How to write a proportion as a ratio, what a percentage is, how to convert a fraction into a percentage and how to find a percentage of a given quantity, with a set of review exercises on fractions.

Ratios and percentages are two ways of comparing quantities. A ratio comes from the numerators of fractions brought to a common denominator, and a percentage is simply the numerator of a fraction whose denominator is one hundred. This lesson shows both on worked examples.

Ratios

A ratio is formed by the numerators of fractions with the same denominator.

Write the given proportion as a ratio: 3/4 of quantity A, 1/6 of quantity B, 1/12 of quantity C.

\tfrac{3}{4} \cdot \tfrac{3}{3} = \tfrac{9}{12}

quantity A

\tfrac{1}{12}

quantity C

\tfrac{1}{6} \cdot \tfrac{2}{2} = \tfrac{2}{12}

quantity B

9 : 2 : 1

Percentages

The numerator of a fraction with a denominator of 100 is called a percentage.

\tfrac{5}{100} = 5\%

\tfrac{13}{100} = 13\%

\tfrac{12}{25} = \left(\tfrac{12}{25} \cdot 100\right)\% = \left(\tfrac{12}{25} \cdot 25 \cdot 4\right)\% = (12 \cdot 4)\% = 48\%

A fraction can be converted into a percentage by multiplying it by 100.

How do you find a percentage of a given quantity?

24\% \cdot 75 = \tfrac{24}{100} \cdot 75 = \tfrac{24}{25 \cdot 4} \cdot 25 \cdot 3 = \tfrac{24}{4} \cdot 3 = 6 \cdot 3 = 18

8\% \cdot 25 = \tfrac{8}{100} \cdot 25 = \tfrac{8}{4 \cdot 25} \cdot 25 = \tfrac{8}{4} = 2

Review exercises

1) Simplify each of the following fractions to its lowest terms

\tfrac{24}{30} = \tfrac{4 \cdot 6}{5 \cdot 6} = \tfrac{4}{5}

-\tfrac{52}{65} = -\tfrac{4 \cdot 13}{5 \cdot 13} = -\tfrac{4}{5}

\tfrac{72}{15} = \tfrac{24 \cdot 3}{5 \cdot 3} = \tfrac{24}{5}

\tfrac{32}{8} = \tfrac{4 \cdot 8}{8} = \tfrac{4}{1} = 4

2) Evaluate the following expressions

\tfrac{5}{9} \cdot \tfrac{2}{5} = \tfrac{5 \cdot 2}{9 \cdot 5} = \tfrac{2}{9}

\tfrac{13}{25} : \tfrac{2}{15} = \tfrac{13}{25} \cdot \tfrac{15}{2} = \tfrac{39}{10} = 3\tfrac{9}{10}

\tfrac{5}{9} + \tfrac{3}{14} = \tfrac{5 \cdot 14 + 3 \cdot 9}{9 \cdot 14} = \tfrac{70 + 27}{126} = \tfrac{97}{126}

\tfrac{3}{8} - \tfrac{2}{5} = \tfrac{3 \cdot 5 - 2 \cdot 8}{40} = \tfrac{15 - 16}{40} = -\tfrac{1}{40}

\tfrac{12}{7} \cdot \left(-\tfrac{3}{5}\right) = -\tfrac{12 \cdot 3}{7 \cdot 5} = -\tfrac{36}{35} = -1\tfrac{1}{35}

\left(-\tfrac{3}{4}\right) : \left(-\tfrac{12}{7}\right) = \tfrac{3}{4} \cdot \tfrac{7}{12} = \tfrac{7}{16}

\tfrac{19}{2} + \tfrac{7}{4} = \tfrac{19}{2} \cdot \tfrac{2}{2} + \tfrac{7}{4} = \tfrac{38}{4} + \tfrac{7}{4} = \tfrac{45}{4} = 11\tfrac{1}{4}

\tfrac{1}{4} - \tfrac{3}{8} = \tfrac{1}{4} \cdot \tfrac{2}{2} - \tfrac{3}{8} = \tfrac{2}{8} - \tfrac{3}{8} = -\tfrac{1}{8}

3) Write the following proportions as ratios

a) 1/2 of quantity A, 2/5 of quantity B, 1/10 of quantity C.

\tfrac{1}{2} \cdot \tfrac{5}{5} = \tfrac{5}{10}

quantity A

\tfrac{1}{10}

quantity C

\tfrac{2}{5} \cdot \tfrac{2}{2} = \tfrac{4}{10}

quantity B

5 : 4 : 1

b) 1/3 of quantity P, 1/5 of quantity Q, 1/4 of quantity R, remainder S.

\tfrac{1}{3} \cdot \tfrac{20}{20} = \tfrac{20}{60}

quantity P

\tfrac{1}{4} \cdot \tfrac{15}{15} = \tfrac{15}{60}

quantity R

\tfrac{1}{5} \cdot \tfrac{12}{12} = \tfrac{12}{60}

quantity Q

\tfrac{13}{60}

remainder S

20 + 12 + 15 = 47

47 + 13 = 60

20 : 12 : 15 : 13

4) Complete the following

\tfrac{2}{5} = \left(\tfrac{2}{5} \cdot 100\right)\% = \left(\tfrac{2}{5} \cdot 5 \cdot 20\right)\% = 40\%

58\% \cdot 25 = \tfrac{58}{100} \cdot 25 = \tfrac{58}{4 \cdot 25} \cdot 25 = \tfrac{58}{4} = \tfrac{29}{2} = 14\tfrac{1}{2}

\tfrac{7}{12} = \left(\tfrac{7}{12} \cdot 100\right)\% = \left(\tfrac{7}{3 \cdot 4} \cdot 4 \cdot 25\right)\% = \tfrac{7 \cdot 25}{3}\% = \tfrac{175}{3}\% = 58\tfrac{1}{3}\%

17\% \cdot 50 = \tfrac{17}{100} \cdot 50 = \tfrac{17}{2 \cdot 50} \cdot 50 = \tfrac{17}{2} = 8\tfrac{1}{2}

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Frequently asked questions

How do you convert a fraction to a percentage?

Multiply the fraction by one hundred and add the percent sign. For example 12/25 · 100 = 48, so 12/25 is 48 percent.

How do you find a percentage of a number?

Turn the percentage into a fraction over one hundred and multiply by the number. Twenty-four percent of 75 is 24/100 · 75, which is 18.

How do you write a proportion as a ratio?

Bring all the fractions to a common denominator and list the numerators. For 3/4, 1/6 and 1/12 the common denominator is 12, the numerators are 9, 2 and 1, so the ratio is 9 : 2 : 1.

A lesson from the PhiBoard knowledge base: phiboard.com/en/knowledge-base/from-zero-to-engineer/ratios-and-percentages. On the lesson page you will also find games and practice exercises - free and without an account.

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