Percentages

Choose the base before calculating

If a value rises from 80 to 100, its increase is 20 and its percent increase is $\frac{20}{80}\cdot 100\%=25\%$. The denominator is the original 80. Reversing the change, from 100 to 80, gives a 20% decrease.

The two changes use different bases. A sentence saying that one value is a certain percent of another also names the base: if $a$ is 30% of $b$, then $a=0.30b$.

Use multipliers for successive changes

A 20% increase multiplies by 1.20, and a 20% decrease multiplies by 0.80. Applying both to 100 gives $100(1.20)(0.80)=96$. The changes do not cancel because the decrease acts on 120.

For several changes, multiply the factors before converting the final multiplier into an overall percent change. A multiplier of 0.96 represents a 4% decrease from the original value.

Work backward from the final amount

A jacket costs 72 dollars after a 20% discount. Its original price $p$ satisfies $0.80p=72$, so $p=90$. Adding 20% of 72 would give 86.40 dollars and would not undo the discount.

Write the equation with the original amount as the unknown. This also works for a final amount after tax, growth, or a markup.

Separate percentage points from percent change

If a success rate rises from 40% to 50%, it rises by 10 percentage points. Relative to the original rate, the percent increase is $\frac{50-40}{40}\cdot 100\%=25\%$.

Read the requested unit carefully. A percentage-point difference subtracts the two rates; a relative percent change divides that difference by the original rate.

Check your understanding

Sample percentages questions

Practice this skill

1. Circle the base quantity in the stem.
2. Write each percent as a decimal factor before multiplying.
3. For a reverse problem, substitute the original amount back through the change.

Print the percentages worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.