Percent Decrease Formula - Definition, Steps, Examples

Key Points

• The percent decrease formula is $\frac{\text{old} - \text{new}}{\text{old}} \times 100$, dividing the fall by the original value.

• A percent decrease can never exceed 100 per cent, because a quantity cannot fall by more than all of itself.

• Rearranged, the value after a fall of $p$ per cent is $\text{new} = \text{old}\left(1 - \frac{p}{100}\right)$.

• A rise of 50 per cent followed by a fall of 50 per cent does not return to the start, because the two percentages act on different bases.

• Reversing a percent decrease means dividing by $1 - \frac{p}{100}$, never adding the same percentage back.

What Is the Percent Decrease Formula?

The percent decrease formula expresses a fall as a proportion of what came before:

$$\text{percent decrease} = \frac{\text{old} - \text{new}}{\text{old}} \times 100$$

The numerator is the fall and the denominator is the starting value. Writing it the other way round, the surviving quantity is $\text{new} = \text{old}\left(1 - \frac{p}{100}\right)$, where the bracket is the fraction that remains.

Why a Decrease Cannot Exceed 100 Per Cent

The largest possible fall is to zero, and a fall to zero removes exactly the whole original value. So the numerator can at most equal the denominator, which caps the ratio at 1, or 100 per cent.

That is a real asymmetry with percentage increase, which has no ceiling at all. A price can rise by 300 per cent but can never fall by 300 per cent.

A reported decrease above 100 per cent therefore signals an error, usually a sign slip or a swapped numerator and denominator. The only exception is a quantity that can genuinely go negative, such as a profit becoming a loss, where the language of percentage change stops being useful.

Variable Key

Examples of the Percent Decrease Formula

Example 1

A price falls from 80 to 60. Find the percent decrease.

$$\frac{80 - 60}{80} \times 100 = \frac{20}{80} \times 100 = 25$$

Final answer: 25 per cent

Example 2

A value rises 50 per cent, then falls 50 per cent. Find the overall change, starting from 200.

The two moves are the same size in opposite directions, so they look as though they cancel. Test it.

The rise gives $200 \times 1.5 = 300$. The fall is then taken on 300, not on 200, so it removes 150 rather than 100, leaving 150.

The starting value was 200, so the quantity has fallen overall:

$$\frac{200 - 150}{200} \times 100 = 25$$

Final answer: A net decrease of 25 per cent, not zero. The fall acted on a larger base than the rise.

Example 3

A machine worth 45,000 depreciates by 12 per cent. Find its new value.

$$45000\left(1 - \frac{12}{100}\right) = 45000 \times 0.88 = 39600$$

Final answer: 39,600

Example 4

A quantity fell by 30 per cent to reach 1,400. Find the original value.

The final value is 70 per cent of the original, so divide:

$$\frac{1400}{0.70} = 2000$$

Final answer: 2,000

Adding 30 per cent to 1,400 would give 1,820, which is wrong because that takes the percentage on the reduced value.

Example 5

A town's population falls from 12,500 to 11,000. Find the percent decrease to one decimal place.

$$\frac{12500 - 11000}{12500} \times 100 = \frac{1500}{12500} \times 100 = 12$$

Final answer: exactly 12 per cent

Don't Confuse These Four Measures

Percent decrease and discount percentage are the same calculation, with the marked price playing the role of the original value. Depreciation is the same again, applied repeatedly.

Percent Decrease Pitfalls and How to Avoid Each One

Mistake 1: Assuming an equal rise and fall cancel

Where it slips in: Two moves of the same percentage in opposite directions feel symmetric.

Don't do this: Treat 50 per cent up then 50 per cent down as returning to the start.

The correct way: Multiply the factors, $1.5 \times 0.5 = 0.75$, so the quantity ends 25 per cent down. The fall is applied to a larger base than the rise.

Students who turn both percentages into factors before doing anything see the non-cancellation immediately, because 1.5 times 0.5 is visibly not 1. The same move handles any chain of percentage changes.

Mistake 2: Dividing by the new value

Where it slips in: The new figure is the one being reported, so it becomes the reference.

Don't do this: $\frac{20}{60} \times 100 = 33.3$ per cent for a fall from 80 to 60.

The correct way: $\frac{20}{80} \times 100 = 25$ per cent. The base is the original value.

Mistake 3: Adding the percentage back to reverse a decrease

Where it slips in: Undoing a 30 per cent fall feels like adding 30 per cent.

Don't do this: $1400 + 30$ of $1400 = 1820$.

The correct way: Divide by the surviving fraction, $\frac{1400}{0.70} = 2000$. Adding always undershoots, and the shortfall grows with the percentage.

Wrapping Up

• The percent decrease formula is $\frac{\text{old} - \text{new}}{\text{old}} \times 100$, with the original value as the base.

• A decrease is capped at 100 per cent, unlike an increase, which has no upper limit.

• Equal percentage rises and falls never cancel, because each acts on a different base.

• Reversing a decrease means dividing by the surviving fraction, never adding the percentage back.

• To work through percentages with a teacher, explore a 7th grade math tutor or math tutoring.

Five Minutes of Practice

Find the percent decrease from 250 to 180. Then take 500, apply a 20 per cent rise followed by a 20 per cent fall, and state the net percentage change.

Want a live Bhanzu trainer to walk through more percentage problems? Book a free demo class.

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