GMAT Weighted Averages

The weighted-average idea

An ordinary average gives every observation equal weight. A weighted average gives some values greater influence because they represent more people, units or time. If two groups have different sizes, their combined average is not usually the simple average of the two group averages. The larger group contributes more observations and therefore has more influence.

The general formula is: weighted average = (sum of each value times its weight) divided by (sum of the weights). If the weights are percentages that sum to 100 percent, the weighted average is the sum of value times percentage share. If weights are counts, divide by the total count.

For example, if 10 items cost $4 each and 30 items cost $8 each, the average cost per item is (10 x 4 + 30 x 8) / (10 + 30) = 280 / 40 = $7. The simple average of $4 and $8 is $6, which is wrong because it treats the two groups as equal sizes.

Translate the story into weights

Before multiplying, identify the quantity represented by each value. A class average may be weighted by number of students. A travel average may be weighted by time or distance. A portfolio return may be weighted by the share of money invested. A mixture concentration may be weighted by volume. The word 'average' alone does not tell you which denominator is right.

Write a small table with value, weight and product. This prevents confusion between the percentage grade and the number of assignments, or between price per unit and number of units. Add the weights last and check that they describe the full population in the question.

If weights are percentages, use them consistently as fractions or percentage points. For a portfolio with 40 percent in a fund returning 5 percent and 60 percent in a fund returning 10 percent, the return is 0.40 x 5% + 0.60 x 10% = 8%. Using 40 and 60 instead of 0.40 and 0.60 produces 800 in percentage-point units, a clear signal that the scale is wrong.

Combine group averages

If group A has nA members and mean a, its total is nA times a. Group B contributes nB times b. The combined mean is (nA times a + nB times b) divided by (nA + nB). This 'total first, divide once' method is reliable for test questions.

A class of 18 students has an average score of 72, and a class of 12 has an average of 82. The combined average is (18 x 72 + 12 x 82) / 30 = (1,296 + 984) / 30 = 2,280 / 30 = 76. The answer lies closer to 72 because the lower-scoring class is larger.

The simple mean of 72 and 82 is 77. That value would be correct only if the groups had equal sizes. A quick reasonableness check helps: the combined average must lie between the group means, and it should be closer to the mean for the larger group. A result outside 72 to 82 signals an arithmetic or setup error.

Solve for a missing group

A common question gives a combined average and asks for a missing group's average. First find the combined total by multiplying the overall average by the overall count. Subtract the known group's total, then divide by the missing group's count.

A club has 20 members with an average age of 31. Twelve members have an average age of 28. What is the average age of the other eight? The full age total is 20 x 31 = 620. The known group contributes 12 x 28 = 336. The remaining total is 284, so the remaining average is 284 / 8 = 35.5 years.

A common wrong approach is to subtract 28 from 31 and assume the missing group's average is 34. That ignores group sizes. Another wrong approach averages the known mean and overall mean. The totals method works because averages are totals divided by counts.

Work with percentages and mixtures

For mixtures, weight concentration by the amount of mixture. Suppose 5 liters of a 20 percent solution are combined with 15 liters of a 40 percent solution. The amount of active ingredient is 5 x 0.20 + 15 x 0.40 = 1 + 6 = 7 liters. The combined concentration is 7 / 20 = 35 percent.

A simple average of 20 and 40 gives 30 percent, but the stronger solution contributes three times as much volume. The combined value should be closer to 40 percent. This same structure applies to weighted grades and portfolio returns: calculate each group's contribution, add contributions, then divide by total weight when weights are counts or volumes.

If a question asks for a blend using percentage shares that already add to 100 percent, division by total weight is unnecessary because that total is 1. For example, a 25 percent allocation with return 4 percent and a 75 percent allocation with return 8 percent gives 0.25 x 4 + 0.75 x 8 = 7 percent. If the shares sum to 90 percent because one asset is omitted, normalize only if the question asks for the average of the included portion.

Rates and time require the right weight

Averages involving speed are a frequent source of errors. Average speed is total distance divided by total time, not always the simple mean of speeds. If a driver travels 60 miles at 30 miles per hour, the trip takes two hours. Then the driver travels 60 miles at 60 miles per hour, taking one hour. Average speed is 120 miles / 3 hours = 40 mph, not 45 mph.

When equal distances are traveled, the slower speed receives more time weight. For equal time intervals, by contrast, the average speed is the arithmetic mean of the speeds because each speed applies for the same duration. Always calculate the quantity the question asks for from total distance and total time if uncertain.

A production-rate average may use hours as weights. If one machine produces 12 units per hour for 2 hours and another period runs at 18 units per hour for 5 hours, the overall rate is total units divided by total hours: (12 x 2 + 18 x 5) / 7 = 114 / 7, about 16.29 units per hour. The two rates are not weighted by number of machines unless the question specifically defines that weight.

Original GMAT-style worked problems

Two teams

Team A has 8 analysts averaging 74 units of output. Team B has 12 analysts averaging 84. What is the combined average? Total output is 8 x 74 + 12 x 84 = 592 + 1,008 = 1,600. There are 20 analysts, so the combined average is 80.

The simple average, 79, is too low because Team B is larger and has the higher mean. A quick check shows the answer must be between 74 and 84 and closer to 84. This check catches common multiplication or denominator mistakes.

Find the missing percentage

A portfolio invests 30 percent in an asset returning 2 percent and 50 percent in an asset returning 8 percent. The remaining 20 percent earns what return if the total portfolio return is 6 percent? Let r be the missing return. Then 0.30(2) + 0.50(8) + 0.20r = 6. The known contribution is 0.6 + 4 = 4.6, so 0.20r = 1.4 and r = 7 percent.

A distractor may divide the remaining contribution by 0.20 twice. After subtracting the known contribution, 1.4 is already expressed in percentage-point contribution units; dividing by the 20 percent weight once gives 7. Another trap is treating the remaining allocation as 30 percent rather than subtracting 30 plus 50 from 100.

Average price with a fixed quantity

A buyer purchases 4 shares at $15 and 6 shares at $25. The average price per share is (4 x 15 + 6 x 25) / 10 = (60 + 150) / 10 = $21. The result lies closer to $25 because more shares were purchased at that price.

If instead the question asks for the average of the two transaction prices, it may mean $20, but that is a different quantity and usually needs clarification in the prompt. In test problems, pay attention to whether the counted unit is dollars per share, transactions or shares. The weight follows the observations whose average is requested.

Use algebra to avoid repeated guessing

When one group's size is unknown, represent it with a variable. If 10 people average 70 and a group of x people averages 80, while the combined mean is 76, write (10 x 70 + 80x) / (10 + x) = 76. Then solve 700 + 80x = 760 + 76x, so 4x = 60 and x = 15.

The equation is easier to check than trying candidate group sizes. It also makes the relationship clear: the combined mean of 76 is six points above 70 and four below 80, so the higher group must be larger. The ratio of group sizes is 6:4, or 3:2, which gives 15 to 10.

Common mistakes and checks

• Averaging group means without considering group sizes.
• Multiplying a percentage by a count but forgetting to divide by the total count.
• Using percent values as whole numbers inconsistently.
• Weighting a rate by the wrong quantity, such as distance instead of time.
• Confusing the mean of transaction prices with the mean price per item.
• Accepting a combined mean outside the range of the group means.
• Forgetting a group when calculating the total weight.

Check dimensions. If each group value is dollars per item and its weight is a number of items, their product is dollars. Dividing total dollars by total items returns dollars per item. If your units do not cancel to the unit requested, the setup is probably wrong.

Use a weighted-average formula when the prompt gives group counts or shares. Use total divided by count for the ordinary mean. For rates, reconstruct the total numerator and denominator. If the value already represents an average, recover its total by multiplying by the group size before combining.

Practice and review

Practice weighted averages in multiple settings so you learn the underlying structure rather than one formula cue. Mix class scores, prices, mixtures, portfolios and travel rates. Before calculating, state the value, the weight and the total weight. After calculating, check that the result lies between the component values unless the weights or conditions make that impossible.

In Quant, work without a calculator and simplify arithmetic. In Data Insights, the calculator is available, but the same setup matters. Review errors by asking whether the mistake came from a wrong weight, wrong total, incorrect units or arithmetic. Then solve a new problem in the same family. Repeating the original numbers only tests memory.

The GMAT's adaptive scoring does not award a special score for a weighted-average topic. It is one mathematical tool within Quantitative Reasoning and may support Data Insights interpretation. Learn it because it lets you model group data correctly, not because a narrow topic promises a particular score gain.