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Percent means per hundred

Ratio, rates and money · level 4, lesson 1 of 6

A question to start

At Maple School, \(\tfrac35\) of the pupils walk to school. At Birch School, \(\tfrac{13}{20}\) of the pupils walk. Which school has the bigger share of walkers?

You could find a common denominator and compare. But a newspaper would never print "three fifths" against "thirteen twentieths". It would print 60% against 65%, and you would see the answer in a blink: Birch.

Why does that work so well? Because percent is a way of writing every fraction with the same bottom number, 100. In this lesson you will see what that means in pictures, and learn to move freely between fractions, decimals and percents, which are three ways of writing the same amount.

The idea

"Per cent" means "for each hundred"

The word percent comes from the Latin per centum, "for each hundred". The sign % is short for it.

\[35\% \text{ means } 35 \text{ out of every } 100, \text{ or } \frac{35}{100}.\]

So a percent is a ratio where the second number is always 100: \(35\%\) is the ratio \(35 : 100\). Fixing that number at 100 is the whole trick. It is like measuring everything with the same ruler.

The hundred grid

Picture a square cut into \(10 \times 10 = 100\) small squares. The whole square is 100%. Each small square is 1%.

50% 25% 10% 1%
  • 50% is 50 squares out of 100: half the grid. So \(50\% = \tfrac12\).
  • 25% is 25 squares: a quarter of the grid. So \(25\% = \tfrac14\).
  • 10% is 10 squares, one column: a tenth of the grid. So \(10\% = \tfrac1{10}\).
  • 1% is one square: a hundredth. So \(1\% = \tfrac1{100}\).

One amount, three names

Shade 35 squares. That is \(\tfrac{35}{100}\) of the grid. In decimals, \(\tfrac{35}{100}\) is 0.35: 3 tenths and 5 hundredths. (3 full columns are 3 tenths, and 5 extra squares are 5 hundredths.) And in percent it is 35%.

35% = 35/100 = 0.35
\[\frac{35}{100} = 0.35 = 35\%.\]

These are not three different amounts. They are three names for the same shaded part, like "a dozen", "12" and "twelve". The decimal and the percent use the same digits because a percent counts hundredths, and the second decimal place is the hundredths place.

Moving between the three

Percent to decimal: divide by 100. That moves the digits two places right, or the decimal point two places left: \(35\% = 0.35\), \(7\% = 0.07\), \(120\% = 1.2\), \(0.5\% = 0.005\).

Decimal to percent: multiply by 100. \(0.6 = 60\%\), \(0.125 = 12.5\%\), \(2 = 200\%\).

Watch \(7\%\). It is 7 hundredths, so the decimal is \(0.07\), not \(0.7\) (which is 70 hundredths, \(70\%\)).

Percent to fraction: put it over 100 and simplify. \(35\% = \tfrac{35}{100} = \tfrac{7}{20}\). \(12.5\% = \tfrac{12.5}{100} = \tfrac{125}{1000} = \tfrac18\).

Fraction to percent. Two ways:

  1. If you can, make the bottom 100 with an equivalent fraction: \(\tfrac35 = \tfrac{60}{100} = 60\%\) (times 20 top and bottom), and \(\tfrac{13}{20} = \tfrac{65}{100} = 65\%\) (times 5).
  2. If not, divide top by bottom to get a decimal, then multiply by 100: \(\tfrac38 = 3 \div 8 = 0.375 = 37.5\%\).

Here is the opening question in grids:

3/5 = 60% 13/20 = 65%

Benchmarks worth knowing by heart

Fraction Decimal Percent
\(\tfrac12\) 0.5 50%
\(\tfrac14\) 0.25 25%
\(\tfrac34\) 0.75 75%
\(\tfrac15\) 0.2 20%
\(\tfrac1{10}\) 0.1 10%
\(\tfrac18\) 0.125 12.5%
\(\tfrac13\) 0.333... about 33.3%
\(\tfrac23\) 0.666... about 66.7%

The last two never come out exactly, because 3 does not divide 100 evenly. \(\tfrac13\) is \(33\tfrac13\%\), which you can round to 33.3%.

Less than 1%, more than 100%

Less than 1% is part of one small square. \(0.5\%\) is half a square: 1 out of every 200.

More than 100% means more than the whole. If this year a school has 150% of last year's pupils, it has the whole of last year's number plus half as many again: one full grid and half of another. \(150\% = 1.5 = 1\tfrac12\).

Exactly 100% is the whole thing. "Give 110%" is a figure of speech; when you are sharing one whole into parts, the parts can never add to more than 100%.

Why percent is useful

Fractions with different bottoms are hard to compare. Percents all have the same bottom, so comparing them is just comparing whole numbers. That is why percents appear on test results, battery icons, nutrition labels, weather forecasts ("40% chance of rain") and in the news.

A percent always has an "of what?". Mars's surface gravity is about 38% of Earth's. That means a rock that weighs 100 newtons on Earth would weigh about 38 newtons on Mars: 38 for every 100. The "of what" here is Earth's gravity.

Worked examples

1. A fraction with an easy bottom. Write \(\tfrac{9}{25}\) as a percent.

  • \(25 \times 4 = 100\), so multiply top and bottom by 4: \(\tfrac{9}{25} = \tfrac{36}{100}\).
  • That is 36% (and 0.36).

2. A fraction with a harder bottom. Write \(\tfrac78\) as a percent.

  • 8 does not go into 100 evenly, so divide: \(7 \div 8 = 0.875\).
  • Multiply by 100: 87.5%.
  • Check with the benchmark: \(\tfrac18 = 12.5\%\), so \(\tfrac78 = 100\% - 12.5\% = 87.5\%\). ✓

3. A percent to a fraction. Write \(45\%\) as a fraction in simplest form.

  • \(45\% = \tfrac{45}{100}\).
  • Both divide by 5: \(\tfrac{9}{20}\).

4. Small percents. Write \(3\%\) and \(0.4\%\) as decimals.

  • \(3\% = 3 \div 100 = 0.03\).
  • \(0.4\% = 0.4 \div 100 = 0.004\).

5. Comparing. Put in order from smallest to largest: \(\tfrac{2}{5}\), \(0.38\), \(41\%\), \(\tfrac{3}{8}\).

  • Write each as a percent: \(\tfrac25 = 40\%\); \(0.38 = 38\%\); \(41\%\); \(\tfrac38 = 37.5\%\).
  • Order: \(\tfrac38\) (37.5%), \(0.38\) (38%), \(\tfrac25\) (40%), \(41\%\).

Try it

  1. Shade-in-your-head: how many small squares of a hundred grid are 20%? 75%? 5%?
  2. Write as decimals: 45%, 8%, 130%.
  3. Write as percents: 0.9, 0.04, 1.75.
  4. Write as percents: \(\tfrac{7}{10}\), \(\tfrac{3}{4}\), \(\tfrac{11}{20}\), \(\tfrac{1}{8}\).
  5. Write as fractions in simplest form: 60%, 15%, 2%.
  6. A phone battery shows 0.6 of its charge left. A tablet shows \(\tfrac58\). Which has more charge left, as a fraction of its full charge?
  7. Put in order from smallest to largest: \(\tfrac{1}{3}\), 30%, 0.35, \(\tfrac{3}{10}\).
  8. Raj says \(\tfrac{1}{4} = 4\%\) because "a quarter is 1 out of 4". What would you tell him?
Answers
  1. 20% is 20 squares (2 columns); 75% is 75 squares (7 and a half columns); 5% is 5 squares (half a column).
  2. \(45\% = 0.45\); \(8\% = 0.08\); \(130\% = 1.3\).
  3. \(0.9 = 90\%\); \(0.04 = 4\%\); \(1.75 = 175\%\).
  4. \(\tfrac{7}{10} = \tfrac{70}{100} = 70\%\); \(\tfrac34 = \tfrac{75}{100} = 75\%\); \(\tfrac{11}{20} = \tfrac{55}{100} = 55\%\); \(\tfrac18 = 1 \div 8 = 0.125 = 12.5\%\).
  5. \(60\% = \tfrac{60}{100} = \tfrac35\); \(15\% = \tfrac{15}{100} = \tfrac{3}{20}\); \(2\% = \tfrac{2}{100} = \tfrac{1}{50}\).
  6. Phone: \(0.6 = 60\%\). Tablet: \(\tfrac58 = 5 \div 8 = 0.625 = 62.5\%\). The tablet has a bigger share of its charge left.
  7. \(\tfrac13 \approx 33.3\%\); \(30\%\); \(0.35 = 35\%\); \(\tfrac{3}{10} = 30\%\). Order: \(30\%\) and \(\tfrac{3}{10}\) (equal, both 30%), then \(\tfrac13\), then \(0.35\).
  8. Percent means out of 100, not out of 4. A quarter of the hundred grid is 25 squares, so \(\tfrac14 = \tfrac{25}{100} = 25\%\). And 4% is only 4 squares, which is \(\tfrac{1}{25}\).

Common mistake

Moving the decimal point the wrong number of places. \(5\%\) is \(0.05\), not \(0.5\). Check with the grid: 5% is 5 small squares, half a column, and \(0.5\) would be 5 whole columns.

Reading the digits of a fraction as a percent. \(\tfrac{3}{5}\) is not 35%, and \(\tfrac14\) is not 14% or 4%. The percent is how many hundredths, so change the fraction to hundredths first.

Where it leads