026)](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/problem-solving)
* [College Board: Advanced Math content (checked 5 October 2026)](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/advanced)
* [College Board: using the Student Question Bank (checked 5 October 2026)](https://satsuite.collegeboard.org/practice/student-question-bank)

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