Percent change between two numbers
Percentage increase compares a new number with an old one. Subtract the old value from the new one, divide by the old value, and multiply by 100. A bill that goes from $1,600 to $1,760 has increased by 160. Then 160 ÷ 1,600 × 100 = 10%. The formula is (new − old) ÷ old × 100. The old value has to be the starting point. Dividing by the new value instead answers a different question.
The same formula reports a decrease as a negative number. From 100 down to 80, the change is −20, and −20 ÷ 100 × 100 = −20%, a 20% decrease. From 80 back to 100 is (100 − 80) ÷ 80 × 100 = 25%. A 20% drop is not undone by a 20% rise. The reverse of a 10% increase is a decrease of about 9.09%, because you divide by 1.10 and the extra tenth is now part of a larger base.
Worked example: applying the increase
Once you know the rate, applying it is a multiplication. A 10% increase means the new value is 110% of the old one, or the old value times 1.10.
- Start at $1,600.
- Ten percent of 1,600 is 160.
- Add it: 1,600 + 160 = 1,760. Equivalently, 1,600 × 1.10 = 1,760.
- A further 10% increase is not 20% overall. 1,760 × 1.10 = 1,936, and the change from the original 1,600 is 21%, not 20%.
Finding the number before the change
If a price is $115 after a 15% increase, the original is not 115 minus 15% of 115. That would subtract the percent from the wrong base. Divide by 1.15 instead: 115 ÷ 1.15 = 100. In general, the original equals the final value divided by (1 + rate/100). For a decrease, the rate is negative: a price of $80 after a 20% decrease started at 80 ÷ 0.80 = 100.
Keep percent change separate from percentage points. A rate that moves from 4% to 5% has gained 1 percentage point. Relative to 4, that is a 25% increase in the rate itself. Headlines mix those up. This calculator does the relative change, (new − old) ÷ old.