Percentages: complete GMAT guide

A percent is a fraction with denominator 100. $45\%=45/100=0.45$. To find a percent of a number, multiply by the decimal form. To express one number as a percent of another, divide the part by the whole and multiply by 100.

The word "of" usually signals multiplication. The phrase "what percent" signals an unknown rate. Translate the sentence into an equation before calculating.

Percent change compares the amount of change to the original value:

The denominator is always the original value, not the new value. If a price rises from $40 to $50, the change is $10 and the percent increase is $10/40=25\%$. If the price falls from $50 to $40, the percent decrease is $10/50=20\%$. The same dollar change produces different percentages because the bases differ.

A percent increase of $r$ percent corresponds to a multiplier of $1+r/100$. A 25% increase is a multiplier of 1.25. A 25% decrease is a multiplier of 0.75. This form is faster for successive changes.

If a price increases by 20% and then decreases by 20%, the final multiplier is $1.20 \times 0.80=0.96$. The final price is 96% of the original, a 4% decrease. The two changes do not cancel because the second change applies to the increased price.

If a value after a change is known, divide by the multiplier to recover the original value. If a price after a 20% increase is $96, the original price is $96/1.20=80$. Do not subtract 20% of 96, because 96 is not the original base.

A markup is an increase from cost to selling price. A discount is a decrease from selling price. If a store buys an item for $50 and marks it up 40%, the selling price is $50 \times 1.40=70$. If it then offers a 20% discount, the customer pays $70 \times 0.80=56$. The final price is 12% above cost, not 20% above cost.

Profit is selling price minus cost. Profit percent is usually calculated as a percentage of cost, but read the question carefully. Some questions define profit as a percentage of selling price.

A change from 20% to 25% is an increase of 5 percentage points, but it is a 25% increase in the rate itself, because $(25-20)/20=25\%$. The GMAT tests this distinction. If a question says "the unemployment rate rose by 2 percentage points," it does not mean the rate rose by 2%.

Simple interest is calculated only on the principal:

where $P$ is the principal, $r$ is the annual rate as a decimal, and $t$ is the time in years. For $1,000 at 5% for 3 years, the interest is $1000 \times 0.05 \times 3=150$.

Compound interest applies interest to the accumulated amount. The formula is

where $n$ is the number of compounding periods per year. For annual compounding, $A=P(1+r)^t$. The GMAT usually asks you to compare the two or to apply one step at a time rather than compute a complicated exponent.

Real-world percentage problems often stack changes. A restaurant bill may include a tax and a tip. If the tax is 8% and the tip is 20% of the pre-tax bill, the total multiplier is $1.08+0.20=1.28$, because both percentages apply to the same base. If the tip is calculated on the post-tax amount, the multiplier is $1.08 \times 1.20=1.296$. Read the wording to see which base each percentage uses.

A markup followed by a discount can be modeled with multipliers. A 30% markup and a 10% discount give $1.30 \times 0.90=1.17$, a net increase of 17% over cost. Do not subtract 10% from 30% and call it 20%, because the discount applies to the marked-up price.

Two quantities can have the same absolute change and different percentage changes. If A rises from 50 to 60 and B rises from 200 to 210, both increase by 10, but A rises by 20% and B rises by 5%. The base determines the percentage.

When a question asks which change is larger, check whether it means absolute change or percent change. A large percentage increase from a small base can be smaller in absolute terms than a small percentage increase from a large base.

When a question gives a percent increase and a percent decrease of the same size, the net result is always a decrease unless the percentages are zero. The multiplier $(1+r)(1-r)=1-r^2$ is less than 1. This is a quick way to check answers that claim the changes cancel.

1. Identify the original value, the new value, and the base of the percentage.
2. Translate the sentence into an equation using decimals.
3. For successive changes, multiply the multipliers.
4. For reverse questions, divide by the multiplier.
5. Distinguish percent change from percentage points.
6. Check whether profit is based on cost or selling price.

• Using the new value as the denominator for percent change.
• Adding successive percentages instead of multiplying multipliers.
• Subtracting a discount from the final price instead of the original price.
• Confusing percent change with percentage points.
• Forgetting to convert a percent to a decimal.
• Assuming a 20% increase cancels a 20% decrease.
• Calculating profit as a percentage of the wrong base.