8.2 Fractions as Percentages

We have just seen that percentages are fractions. Travelling the other direction: given any fraction, can we dress it up as a percentage? Yes, we rebuild it as an equivalent fraction whose denominator is 100.

Worked Example

Kabir mixes blue and white paint for a winter sky. Blue makes up \tfrac{4}{5} of the blend. What percentage of the colour is blue?

We climb to a denominator of 100 using equivalent fractions:

\frac{4}{5} = \frac{8}{10} = \frac{40}{50} = \frac{80}{100} = 80\%.

Or, in one stroke, multiply top and bottom by 20: \dfrac{4}{5} = \dfrac{4\times 20}{5 \times 20} = \dfrac{80}{100} = 80\%. So blue is 80%, and the white that fills the rest is 100\% - 80\% = \mathbf{20\%}.

The Key Idea A fraction measures a part of one unit; a percentage measures a part per 100. So to dress any fraction as a percentage, simply multiply it by 100: \text{fraction as a percentage} = \Big(\text{fraction}\Big)\times 100\,\%. For instance \dfrac{2}{5}\times 100 = 40\%.

To strip the costume off again and recover the fraction, write the percentage over 100 and reduce:

24\% = \frac{24}{100} = \frac{12}{50} = \frac{6}{25}.

More generally, z\% is any fraction equivalent to \dfrac{z}{100}.

Try This: a quick converter Type any fraction below and watch it slip into its decimal and percentage disguises.

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Why 100? Why not just stick with fractions?

If percentages are only fractions in fancy dress, why invent them at all? Because they make comparison effortless. Imagine a snack company testing two cereal bars: sugar is \tfrac{11}{41} of Recipe A and \tfrac{16}{53} of Recipe B. Which is sweeter? Glaring at the bare fractions tells you almost nothing. But rewrite them as percentages,

\frac{11}{41} = 26.83\%, \qquad \frac{16}{53} = 30.19\%

and it is obvious in a heartbeat that Recipe B is sweeter. Pinning every fraction to a shared denominator turns a stubborn comparison into a trivial one, and 100 happens to be the most cooperative denominator going. Because our number system runs on base ten, 31\% = \tfrac{31}{100} = 0.31 glides straight into a decimal. One hundred is big enough to hold meaningful detail, yet round enough to keep in your head.

Did You Know: “Per Hundred” Through the Ages Thinking in hundredths is genuinely old. In the 4th century BCE, Kautilya’s Arthaśhāstra laid out lending rates quoted “per month per cent” for different classes of traders. In the same era the Romans levied duties such as \tfrac{1}{20} and \tfrac{1}{100} on sales and auctions. By the 15th century, Italian merchants’ ledgers carried entries like “xx p cento”, “x p cento”, “vii p cento”, our 20%, 10% and 7% exactly. Across the centuries per cento was clipped shorter and shorter, and the leftover strokes eventually curled into the modern sign %.

Percentages Around Us Percentages crop up everywhere once you start noticing. The human body is roughly 60% water by weight. Ice cream is somewhere between 30% and 50% air by volume. Close to 45% of the planet caught some part of the 2022 FIFA World Cup, and an astonishing 99.86% of the entire Solar System’s mass is parked inside the Sun. Which figure surprises you most?

Figure it Out: Fractions and Percentages

Practice

  1. Express as percentages:   (i) \tfrac{35}{100}   (ii) \tfrac{7}{14}   (iii) \tfrac{11}{20}   (iv) \tfrac{63}{150}   (v) \tfrac{5}{13}   (vi) \tfrac{5}{11}.
  2. Meera has 30 beads, of which 18 are blue. What percentage are blue?   (i) 18%   (ii) 30%   (iii) 40%   (iv) 60%   (v) 50%   (vi) None of these.
  3. In a colony, 24 of the 96 households own a bicycle. What percentage own a bicycle?
  4. Without calculating, fill the blank with >,< or =:   (i) 50\% \;\underline{\quad}\; 5\%   (ii) \tfrac{2}{4}\;\underline{\quad}\;50\%   (iii) \tfrac{3}{8}\;\underline{\quad}\;61\%   (iv) 30\%\;\underline{\quad}\;\tfrac{1}{3}.
    1. \mathbf{35\%}.   (ii) \tfrac{7}{14}=\tfrac12 = \mathbf{50\%}.   (iii) \tfrac{11}{20}=\tfrac{55}{100}=\mathbf{55\%}.   (iv) \tfrac{63}{150}=\tfrac{42}{100}=\mathbf{42\%}.   (v) \tfrac{5}{13}\times 100 \approx \mathbf{38.46\%}.   (vi) \tfrac{5}{11}\times 100 \approx \mathbf{45.45\%}.
  2. \tfrac{18}{30}=\tfrac{60}{100}=\mathbf{60\%}, so option (iv).
  3. \tfrac{24}{96}=\tfrac14=\mathbf{25\%}.
    1. 50\% \;\mathbf{>}\; 5\%.   (ii) \tfrac{2}{4}=50\%, so \mathbf{=}.   (iii) \tfrac{3}{8}= 37.5\% \;\mathbf{<}\; 61\%.   (iv) 30\% \;\mathbf{<}\; \tfrac13 \approx 33.3\%.