How to calculate a percentage increase or decrease

The most common percentage-change error is using the new amount as the denominator. The question asks how large the change is compared with where you started, so the original amount is the base.

The formula

A positive answer is an increase; a negative answer is a decrease. You can also calculate the size of the decrease as original minus new, divide by original, and state that the result is a decrease. Keep the direction in your final answer.

Worked examples

1. A value rises from 48 to 60. Change = 60 − 48 = 12. Divide by the original 48: 12 ÷ 48 = 0.25. Convert to a percent: 25% increase.
2. A value falls from 75 to 60. Change = 60 − 75 = −15. Divide by 75: −15 ÷ 75 = −0.20. The value decreased by 20%.
3. A price rises by $12 from an original $48. The increase is 25%, because the $12 change is one quarter of the starting amount.

Percentage change is not percentage points

If a rate moves from 20% to 25%, it rises by 5 percentage points. Its relative percentage increase is (25 − 20) ÷ 20 = 25%. These answer different questions: percentage points subtract the rates directly; percentage change compares the size of the difference with the original rate.

Why reversing the change gives a different percent

A 20% decrease from 75 produces 60. Returning from 60 to 75 is a 15-unit increase on a base of 60, so it is a 25% increase. The dollar change is the same, but the base changed. Percentages describe a change relative to a specified starting value.

A reliable routine for word problems

• Circle the original amount and label it as the base.
• Find the numerical change and decide whether the value went up or down.
• Divide the change by the original amount, not by the new amount.
• Convert the decimal to a percent by multiplying by 100, or move the decimal point two places right.
• Write “increase” or “decrease” so the answer keeps its direction.
• Estimate first: a change smaller than the original should usually produce a percentage under 100%.

When the question gives a target percent

To find the new amount after a change, convert the rate to a decimal multiplier. An increase of 15% uses 1.15 times the original; a decrease of 15% uses 0.85 times the original. For an original amount of 80, a 15% increase gives 80 × 1.15 = 92. A 15% decrease gives 80 × 0.85 = 68.

For the FTCE, keep the base visible in your work. Most wrong answers come from dividing by the wrong amount, mixing a percentage-point difference with a percentage change, or losing the direction of the change.

Use the original amount as the reference

Percent change compares the difference with the original amount: percent change = (new − original) ÷ original × 100%. If a price rises from $40 to $50, the increase is $10 and the percent increase is 10/40 × 100% = 25%. Dividing by the new amount, 50, would answer a different question and understate the increase.

For a decrease from 80 to 68, the change is 12 and the percent decrease is 12/80 × 100% = 15%. Use the magnitude of the decrease when reporting the percent decrease, or keep the negative sign when describing signed percent change. State whether the question asks for an increase, decrease, or signed change.

Apply the percent multiplier

An increase of r percent multiplies the original by 1 + r, where r is written as a decimal. A 12% increase gives a multiplier of 1.12; a 12% decrease gives 0.88. For an original value of 250, a 12% increase yields 250 × 1.12 = 280, while a 12% decrease yields 250 × 0.88 = 220. This shortcut includes the original amount plus or minus the change.

To find only the change, multiply the original amount by the rate: 250 × 0.12 = 30. Then add or subtract it. Keep the distinction clear between the new amount and the amount of change. A problem may ask either, and giving $30 instead of $280 would answer only the increase component.

Handle successive changes carefully

Successive percentage changes apply to the current amount each time, not repeatedly to the original. A 10% increase followed by a 10% decrease returns 100 to 99: 100 × 1.10 × 0.90 = 99. The percentages do not cancel because the second change is calculated from 110. For two increases, multiply the factors, such as 1.05 × 1.08, rather than adding percentages when the base changes.

A percent change is undefined when the original amount is zero because the formula divides by the original. In that situation, describe the absolute change or use another stated comparison. For negative quantities, follow the context’s definition carefully; the ordinary increase/decrease language may not map intuitively onto a signed value.

• Subtract original from new to find the signed change.
• Divide by the original amount, then multiply by 100%.
• Distinguish the change amount from the new total.
• Use 1 + rate for an increase and 1 − rate for a decrease.
• Apply successive changes to each new amount, not the initial base.

Choose the baseline before calculating

The original value answers the question ‘compared with what?’ It remains the denominator even when the amount changes dramatically. If a fee rises from $40 to $50, the change is $10 and the rate is 10 ÷ 40 = 25%. Reversing the comparison gives a $10 decrease from $50, or 10 ÷ 50 = 20%. These percentages differ because the starting bases differ. Label the original and new amounts before substituting to avoid reversing them.

A multiplier is a quick check. An increase of r percent multiplies the original by 1 + r; a decrease multiplies by 1 − r, with r written as a decimal. A 12% increase uses 1.12, while a 12% decrease uses 0.88. To recover the starting value from a final value, divide by the multiplier: a price of $88 after a 12% decrease came from $88 ÷ 0.88 = $100. Adding 12% to $88 would not reverse the discount because it uses the wrong base.

Separate percent change from percentage points

When a rate moves from 30% to 36%, it increases by 6 percentage points. Its relative percentage increase is 6 ÷ 30 = 20%. Questions about rates may ask for either measure, so preserve the wording and units. A change from 30 to 36 units is a 20% increase, but a change from 30% to 36% is often most clearly stated as six percentage points, with the relative increase reported only if requested.

For successive changes, apply each multiplier to the updated amount. A 10% increase followed by a 10% decrease gives 1.10 × 0.90 = 0.99, leaving 99% of the starting amount. The net result is a 1% decrease, not zero. In word problems, keep currency or count units attached to the values and round only after the final calculation unless the directions specify otherwise.