Skip to content

Percent of an amount

Ratio, rates and money · level 4, lesson 2 of 6

A question to start

Which is bigger: 8% of 25, or 25% of 8?

Most people reach for a calculator. But you can do both in your head in a few seconds, and the answer is a small surprise: they are exactly the same. Both are 2.

Is that a coincidence? Would 7% of 40 and 40% of 7 also match? By the end of this lesson you will know a few fast ways to find any percent of any amount, and you will be able to explain why the swap always works.

The idea

"Of" means multiply

You already know that "half of 10" is 5, which is \(\tfrac12 \times 10\). In math, the little word of between a fraction and an amount means multiply. Percents are fractions with 100 at the bottom, so

\[35\% \text{ of } 80 = \frac{35}{100} \times 80.\]

That is the whole idea. The rest of this lesson is about doing it quickly and sensibly.

Start from 100%

100% of an amount is the whole amount. From there, a few easy steps get you anywhere.

  • 50%: halve it. 50% of 80 is 40.
  • 25%: halve again. 25% of 80 is 20.
  • 10%: divide by 10. 10% of 80 is 8.
  • 5%: halve the 10%. 5% of 80 is 4.
  • 1%: divide by 100. 1% of 80 is 0.8.

Why is 10% "divide by 10"? Because 10% is \(\tfrac{10}{100} = \tfrac{1}{10}\). On the hundred grid, 10% is one column out of ten equal columns. Sharing 80 equally into 10 columns puts 8 in each.

Build up from pieces

Any percent can be built from these pieces, the way you build an amount of money from coins.

\[35\% = 10\% + 10\% + 10\% + 5\%\]

So 35% of 80 is \(8 + 8 + 8 + 4 = 28\).

A double number line shows it. The top line is the amount, the bottom line is the percent, and marks that line up go together.

0 8 24 28 40 80 0% 10% 30% 35% 50% 100%

Some other useful builds:

  • \(15\% = 10\% + 5\%\).
  • \(20\% = 10\% + 10\%\), or \(10\%\) doubled.
  • \(90\% = 100\% - 10\%\) (sometimes taking away is faster than adding).
  • \(12.5\%\) is half of \(25\%\).
  • \(2\% = 1\% + 1\%\).

Using decimals

The build-up method is great in your head. With a calculator, or with awkward percents like 37%, change the percent to a decimal and multiply:

\[35\% \text{ of } 80 = 0.35 \times 80 = 28.\]

This is the same calculation as before, because \(0.35 = \tfrac{35}{100}\). Two different routes, one answer.

Why the swap works

Back to the opening puzzle. Write both with "of means multiply":

\[8\% \text{ of } 25 = \frac{8}{100} \times 25 = \frac{8 \times 25}{100}, \qquad 25\% \text{ of } 8 = \frac{25}{100} \times 8 = \frac{25 \times 8}{100}.\]

Both are "the two numbers multiplied together, then divided by 100". And \(8 \times 25 = 25 \times 8\), because multiplying works in either order. So the answers must be equal: both are \(\tfrac{200}{100} = 2\).

So you can always swap: \(a\%\) of \(b\) equals \(b\%\) of \(a\). Use whichever is easier. 4% of 75 sounds hard; 75% of 4 is obviously 3. And 7% of 40 is 40% of 7, which is \(0.4 \times 7 = 2.8\).

Percents over 100

The same tools work for percents bigger than 100. 150% of 40 is \(100\% + 50\%\), so \(40 + 20 = 60\). With a decimal: \(1.5 \times 40 = 60\). Taking more than 100% of something gives more than you started with.

Does the answer make sense?

Always do a quick sense check:

  • A percent under 100% gives an answer smaller than the amount.
  • 50% is half, 25% is a quarter, 10% is a tenth: compare your answer with these.
  • 35% is a bit more than a third, so 35% of 80 should be a bit more than \(80 \div 3 \approx 27\). And 28 is. ✓

Worked examples

1. A mental build. Find 15% of 60.

  • 10% of 60 is 6. 5% is half of that, 3.
  • 15% is \(6 + 3 = 9\).

2. Taking away. Find 90% of 250.

  • 10% of 250 is 25.
  • 90% is \(250 - 25 = 225\).

3. Using a decimal. Find 37% of 450.

  • \(0.37 \times 450 = 166.5\).
  • Sense check: 37% is a bit under 40%, and 40% of 450 is \(4 \times 45 = 180\). Our answer is a bit under 180. ✓

4. The swap. Find 4% of 75 and 16% of 25.

  • 4% of 75 is 75% of 4, which is 3.
  • 16% of 25 is 25% of 16, a quarter of 16, which is 4.

5. A small percent. Find 2.5% of 360.

  • 10% is 36, so 5% is 18, and 2.5% is 9.

6. A survey. In a class of 30, 40% walk to school, 30% take the bus, and the rest come by car. How many come by car?

  • The percents must add to 100%, so by car is \(100\% - 40\% - 30\% = 30\%\).
  • 10% of 30 is 3, so 30% is 9 children.
  • Check the whole class: walking is 40% of 30, which is 12; bus is 9; car is 9. \(12 + 9 + 9 = 30\). ✓

Try it

  1. Find 10% of 70, of 340 and of 5.
  2. Find 50%, 25% and 1% of 240.
  3. Find 15% of 80 without a calculator.
  4. Find 45% of 60 without a calculator.
  5. Find 65% of 120 using a decimal.
  6. Find 4% of 75 and 8% of 50. (Try the swap.)
  7. Find 120% of 35.
  8. A shop gives 20% of the price of every book to a library fund. How much goes to the fund from a book costing 14.50 dollars? From three books at 14.50 dollars each?
Answers
  1. \(70 \div 10 = 7\); \(340 \div 10 = 34\); \(5 \div 10 = 0.5\).
  2. 50% is 120; 25% is 60; 1% is 2.4.
  3. 10% is 8 and 5% is 4, so 15% is \(8 + 4 = 12\).
  4. 10% is 6, so 40% is 24; 5% is 3. 45% is \(24 + 3 = 27\). (Or: 50% is 30, minus 5% which is 3, gives 27.)
  5. \(0.65 \times 120 = 78\). Sense check: a bit under two thirds of 120 (which is 80). ✓
  6. 4% of 75 is 75% of 4, which is 3. 8% of 50 is 50% of 8, which is 4.
  7. 100% is 35 and 20% is 7, so 120% is \(35 + 7 = 42\). (Decimal: \(1.2 \times 35 = 42\).)
  8. 10% of 14.50 is 1.45, so 20% is 2.90 dollars. For three books: \(3 \times 2.90 = 8.70\) dollars (or 20% of \(3 \times 14.50 = 43.50\), which is also 8.70 dollars).

Common mistake

Dividing by the percent. "15% of 60" is not \(60 \div 15 = 4\). A percent is a number of hundredths, so you multiply by \(\tfrac{15}{100}\). The only percent that works by dividing by its own number is 10% (divide by 10), and that is because \(10\% = \tfrac{1}{10}\).

Writing the decimal wrong. 5% of 80 is \(0.05 \times 80 = 4\), not \(0.5 \times 80 = 40\). A sense check catches it: 5% is tiny, so the answer must be small.

Where it leads