A number is increased by 25% then decreased by 20% — net change in percent

Question

A number is first increased by 25% and then decreased by 20%. What is the net percentage change in the number?

Solution — Step by Step

Let the original number = 100. Using 100 makes the percentage arithmetic clean — percentages directly represent the actual change in value.

Increase of 25% on 100:

\text{New value} = 100 + 25\% \text{ of } 100 = 100 + 25 = 125

Now decrease 125 by 20%:

20\% \text{ of } 125 = \frac{20}{100} \times 125 = 25

\text{Final value} = 125 - 25 = 100

\text{Net change} = \frac{\text{Final} - \text{Original}}{\text{Original}} \times 100 = \frac{100 - 100}{100} \times 100 = \mathbf{0\%}

The net change is zero — the number returns to its original value.

Why This Works

The 25% increase multiplies by $1.25$ , and the 20% decrease multiplies by $0.80$ :

1.25 \times 0.80 = 1.00

The product is exactly 1, so the combined effect is no change. This isn’t a coincidence — it reflects the algebraic relationship between the two percentages.

In general, if a number is increased by $a\%$ and then decreased by $b\%$ , the net percentage change is:

\text{Net \%} = a - b - \frac{ab}{100}

Here: $25 - 20 - \frac{25 \times 20}{100} = 5 - 5 = 0\%$ . ✓

If a value is first increased by $a\%$ and then decreased by $b\%$ :

\text{Net \%} = a - b - \frac{ab}{100}

Positive result = net increase. Negative result = net decrease. Zero = no change.

The formula works for any combination: increase then increase, decrease then decrease, or any order. For increase followed by increase: both $a$ and $b$ are positive in $a + b + ab/100$ (the sign before $ab/100$ is positive).

Alternative Method

Multiply the scale factors directly:

\text{Final} = \text{Original} \times \left(1 + \frac{25}{100}\right) \times \left(1 - \frac{20}{100}\right) = P \times 1.25 \times 0.80 = P \times 1.0 = P

Final = Original → 0% net change.

This multiplicative approach is the most general method. For any sequence of percentage changes, multiply all the scale factors.

Common Mistake

The most common error: simply adding the percentages: $+25\% - 20\% = +5\%$ net change. This is wrong. The second percentage is applied to the new (changed) value, not the original. Since the base changes after the first operation, the percentages don’t simply add. Always apply percentages sequentially to the running value, or use the net formula $a - b - ab/100$ .

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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