How to Calculate Percentages by Hand, Backwards and Over Several Steps

“Per cent” means “per hundred”: 25% is 25 out of every 100, or 0.25. The four basic formulas are short. The mistakes come from mixing them up, such as taking tax off a price the wrong way, or expecting two 10% rises to add up to 20%.

The four everyday formulas

A quick way to do the first one in your head: 10% of a number moves the decimal point one place left, and 1% moves it two. For 12% of 250, that is 25 + 2.5 + 2.5 = 30.

The change formula is measured from the starting value, so it isn’t symmetric: 1,250 to 1,500 is +20%, but 1,500 back to 1,250 is −16.67%.

How do you find the original amount?

Many real questions run the other way: you know the result and the percentage, and need the starting number. Divide by the percentage as a decimal instead of multiplying.

• 12 is 15% of what? 12 ÷ 0.15 = 80.
• You paid 48 after 20% off. What was the price? 48 is 80% of it, so 48 ÷ 0.8 = 60.
• A price of 119 includes 19% tax. What was it before tax? 119 ÷ 1.19 = 100. Subtracting 19% of 119 would give 96.39, which is wrong, because the tax was 19% of the smaller price. The tax calculator does this division for any rate.

Percentages over several steps

Successive percentages multiply; they don’t add. Each step is taken of the amount after the step before.

• Two rises of 10% are 1.1 × 1.1 = 1.21, a 21% rise in total, not 20%.
• Up 50%, then down 50% is 1.5 × 0.5 = 0.75, a 25% loss overall.
• Stacked discounts work the same way: 20% off and then 10% off is 0.8 × 0.9 = 0.72, so 28% off. The discount guide has more examples.

The same rule catches averages of growth rates. A shop that grows 10%, then 20%, then shrinks 15% over three years has grown 1.1 × 1.2 × 0.85 = 1.122, or 12.2%, in total. That is about 3.9% a year compounded, not the 5% you get by averaging 10, 20 and −15. Interest that is added to the balance works like this too, which is how compound interest works.

Percent or percentage points?

When the thing that changes is itself a percentage, say which change you mean. An interest rate that goes from 2% to 3% has risen by 1 percentage point, but by 50% of its old value. Both are correct, and news reports often pick whichever sounds bigger.

In a spreadsheet

Excel, Google Sheets and Numbers all accept a % sign in formulas. With the amount in A1 and the total in B1:

• X% of a number: =A1*15%
• What percent of a total: =A1/B1, then format the cell as a percentage
• Change from A1 to B1: =(B1-A1)/A1
• Price before 19% tax: =A1/(1+19%)

Rounding so the parts add up to 100%

Three equal shares are 33.3% each when rounded to one decimal, which adds up to 99.9%. That isn’t an error in the maths, only in the rounding. Either round one share the other way (33.4%), or add a note that totals may not be exactly 100% because of rounding.