Calculator of Alexandria

Percentage calculator

Find a percent of a number, what percent one number is of another, or the whole from a part and a percent, with the reasoning shown.

Solve for

As a number, so 15 for 15%.

Formula

X=p100 WX = \frac{p}{100}\,W
How it's derived

The formula

X=p100 WX = \frac{p}{100}\,W

XX is the part, pp the percent and WW the whole. Every everyday percentage question is this one equation solved for a different letter:

  • What is 15% of 240? Solve for XX.
  • 36 is what percent of 240? Solve for pp.
  • 36 is 15% of what? Solve for WW.

Where it comes from

“Per cent” is Latin for “per hundred”. Saying 15% means 15 out of every 100, the fraction 15/100=0.1515/100 = 0.15. Taking a fraction of something means multiplying by it, so 15% of 240 is 0.15×2400.15 \times 240.

Rearranging gives the other two questions. Divide both sides by WW and multiply by 100 to find the percent:

p=XW×100p = \frac{X}{W} \times 100

Or divide both sides by p/100p/100 to find the whole:

W=Xp/100=100 XpW = \frac{X}{p/100} = \frac{100\,X}{p}

Worked example

A $240 jacket is 15% off. The discount is

X=15100×240=36X = \frac{15}{100} \times 240 = 36

so you save $36. Going the other way, if you saved $36 on a $240 jacket, p=36/240×100=15%p = 36/240 \times 100 = 15\%.

Mental shortcuts

  • Percentages swap. 8% of 50 equals 50% of 8, which is 4, because both are 8×50/1008 \times 50 / 100.
  • 10% is a decimal shift. Build other percentages from it. 15% is 10% plus half of 10%.

Watch out for

  • “Of what?” questions. If a sale price of $204 is 15% off, the original is not 204×1.15204 \times 1.15. The 15% was taken from the original, so the sale price is 85% of it: 204/0.85=240204 / 0.85 = 240.
  • Percent vs percentage points. A rate going from 4% to 5% rose by 1 percentage point, but by 25 percent.

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