Percentage Calculation Basics: Increases, Discounts, and the Base Value
The formulas for a percent increase and a percent discount, why the base value shifts between successive calculations, and percentage points vs. percent.
What does "percent" mean?
Percent means "per hundred": a ratio multiplied by 100, written with a % sign. An 8% sales tax on a 100-yen item, for instance, means 8 yen of tax for every 100 yen of price. Before using a percentage in a calculation, convert it back to a decimal ratio by dividing by 100, then multiply.
Pin down the base value first
The most important step in any percentage calculation is identifying exactly what the percentage is "of" -- the base value. The same "10%" produces a completely different result depending on what base amount it's applied to. This matters most when a price is raised and then discounted in sequence: the second calculation's base is the result of the first, not the original amount.
The formula for a percent increase
A percent increase equals base value times (1 plus the percentage divided by 100). Adding 8% sales tax to a 100-yen item gives 100 x (1 + 0.08) = 100 x 1.08 = 108 yen. Converting the percentage to a decimal and adding 1 before multiplying keeps the calculation reliable even by hand.
The formula for a percent discount
A percent discount equals base value times (1 minus the percentage divided by 100). A 1,000-yen item at 20% off costs 1,000 x (1 - 0.20) = 1,000 x 0.8 = 800 yen. Because the base shifts with each step, raising a price by 10% and then discounting it by 10% doesn't return to the original: it lands on 99, since the discount applies to 110, not 100.
Percentage points vs. percent change
If a rate rises "from 10% to 15%," there are two valid ways to describe the change: a straightforward difference of 5 percentage points (pp), or a relative increase of 50% (15 divided by 10 equals 1.5). Reading a reported change without noticing which framing is used is a common way to misjudge how large the shift actually is.
Common mistakes in percentage math
Frequent errors include forgetting that the base value changes across a sequence of increases and discounts, mixing up percentage points with percent change, and multiplying by the raw percentage number instead of converting it to a decimal first (using 8 instead of 0.08). Checking the base value and the decimal conversion each time heads off most mistakes.
Frequently asked questions
If a 100-yen item goes up 10% and then down 10%, does it return to the original price?
No. 100 x 1.10 x 0.90 = 99 yen, one yen less than the original, because the discount applies to the raised price of 110, not the original 100.
What's the difference between a percentage point and a percent?
A percentage point is a plain difference between two rates (15% - 10% = 5pp); a percent change is relative to the starting rate (15 / 10 = 1.5, a 50% increase).
What does a 1,000-yen item cost at 20% off?
1,000 x (1 - 0.20) = 800 yen.