# GRE Percentages and Ratios: Worked Problems | topin

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## Name the base before choosing an operation

[ETS](https://www.ets.org/gre/test-takers/general-test/prepare/content/quantitative-reasoning.html) includes percent and ratio in the arithmetic content of GRE Quant, and its [Math Review](https://www.ets.org/content/dam/ets-org/pdfs/gre/gre-math-review.pdf) covers the foundations. Percentages turn up in every Quant format: as a bar or table change in [Data Interpretation](/gre/quant/data-interpretation), as a box with a % sign in Numeric Entry, and as two expressions to compare in Quantitative Comparison. Our format pages each work one percent example; this page is about the decision those examples share. Write “part / whole” or “change / original” as your first line of scratch work. The [GRE Quant section guide](/gre/quant) sets out all five question formats.

Consider the original statement “36 of the 90 applications were complete”. The proportion complete is 36/90 = 0.4, so 40% were complete. The same 36 applications are 66⅔% of the 54 incomplete applications. Both calculations are correct, but they answer different questions. Underline the words after “of” or “than”, because they usually identify the comparison base.

A ratio such as complete:incomplete = 2:3 compares two parts. It does not mean that the complete fraction is 2/3 of all applications. The total has five ratio parts, so the complete fraction is 2/5\. Write the labels above the numbers and add the ratio parts only when they exhaust the total. A third category would make that shortcut invalid.

## Translate the wording into one labelled equation

Our recommended decision table separates phrases that look similar but require different denominators. Read the direction of the comparison aloud during practice, then stop doing so when working under test conditions. “A exceeds B by 20%” means the extra amount is one fifth of B, so A = 1.2B. It does not mean B is 20% below A.

When the original total is unspecified and only percentages matter, choosing 100 can make the model transparent. When actual counts or divisibility restrictions are supplied, use the stated total instead. If 37 people are involved, a fictional total of 100 must not replace that integer constraint. Convenient numbers are a way to expose proportional structure, not permission to change the problem.

__Choose the equation from the exact comparison__
| Wording                    | Equation or interpretation | Check                               |
| -------------------------- | -------------------------- | ----------------------------------- |
| p% of B                    | (p/100) × B                | B is the base                       |
| A is p% greater than B     | A = (1 + p/100)B           | Compare the increase with B         |
| A is p% less than B        | A = (1 − p/100)B           | Compare the decrease with B         |
| A:B = m:n                  | A = mk; B = nk             | Total is (m + n)k if only two parts |
| Percent change from B to A | 100(A − B)/B               | Original value is the denominator   |

## Original problem: reverse a percentage comparison

Original question: a laboratory processed 150 samples on Monday and 180 on Tuesday. By what percentage did the number increase? By what percentage was Monday’s number less than Tuesday’s? These two questions describe the same difference of 30 samples, but they deliberately change the reference quantity. Solve both before assuming that opposite wording gives opposite signs with the same magnitude.

For the increase from Monday, use 30/150 × 100 = 20%. For Monday relative to Tuesday, use 30/180 × 100 = 16⅔%. Check the language with equations: 180 = 1.2 × 150, while 150 = (5/6) × 180\. A useful error-log entry is “reversed comparison without changing denominator”, because that tells you exactly what to inspect next time.

Now suppose a cost is 84 after a 30% reduction. If the original cost is C, the remaining amount is 0.7C, so C = 84/0.7 = 120\. Adding 30% to 84 gives 109.2 and fails the forward check. Recovery after a decrease requires dividing by the remaining multiplier. Always substitute the recovered original into the stated forward operation.

## Original problem: separate successive changes from points

Original question: a quantity of 250 rises by 12%, then falls by 20%. The first change produces 250 × 1.12 = 280; the second produces 280 × 0.8 = 224\. Relative to 250, the difference is −26, a 10.4% decrease. The net multiplier is 1.12 × 0.8 = 0.896, which explains the answer without treating the changes as additive.

For equal opposite percentage changes of p%, the multiplier is (1 + p/100)(1 − p/100) = 1 − (p/100)^2\. A 10% rise followed by a 10% fall therefore leaves 99% of the original quantity. The second operation applies to the changed value. This identity is a derived shortcut; understanding the two bases is more useful than memorising it in isolation.

Percentage points measure a difference between percentage figures. If completion rises from 40% to 50%, the increase is ten percentage points but 25% relative to the original completion proportion, because 10/40 = 0.25\. If the underlying populations also change, a higher completion percentage need not mean more completed applications. Keep percentage, percentage points and actual counts in separate labelled columns.

## Original problem: a ratio changes when people join

Original question: the ratio of research students to taught students in a group is 3:5\. After six research students join and two taught students leave, the ratio is 3:4\. How many students were originally in the group? Let the initial counts be 3k and 5k. The new counts are 3k + 6 and 5k − 2, giving (3k + 6)/(5k − 2) = 3/4.

Cross-multiplying gives 12k + 24 = 15k − 6, so k = 10\. The initial counts were 30 and 50, with total 80\. The final counts are 36 and 48, which reduce to 3:4\. Notice that six was added to an actual count, not to the ratio number 3\. Ratio parts become counts only after multiplying by the common scale factor.

If the question supplies an initial total instead, divide it by the sum of the ratio parts first. For a 4:7 split of 66 people, one part is six people. If the total does not give integer category counts, recheck the assumptions or whether the categories represent people. A ratio of measured lengths can legitimately use noninteger values, while headcounts require integers.

## Original problem: combine group proportions correctly

Original question: 40% of 80 morning applicants and 25% of 120 afternoon applicants passed a screening exercise. What percentage of all applicants passed? The pass counts are 32 and 30, so 62 of 200 passed, or 31%. Averaging 40% and 25% gives 32.5%, which wrongly grants the smaller and larger groups equal influence.

This is the same total-over-total structure used in [weighted averages and mixtures](/articles/gre-weighted-averages-mixtures). Work with pass counts if the percentages and totals are easy. If a question gives the group-size ratio instead, that ratio can serve as the weight: a 2:3 split gives (2 × 40 + 3 × 25)/5 = 31\. The numerical total is unnecessary when the relative weights are known.

For [Numeric Entry](/gre/quant/numeric-entry), inspect the requested unit after finishing the maths. A proportion of 0.31 corresponds to 31%, and the expected entry depends on the displayed request. ETS advises keeping intermediate calculations exact and rounding only as instructed. Do not turn recurring fractions into short decimal approximations before a later subtraction or percentage comparison.

A useful original boundary check uses a group of ten with six successful applicants and a group of 90 with 18 successes. The combined success proportion is 24/100 = 24%, despite subgroup rates of 60% and 20%. The large lower-rate group dominates. An answer above 60% or below 20% is impossible under these positive weights.

## Practise denominator decisions before adding a timer

Use a short practice set containing one direct percentage, one reverse percentage, one part-to-part ratio, one changing ratio and one combined proportion. Our recommendation is to write the denominator label before every calculation. During review, cover the solution and explain why a plausible alternative denominator is wrong. That explanation checks understanding more effectively than simply repeating the same arithmetic.

After the isolated set is reliable, use [topin’s free GRE mock](/gre/practice-test), marked on the official scale, to see whether the same choices remain accurate among other topics. Record whether each miss came from the base, translation, arithmetic or final requested quantity. A full mock is useful for transfer; the small concept set is useful for locating the cause of failure.

Use the calculator after setting up the relationship, especially for awkward division. ETS recommends estimating beforehand. An answer of 400% to a question asking what fraction of an ordinary two-category group belongs to one category should trigger a check, whereas “400% of another quantity” can be legitimate. The context, rather than a blanket rule that percentages cannot exceed 100, decides plausibility.

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## FAQs

How do I know which number is the percentage base?

Look for the whole after “of”, the comparison after “than”, or the original value in a change. Rewrite the phrase as a labelled fraction before substituting numbers. The same difference can yield different percentages when the base changes.

Can I add successive percentage changes?

Usually not. Multiply the successive remaining or growth factors because each change acts on the current value. A 12% rise and 20% fall give 1.12 × 0.8 = 0.896, a 10.4% overall decrease.

Does a 3:5 ratio mean 3/5 of the total?

No. For two categories that make up the entire total, 3:5 means eight parts altogether and the first category is 3/8 of the total. It is 3/5 of the second category.

When is choosing a starting value of 100 useful?

Use 100 when only proportional relationships matter and no actual total or integer restriction is given. Keep the stated number when the question supplies one, and use a variable for unknown counts that change by fixed amounts.

Are these official GRE questions?

No. Every worked question on this page was written for this guide. ETS sources establish the mathematical scope and conventions; the examples, checking routines and practice sequence are our recommendations.

## Sources (checked 5 October 2026)

* [ETS: Quantitative Reasoning overview and calculator guidance (checked 5 October 2026)](https://www.ets.org/gre/test-takers/general-test/prepare/content/quantitative-reasoning.html)
* [ETS: GRE Math Review (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/gre-math-review.pdf)
* [ETS: GRE Mathematical Conventions (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/gre-math-conventions.pdf)
* [ETS: Guidelines specific to the on-screen calculator (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/on-screen-calculator-guidelines.pdf)

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