# GRE Weighted Averages and Mixtures Explained | topin

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## Start with the quantity that can actually be added

A class mean represents total marks divided by the number of students. A solution concentration represents ingredient amount divided by total solution amount. Neither percentage nor mean is usually the quantity you add directly. First recover the underlying totals, add those, then divide by the new total weight. [ETS](https://www.ets.org/gre/test-takers/general-test/prepare/content/quantitative-reasoning.html) covers means, ratios and algebraic word problems in its Quant resources.

For two groups of sizes n and m with means a and b, the combined mean is (na + mb)/(n + m). Here na and mb are the group totals. The formula is not an extra statistical assumption: it comes from adding the individual values in both groups. If the question describes overlapping groups, some individual values might be counted twice and this simple combination needs adjustment.

The mean of the two group means, (a + b)/2, is correct only when the group sizes are equal or the means are equal. This is where weighted averages often appear in [Quantitative Comparison](/gre/quant/quantitative-comparison): Quantity A is the combined mean of 20 students averaging 70 and 30 averaging 80, Quantity B is 75\. You need no arithmetic. The larger group sits at 80, so the combined mean is pulled above the midpoint and A is greater (it is 76). Ask which side of the midpoint the weights push the answer before calculating anything. The [GRE Quant section guide](/gre/quant) sets out all five question formats.

## Choose the weight from the units

The correct weight is the denominator behind each given average. For marks per student, use students. For distance per hour, use time. For ingredient per litre, use litres, provided the problem specifies additive volumes or its model makes that assumption explicit. Do not use a familiar formula until you have written the units in both numerator and denominator.

Our decision table uses units as the main test. If multiplying a mean by its weight does not produce a quantity you can add, the model is probably wrong. Currency per item multiplied by items gives currency; kilometres per hour multiplied by hours gives kilometres. This small unit check also catches a swap between distance-weighted and time-weighted average speeds.

__The weight must match the denominator__
| Given quantity              | Appropriate weight                          | Add this total     |
| --------------------------- | ------------------------------------------- | ------------------ |
| Mean marks                  | Number of students                          | Total marks        |
| Mean item price             | Number of items                             | Total cost         |
| Concentration by volume     | Solution volume under additive-volume model | Ingredient volume  |
| Speed                       | Time at that speed                          | Distance travelled |
| Frequency distribution mean | Frequency of each value                     | Value × frequency  |

## Original problem: combine unequal groups

Original question: 18 students have a mean score of 72 and 12 students have a mean score of 87\. What is the mean score of all 30 students? The totals are 18 × 72 = 1,296 and 12 × 87 = 1,044\. Their sum is 2,340, giving 2,340/30 = 78\. The result is closer to 72 because that group is larger.

A shorter calculation starts from the lower mean. If everyone scored 72, the overall mean would be 72\. The higher group contributes an extra 15 marks each across 12 students, or 180 extra marks spread over 30 students. The mean therefore rises by six to 78\. This deviation method preserves the same totals while using smaller numbers.

Check the result against the bounds: with positive group sizes, a combined mean must lie strictly between the component means, here 72 and 87\. Then check the side: the larger group has the lower mean, so the answer must sit below the midpoint of 79.5\. A bounds check rejects impossible answers; it does not prove a plausible one correct.

## Original problem: find an unknown group or weight

Original question: a 25-person group has mean age 28\. Ten members have mean age 31\. What is the mean age of the other 15? The whole group’s age total is 25 × 28 = 700, and the known subgroup contributes 10 × 31 = 310\. The remaining total is 390, so the requested mean is 390/15 = 26.

Do not subtract the averages and report −3 or treat the difference as an age. A mean is a ratio, so subtraction must happen at the total level first. The answer of 26 makes sense because the remaining group needs to pull the combined mean below 31\. Recombining 10 × 31 and 15 × 26 recovers 700, an independent forward check.

A second original question gives means 60 and 84 and a combined mean of 69, but no group counts. Let the weights be n and m: 60n + 84m = 69(n + m), giving 9n = 15m and n:m = 5:3\. This determines a ratio, not an actual headcount. Counts of five and three work, but so do ten and six.

For another original check, three items averaging 14 have total value 42\. Adding one item with value 22 gives total 64 over four items, mean 16\. The mean rises by two, not eight, because the added item’s eight-unit excess is shared across all four observations. This explains why a new item does not change a group mean by its full difference.

## Original problem: solve a concentration mixture

Original question: eight litres of a 15% salt solution are mixed with x litres of a 35% salt solution to make a 25% solution. Assume volumes add and no ingredient is lost. The initial salt amount is 0.15 × 8 = 1.2 litres under this stated volume model, while the added salt amount is 0.35x. Total volume is 8 + x.

Set 1.2 + 0.35x = 0.25(8 + x). This gives 0.10x = 0.8 and x = 8 litres. Equal volumes of equally distant concentrations, 15% and 35%, produce the midpoint 25%. That symmetry checks this particular answer. It is not a rule that mixtures always require equal volumes; the target position determines the weights.

If the target were 20% instead, the distances from the low and high concentrations would be five and fifteen percentage points. The low:high volume ratio must be 15:5 = 3:1\. With eight litres of low solution, the high volume would be 8/3 litres. This inverse-distance shortcut follows by balancing departures from the target, so write the target equation when the ratio direction feels uncertain.

## Original problem: dilution and removal change the model

Original question: a container holds 20 litres of a uniformly mixed 30% solution. Four litres are removed and replaced with four litres of pure water. Assume additive volumes. Initially there are six litres of ingredient. The removed portion contains 0.3 × 4 = 1.2 litres, leaving 4.8 litres. Replacement restores the volume to 20, so concentration becomes 24%.

The statement “uniformly mixed” matters: it makes the removed portion have the same concentration as the container. Removing four litres of pure ingredient instead would give a different answer. Adding water without removal would also change the denominator to 24 litres. Draw a before–removed–added–after line whenever a question contains both removal and replacement.

If the same four-litre replacement occurs again after remixing, each cycle retains 16/20 = 0.8 of the ingredient. The concentration becomes 30% × 0.8 × 0.8 = 19.2%. Do not subtract six percentage points repeatedly: the second removal acts on a weaker solution. This changing-base idea also appears in [GRE percentages and ratios](/articles/gre-percentages-ratios-word-problems).

## Check bounds, conservation and the requested quantity

Use three checks in that order. The combined mean or concentration should fit the component bounds. Your ingredient or value totals should balance. The final number should answer the requested quantity, which might be a group size rather than a mean. These are our solving recommendations. They help distinguish a correct equation with the wrong final interpretation from a genuinely incorrect model.

Practise a combined mean, a missing mean, a missing weight, a two-solution mixture and a removal problem as separate items. Then mix them so the wording has to choose the model. Use [topin’s free GRE mock](/gre/practice-test), marked on the official scale, for a later transfer check. Record “wrong weight” separately from “wrong arithmetic” because the next practice action differs.

ETS recommends using the calculator for tedious computations after estimating the result. Keep fractions exact during short algebra and use the calculator for larger totals when helpful. If the final answer is 8/3 litres, follow the question’s answer format and rounding instruction. The [Numeric Entry guide](/gre/quant/numeric-entry) explains the input step; it does not change the conservation equation.

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## FAQs

When can I average two averages directly?

When the two groups have equal weights, or when both averages are identical. Otherwise multiply each mean by its group size, add the totals and divide by the combined size.

What is the quickest weighted-average check?

Check that the result lies between the component means and nearer the mean of the larger group. This identifies impossible values but does not replace calculating or checking the totals.

Can a mixture concentration exceed both inputs?

Under a simple additive mixture with positive amounts and no ingredient gain or loss, it cannot. A concentration outside the input range means the equation or assumptions need checking.

Why are mixture ratios opposite the concentration gaps?

The ingredient surplus above the target must equal the deficit below it. A component farther from the target needs a smaller weight to balance the other component. Derive the equation if you forget the direction.

Does removing solution change its concentration immediately?

Removing a representative portion from a uniformly mixed solution leaves the concentration unchanged, although the amount and volume both shrink. Adding replacement water then lowers concentration. The uniform-mixing condition is essential.

## 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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