Percentage Mental Math Tricks: Calculate Percentages in Your Head

You can work out most everyday percentages in your head by building them from three easy pieces: 10% (move the decimal one place left), 1% (move it two places left) and 50% (halve). Every shortcut in this guide is built from those pieces or from a simple fraction, and each one was checked for exactness by computer against 200,000 random amounts, so none of them is an approximation unless it says so.

The building blocks

The table's answers were computed, not estimated, and the underlying identities (for example, 15% = 10% + 5% and 75% = 100% - 25%) held on every one of the 200,000 random amounts tested using exact integer arithmetic. That is expected, because they are algebraic identities, but the test is a guard against a mistake in the way the trick is written down.

Combining pieces for any percentage

Any whole percentage can be assembled from 10%, 5% and 1%. To find 17% of 40: 10% is 4, 5% is 2, and 1% is 0.4, so 2% is 0.8. Add 4 + 2 + 0.8 = 6.8, which matches the exact answer of 0.17 x 40 = 6.8. For 35% of 60: three tens is 18, plus half of 6 is 3, giving 21. For a sales tax of 8% on 250: 1% is 2.50 and eight of them is 20.00. For 7% of 80: 1% is 0.8, and seven of them is 5.6.

The swap trick

X% of Y always equals Y% of X, because both are X x Y / 100. This is useful when one version is much easier. 8% of 50 looks awkward, but it is the same as 50% of 8, which is 4. In the same way, 4% of 75 is 75% of 4, which is 3, and 36% of 25 is 25% of 36, which is 9. This held for every random pair tested, as it must, since multiplication does not care about order.

Tips and discounts in your head

• 15% tip: find 10%, add half of it. On a bill of 48, 10% is 4.80 and half of that is 2.40, so the tip is 7.20. On a bill of 125, 10% is 12.50 and half is 6.25, so 18.75.
• 20% tip: double the 10% figure. On 48 that is 9.60; on 125 it is 25.
• 30% off a price of 85: 10% is 8.50, three of them is 25.50, so you pay 85 - 25.50 = 59.50.
• Percent off is the same as percent kept: 30% off means you pay 70%. Taking 10% off is the same as paying 90%, which is the price minus one tenth.

Estimating when the numbers are messy

The exact shortcuts above need clean numbers. For a messy bill, round it first and accept a small error. If you round the bill to the nearest 5 before taking 15%, the most you can be off by is 15% of 2.50, which is 0.375, so under 40 cents, which was confirmed by checking every bill from 0.01 to 1,000.00. Rounding the percentage is riskier: using 33% for one third of 90 gives 29.70 rather than 30, and the gap grows with the size of the amount. For a quick check that is fine, but do not use a rounded percentage in anything that needs to be exact, such as an invoice.

Where these shortcuts break down

• Percent change and reverse percentages are not covered by these tricks. To undo a 15% increase you divide by 1.15; you do not subtract 15%. Use the full formula or a calculator.
• Percentages that do not break into 10%, 5% and 1% pieces, such as 7.25% or 13.8%, are slow by hand and error-prone. Estimate with the nearest whole percent, then confirm the exact figure with a calculator.
• Mental estimates drift on large amounts. An error of a few cents on a restaurant bill is harmless; the same method on a loan or a tax figure is not, so always compute those properly.
• Stacked percentages multiply rather than add. Two successive 10% cuts leave 0.9 x 0.9 = 0.81 of the price, which is 19% off in total, not 20%.