Calculator of Alexandria

Percentage change calculator

Work out the percentage increase or decrease between two values, or find the new value after a percentage change.

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Formula

Δ%=x2−x1∣x1∣×100\Delta\% = \frac{x_2 - x_1}{|x_1|} \times 100
How it's derived

The formula

Δ%=x2−x1∣x1∣×100\Delta\% = \frac{x_2 - x_1}{|x_1|} \times 100

x1x_1 is the original value and x2x_2 the new one. A positive result is an increase, a negative one a decrease.

Where it comes from

A percentage is a fraction scaled so the whole is 100. A change is measured against where you started, so the starting value is the whole:

fraction changed=changeoriginal=x2−x1x1\text{fraction changed} = \frac{\text{change}}{\text{original}} = \frac{x_2 - x_1}{x_1}

Multiplying by 100 turns the fraction into a percentage. Going the other way, a change of p%p\% adds p/100p/100 of the original:

x2=x1+x1⋅p100=x1(1+p100)x_2 = x_1 + x_1 \cdot \frac{p}{100} = x_1\left(1 + \frac{p}{100}\right)

The absolute value in the denominator handles negative starting values, so a move from −10 to −15 reads as a 50% decrease rather than an increase.

Worked example

A price rises from 80 to 92:

Δ%=92−8080×100=15%\Delta\% = \frac{92 - 80}{80} \times 100 = 15\%

Watch out for

  • Up and back down isn’t zero. A 50% rise followed by a 50% fall leaves you at 75% of where you started, because the second change is measured from a bigger base.
  • Percent vs percentage points. An interest rate going from 2% to 3% is a 1 percentage point rise, but a 50% increase.

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