Why Going Up 10% and Down 10% Does Not Cancel Out

The percent change formula

Percent change tells you how big a change is compared with where you started. The formula is the same for increases and decreases.

Percent change = (new value − old value) ÷ old value × 100

A positive result is an increase, and a negative result is a decrease. The important detail is the denominator: it is always the old value, the starting point.

Increase from 50 to 65: (65 − 50) ÷ 50 = 0.30, so +30%.
Decrease from 80 to 60: (60 − 80) ÷ 80 = −0.25, so −25%.
Population from 12,000 to 13,800: (13,800 − 12,000) ÷ 12,000 = 0.15, so +15%.

You can enter any two numbers into the percentage calculator to get the change instantly. This guide explains what the answer means and where people go wrong.

Applying a percent change to a number

To go the other way, from a number and a percentage to the new value, multiply by (1 + change as a decimal). An increase of 12% means multiplying by 1.12, and a decrease of 12% means multiplying by 0.88.

A value of 250 increased by 12%: 250 × 1.12 = 280.
A value of 250 decreased by 12%: 250 × 0.88 = 220.

Each of these is a single step from 250. The trouble starts when you chain changes together, because each new percentage is taken from a different base.

A one-line check: a price rising from 40 to 46 is (46 − 40) ÷ 40 = 0.15, so +15%. And a shop with 500 visitors on Monday that falls to 400 on Tuesday has dropped 20%, but it needs a 25% rise on Wednesday to get back to 500.

Why +10% then −10% does not return you to the start

Start with 100. Increase it by 10%, and you have 110. Now decrease that by 10%. But 10% of 110 is 11, not 10, so you end at 99, not 100. The second percentage is calculated on a different base.

Change sequence	Start	Middle	End
+10% then −10%	100	110	99
−50% then +50%	100	50	75
−20% then +25%	100	80	100

The bigger the first move, the bigger the gap. After a 50% drop you do not need a 50% rise to recover; you need a 100% rise, because you must double the remaining 50. In general, a fall of p% needs a rise of p ÷ (100 − p) × 100 percent to undo it. A 20% drop needs 20 ÷ 80 = 25% back, as the last row shows.

Growth over several periods

When something grows by a fixed percentage each period, the growth compounds: each period's increase is calculated on a bigger base. Multiply by the growth factor once per period.

Example: 1,000 growing 5% a year for 3 years is 1,000 × 1.05 × 1.05 × 1.05 = 1,000 × 1.157625 = 1,157.63 (rounded to two decimals). The total growth is 15.76%, slightly more than the 15% you would get by simply adding 5% three times.

This is pure arithmetic, so it works for anything that grows at a steady rate: a plant's height, a website's visitors, a subscriber count. It does not predict the future, because real quantities rarely grow at an exactly steady rate. To repeat a multiplication many times, use the power button as explained in the scientific calculator guide.

You can also run the growth backwards. If 1,000 became about 1,157.63 over 3 years at a steady rate, the total factor is 1.15763, and its cube root is about 1.05, so the steady rate was 5% a year. In general, the yearly factor is the total factor raised to the power 1 ÷ number of years.

Reading charts and headlines carefully

Percent change is easy to misread in graphs and news stories. Watch for these points.

A truncated axis exaggerates change. Suppose two bars show 52 and 55, but the vertical axis starts at 50. The bars are drawn 2 and 5 units tall, so the second looks 2.5 times as tall, a visual jump of 150%. The real change is (55 − 52) ÷ 52 = 5.8%.
Small bases give big percentages. Going from 2 to 6 is a 200% rise, yet only 4 units. Always ask what the starting number was.
Percent change from zero is undefined. If the old value is 0, you cannot divide by it. Reports usually say "new" instead of giving a percentage.
Percentage points are different. A rate going from 8% to 10% is 2 percentage points, which is a 25% relative increase. See the percentages guide for the details.

A short step-by-step routine

Identify the old value (the starting point) and the new value.
Subtract: new − old.
Divide by the old value.
Multiply by 100 and attach a plus or minus sign.
To chain changes, multiply the factors (1.10 × 0.90) rather than adding percentages.

FAQ

Is a decrease from 200 to 100 the same as an increase from 100 to 200?
No. The fall is −50% because the base is 200. The rise is +100% because the base is 100. Same two numbers, different direction, different percentage.

How do I combine two percent changes?
Do not add them. Multiply the factors instead. +10% followed by +20% is 1.10 × 1.20 = 1.32, a total rise of 32%, not 30%.

Can a percent decrease be more than 100%?
For a quantity that cannot go below zero, no. A fall of 100% means it reached zero. A drop of more than 100% only makes sense if the value can become negative, such as a temperature.

Open the calculator: Percentage Calculator →