Find the base value first. Convert each percentage change to a multiplier, then check whether the question asks for a value or a relative change.
A percentage describes an amount relative to a base. Before calculating, name that base: the original price, the current population, or the whole group. Using the wrong base produces a wrong answer even when the arithmetic is accurate.
Turn each change into a multiplier
A p% increase multiplies the base by 1 + p/100. A p% decrease multiplies it by 1 − p/100. Taking p% of a quantity multiplies by p/100; that is different from decreasing it by p%.
The examples in this guide are original teaching exercises, not released SAT questions.
Worked example: successive changes use changing bases
A price of £200 rises by 15% and then falls by 10%. Its final value is £200 × 1.15 × 0.90 = £207. The combined multiplier is 1.035, so the net increase is 3.5%, not 5%.
The first change adds £30. The second removes 10% of £230, which is £23. The percentages cannot be subtracted as if both used £200. For fixed multipliers applied successively to the current value, their order gives the same final product. That rule does not cover a problem where a charge is a fixed amount or explicitly uses a different base.
Worked example: recover the original value
After a 20% discount, a jacket costs $72. If P is its original price, 0.80P = 72, so P = 90. Adding 20% to $72 gives $86.40, which does not reverse a discount taken from $90.
Likewise, a quantity that becomes 156 after a 30% increase began at 156/1.30 = 120. Reversing a multiplier means dividing by it.
Worked example: “more than” and “less than” use different bases
If A is 25% greater than B, then A = 1.25B. With B = 80, A = 100. B is therefore 20% less than A, since the difference of 20 is 20% of 100. The same absolute difference can have two different relative percentages.
Percentage points versus percentage change
If a rate rises from 40% to 50%, it increases by 10 percentage points. Relative to its original rate, the percentage increase is (50 − 40)/40 × 100 = 25%. Read which quantity the question asks for.
Taxes, tips, and stated bases
A $50 item discounted by 10%, with an 8% tax on the discounted price, costs 50 × 0.90 × 1.08 = $48.60. The tax base is $45 because the prompt says so. A restaurant problem may calculate a tip on the pre-tax amount; follow the stated base rather than assuming every change compounds.
A quick check is to estimate the direction and size: a 20% discount should lower a price to four-fifths, and recovering the original should produce a larger number. Then verify the result by applying the forward change.
For percentage questions in practice, label the error as wrong base, wrong multiplier, addition of successive changes, reverse-change error, or confusion with percentage points. For ratios and units, use the ratios and rates guide.
Official sources
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