How can you translate the wording before calculating?
GMAC lists percentages, ratios, and proportions among quantitative topics. The exercises below practice these concepts without making claims about how often they appear on the test. The English wording of the questions is also part of the practice.
“Increased by 20 percent” means multiply the starting value by 1.20. “Is 20 percent of” means take 0.20 of it. In a 2:3 ratio, the quantities are 2k and 3k, so the first is 2/5 of the total, not 2/3. Naming the base helps you avoid solving the wrong problem.
Before calculating, write a unit next to each quantity: dollars, items, liters, or people. A ratio is unitless when the two quantities being compared share the same unit. A speed or unit price, by contrast, retains its compound unit.
Why can’t you always add two percentage changes?
Each successive change applies to the value from the previous step. Multiply the factors. To find the starting value, divide by the combined factor, provided it is nonzero.
A subscription price is increased by 20 percent, then reduced by 10 percent. Its final price is $108. What was the original price?
Choose an answer.
The solution, step by step
$100.
The method in action
- The combined factor is 1.20 × 0.90 = 1.08. The starting price is 108/1.08 = 100.
- Check: 100 becomes 120, then 108. The net increase is 8% because the 10% reduction applies to 120.
- The other prices do not yield 108: 90 would become 97.20; 96, 103.68; 102, 110.16; and 110, 118.80. Adding +20% and −10% would give a 10% increase, which is incorrect.
How does a change to the total affect the parts?
When quantities are given only as a ratio, define a common unit k. Then you can write down what changes and what stays the same.
A box contains red and blue tokens in the ratio 3:5. After 12 red tokens are added, the numbers of red and blue tokens are equal. How many blue tokens were originally in the box?
Choose an answer.
The solution, step by step
30.
The method in action
- Initially there are 3k red tokens and 5k blue tokens. Adding 12 red tokens closes the 2k gap: 2k = 12, so k = 6.
- There are 5 × 6 = 30 blue tokens. Initially there are 18 red tokens; after adding 12, there are 30. The number of blue tokens does not change.
- 18 is the initial number of red tokens; 24 may result from doubling the added amount; 36 and 40 do not satisfy both the initial ratio and the gap of 12.
The pitfall
Averaging two rates can describe the wrong population
If two groups differ in size, the simple average of their percentages usually does not give the percentage for the combined group. First count the items that meet the condition, then divide by the total number of items.
This mistake can go unnoticed when the result seems reasonable. Checking that the result falls “between the two rates” is useful but insufficient: many incorrect values also fall between them.
How do you calculate a weighted percentage?
The next problem can be solved using absolute counts, without memorizing a weighted-average formula. That is often the clearest approach.
Team A has 40 members, 25 percent of whom work remotely. Team B has 60 members, 50 percent of whom work remotely. What percent of all members work remotely?
Choose an answer.
The solution, step by step
40%.
The method in action
- Team A has 10 remote workers; Team B has 30. That makes 40 out of 100, or 40%.
- 37.5% is the simple average of the 25% and 50% rates. It gives equal weight to two teams of different sizes.
- 25% and 50% each describe only one team; no correct totals produce 35%. Rechecking with 10 + 30 and 40 + 60 confirms the answer.
Key takeaways
- Always write down the rate’s base.
- A ratio of a:b corresponds to quantities ak and bk.
- To combine groups, combine their counts before dividing.