Ratio Notation

Ratio Notation and Simplification

A ratio is a way of comparing two or more quantities of the same type, showing how many times one contains the other. Ratios are written using a colon, e.g. 3 : 5, and — like fractions — should be expressed in their simplest form by dividing all parts by their HCF. Ratios can also be written as unit ratios of the form \( 1 : n \) or \( n : 1 \).

Simplifying Ratios

Divide every part of the ratio by the same value. To reduce to simplest form, divide by the HCF of all parts.

\[ 24 : 36 \xrightarrow{\div 12} 2 : 3 \]

Ratios with fractions or decimals: multiply all parts by the LCM of the denominators (or by 10/100) to clear to integers first, then simplify.

\[ \frac{1}{2} : \frac{3}{4} \xrightarrow{\times 4} 2 : 3 \qquad 0.4 : 1.2 \xrightarrow{\times 10} 4 : 12 \xrightarrow{\div 4} 1 : 3 \]

Unit Ratios

A unit ratio has 1 as one of its parts. Writing a ratio in the form \( 1 : n \) makes comparison easier — it directly shows "for every 1 of the first quantity, there are \(n\) of the second".

\[ 5 : 8 \implies 1 : \frac{8}{5} = 1 : 1.6 \]

Three-Part Ratios

Ratios can have three or more parts: e.g. red : blue : green = 2 : 3 : 5. Simplify by dividing all parts by the HCF.

\[ 12 : 18 : 30 \xrightarrow{\div 6} 2 : 3 : 5 \]

Worked Examples

Simplify: 45 : 60, and 0.6 : 1.5.
\[ 45 : 60 \div 15 = 3 : 4 \] \[ 0.6 : 1.5 \xrightarrow{\times 10} 6 : 15 \xrightarrow{\div 3} 2 : 5 \]
Simplify \( \dfrac{2}{3} : \dfrac{5}{6} \).
\[ \frac{2}{3} : \frac{5}{6} \xrightarrow{\times 6} 4 : 5 \]
Write 3 : 8 as a unit ratio \( 1 : n \). Write 15 : 6 in the form \( n : 1 \).
\[ 3 : 8 \implies 1 : \frac{8}{3} = 1 : 2.\overline{6} \] \[ 15 : 6 \implies \frac{15}{6} : 1 = 2.5 : 1 \]
Simplify 15 : 25 : 35. Write the first quantity as a fraction of the total.
\[ 15 : 25 : 35 \xrightarrow{\div 5} 3 : 5 : 7 \] \[ \text{Total parts} = 3+5+7=15. \quad \text{First as fraction of total} = \frac{3}{15} = \frac{1}{5} \]

 Key Takeaways

  • Simplify a ratio by dividing all parts by the same value — ideally the HCF for a single step.
  • To clear fractions or decimals: multiply all parts by the LCM of denominators (fractions) or an appropriate power of 10 (decimals) first.
  • Unit ratio \( 1 : n \): divide both parts by the first part. Unit ratio \( n : 1 \): divide both parts by the second part.
  • For a three-part ratio, divide all three parts by the same number.
  • Ratio is a comparison of quantities in the same unit — units cancel out. Ratios are dimensionless.