GMAT Percentages From Scratch
Percentages are among the most dangerous topics on GMAT Quant. Not because the maths is hard, but because there are three separate ways to get a question wrong that all look like small slips: confusing what you are taking a percentage of, misplacing a zero, and treating a percentage change like a percentage share.
Name the base, the share, and the percentage share
Every percentage question has three parts. The base is the whole thing you are splitting. The percentage share is the proportion you take, always relative to that base. The share is the actual number you end up with.
Take 70% of a group of 240 people and you get 168. The base is 240, the percentage share is 70%, the share is 168.
The distinction matters because 168 means something on its own. "70%" means nothing without knowing 70% of what: a neighbourhood, a city, or a country are three completely different answers. Before calculating anything, identify which two of the three you have and which one you are solving for.
Do not trust the % sign to tell you which is which
This is the trap. A number with a percent sign attached can be the base, the share, or the percentage share.
Say 40% of survey respondents rank the economy first, and 75% of that 40% intend to vote for candidate X. The 40% is a percentage share of the original group, and it is also the base of the second calculation. The answer, 30%, is a share. All three carry a percent sign and all three play different roles.
Sense check with a different calculation
Redoing the same sum finds nothing, because you will make the same mistake twice. Approximate with a different route instead.
Calculating 15% of 200? 15% is a little under 20%, and 20% is one fifth. One fifth of 200 is 40, so the answer should be a bit under 40. If you got 30, that fits. If you got 3 or 300 you dropped or added a factor of ten, which is the single most common percentage error on the GMAT.
A percentage change is not a percentage of
A 25% increase is not 0.25 × the old price. It is the old price plus 25% of it, which is 1.25 × the old price. A 40% decrease is 0.6 × the old value, because taking 40% away leaves 60% behind.
Getting comfortable with the 1.25 and 0.6 forms is what makes the next part work.
Stacked changes do not add
Increase something by 50%, then decrease it by 50%, and you do not return to where you started. You get 1.5 × 0.5 = 0.75, a 25% decrease overall.
The reason is that the second change acts on a bigger number than the first one did. Same effect upward: a 10% rise followed by a 20% rise gives 1.1 × 1.2 = 1.32, a 32% increase rather than 30%.
Multiply the factors. Never add the percentages.
Work out the digits, then place the decimal
Most percentage arithmetic on the GMAT reduces to a times table fact plus a decision about zeros. For 40% of 2,000: 4 × 2 = 8, then ask whether the answer is 80, 800, or 8,000. Since 40% is a bit under half of 2,000, and half is 1,000, it must be 800.
Get the digits from your times tables, then place the decimal by sense checking. It is faster than long multiplication and it fails loudly rather than quietly.