Skip to content

One amount as a percent of another

Ratio, rates and money · level 4, lesson 3 of 6

A question to start

Ava gets two tests back on the same day.

  • Math: 18 out of 24.
  • Science: 21 out of 30.

She scored more marks in science (21 is more than 18), so was science her better test?

Not so fast. The science test had more marks to win. Getting 21 out of 30 means she missed 9 marks; getting 18 out of 24 means she missed only 6. But even "marks missed" is not a fair comparison, because the tests are different sizes.

To compare fairly, we need to ask: if each test had been out of 100, what would she have scored? That is exactly what a percent tells you. By the end of this lesson you will be able to turn any "this out of that" into a percent, and you will know which test was really better.

The idea

The question "what percent?"

"What percent of 24 is 18?" means: if 24 were stretched to 100, what would 18 become?

The part and the whole make a ratio, part : whole \(= 18 : 24\). We want an equivalent ratio with 100 as the whole: \(? : 100\). A percent is just a part-to-whole ratio scaled so the whole is 100.

0 6 12 18 24 marks 0% 25% 50% 75% 100%

Split 24 into four equal quarters of 6. Then 18 is three quarters, and three quarters of 100% is 75%.

Three ways to find it

Way 1: simplify, then use a benchmark. \(\tfrac{18}{24}\): both divide by 6, giving \(\tfrac34\), which is 75%.

Way 2: scale the whole to 100. Sometimes the whole goes neatly into 100. For 21 out of 30, that is awkward, but for 13 out of 20 it is easy: multiply top and bottom by 5 to get \(\tfrac{65}{100}\), which is 65%.

Way 3: divide, then multiply by 100. This always works:

\[\text{percent} = \frac{\text{part}}{\text{whole}} \times 100\%.\]

For science: \(21 \div 30 = 0.7\), and \(0.7 \times 100 = 70\%\).

So Ava's math test (75%) was better than her science test (70%), even though she scored fewer marks in math.

Why "divide, then times 100" works

Dividing part by whole gives the fraction of the whole as a decimal: \(21 \div 30 = 0.7\) means "seven tenths of the test". Multiplying by 100 counts how many hundredths that is. It is the same move as in lesson 1, where a decimal became a percent.

Which number is the whole?

Everything depends on dividing by the right number. The whole is the number you are comparing with: it comes after "of" or "out of".

  • "18 out of 24": the whole is 24.
  • "What percent of the class walks?": the whole is the class.
  • "What percent of last year's sales is this year's?": the whole is last year's sales.

A helpful habit is to say the question as a sentence: "part is what percent of whole?" The number after "of" goes on the bottom.

Rounding

Many percents do not come out even. 7 out of 9 is \(7 \div 9 = 0.7777...\), which is \(77.77...\%\). We usually round, for example to one decimal place: 77.8%. Say "about 77.8%" so the reader knows it is rounded.

More than 100%

If the part is bigger than the whole, the percent is over 100%. A class aimed to raise 120 dollars and raised 150 dollars: \(150 \div 120 = 1.25\), which is 125% of the target. They beat it by 25%.

Same units first

Before you divide, the part and the whole must be in the same unit. 45 minutes as a percent of 2 hours: change 2 hours to 120 minutes first. Then \(45 \div 120 = 0.375 = 37.5\%\).

Comparing with percents

Percents turn results with different totals into results with the same total, 100. That is why sports statistics, election results and test reports use them. Raw numbers can mislead: a team with 22 wins out of 30 games won more games than a team with 15 wins out of 20, but the second team won a bigger share of its games (75% against about 73.3%).

Worked examples

1. A test score. 34 out of 40 as a percent.

  • Scale: \(40 \times 2.5 = 100\), so \(34 \times 2.5 = 85\). 85%.
  • Or: \(34 \div 40 = 0.85\), which is 85%.

2. A survey. In a survey of 40 children, 14 said soccer was their favourite sport. What percent is that?

  • \(14 \div 40 = 0.35\), so 35%.
  • Sense check: 14 is a bit more than a quarter of 40 (which is 10) and less than half (20). 35% is between 25% and 50%. ✓

3. Rounding. 5 out of 7 as a percent.

  • \(5 \div 7 = 0.714285...\)
  • That is about 71.4%.

4. More than the whole. A plant was 40 cm tall in spring and is 58 cm tall in autumn. What percent of its spring height is its autumn height?

  • \(58 \div 40 = 1.45\), so 145%.
  • It is now its whole spring height plus 45% more.

5. A day on Mars. A day on Earth (from noon to noon) is 24 hours, which is 86 400 seconds. A day on Mars, called a sol, is about 24 hours 39 minutes 35 seconds. What percent of an Earth day is a sol?

  • A sol in seconds: \(24 \times 3600 + 39 \times 60 + 35 = 86\,400 + 2340 + 35 = 88\,775\) seconds.
  • \(88\,775 \div 86\,400 \approx 1.0275\), so a sol is about 102.7% of an Earth day: under 3% longer.

6. Mixed units. 300 g as a percent of 1.2 kg.

  • 1.2 kg is 1200 g.
  • \(300 \div 1200 = 0.25\), so 25%.

Try it

  1. Write 12 out of 50 as a percent.
  2. Write 9 out of 20 as a percent.
  3. Write 17 out of 25 as a percent.
  4. Jun scored 34 out of 40 on one quiz and 42 out of 50 on another. Which was the better result?
  5. A basketball player scores 5 of her 8 free throws. What percent is that?
  6. A tank holds 80 litres and has 30 litres in it. What percent full is it? What percent empty?
  7. What percent of 2 hours is 45 minutes?
  8. A school aimed to collect 400 cans of food and collected 520. What percent of the target did they collect?
Answers
  1. \(\tfrac{12}{50} = \tfrac{24}{100}\) (times 2), so 24%.
  2. \(\tfrac{9}{20} = \tfrac{45}{100}\) (times 5), so 45%.
  3. \(\tfrac{17}{25} = \tfrac{68}{100}\) (times 4), so 68%.
  4. \(34 \div 40 = 0.85 = 85\%\) and \(42 \div 50 = 0.84 = 84\%\). The 34 out of 40 was slightly better, even though it is fewer marks.
  5. \(5 \div 8 = 0.625\), so 62.5%.
  6. \(30 \div 80 = 0.375\), so 37.5% full. Empty: \(100\% - 37.5\% = 62.5\%\) (check: \(50 \div 80 = 0.625\)).
  7. 2 hours is 120 minutes. \(45 \div 120 = 0.375\), so 37.5%.
  8. \(520 \div 400 = 1.3\), so 130% of the target.

Common mistake

Dividing by the wrong number. "What percent of 40 is 14?" is \(14 \div 40\), not \(40 \div 14\). If your answer is over 100% but the part is smaller than the whole, you have divided the wrong way. Ask: "out of what?" That number goes on the bottom.

Comparing raw numbers instead of percents. 21 marks is more than 18 marks, but 21 out of 30 is a lower score than 18 out of 24. When the totals differ, compare percents.

Where it leads