Writing ratios

Ratios are a way to compare two or more quantities. Writing and simplifying ratios is an essential skill in mathematics that allows us to express relationships in their simplest form, making them easier to interpret and use.

1. Writing Ratios:

Ratios compare two or more quantities and can be written in three main ways:

• Using the word "to": $3\text{to}5$
• Using a colon: $3:5$
• As a fraction: $\frac{3}{5}$

Ratios can involve more than two quantities. For example, if a recipe uses 2 parts sugar, 3 parts flour, and 1 part butter, the ratio is $2:3:1$.

2. Simplifying Ratios:

To simplify a ratio, divide all parts of the ratio by their highest common factor (HCF). This reduces the ratio to its simplest form while keeping the relationship between the quantities the same.

Example 1:

Simplify the ratio $12:8$:

• The HCF of 12 and 8 is 4.
• Divide both numbers by 4: $\frac{12}{4}:\frac{8}{4}=3:2$.

The simplified ratio is $3:2$.

Example 2:

Simplify the ratio $18:24:6$:

• The HCF of 18, 24, and 6 is 6.
• Divide each number by 6: $\frac{18}{6}:\frac{24}{6}:\frac{6}{6}=3:4:1$.

The simplified ratio is $3:4:1$.

3. Ratios with Different Units:

When quantities are measured in different units, convert them to the same unit before writing the ratio. For example:

Example: A car travels 120 km, and a bike travels 30 miles. Write the ratio of the distances travelled.

• Convert 30 miles to kilometres (1 mile ≈ 1.609 km): $30\times 1.609=48.27\text{km}$.
• Write the ratio as $120:48.27$.
• Simplify by dividing both numbers by their HCF (approximately 24): $\frac{120}{24}:\frac{48.27}{24}\approx 5:2$.

The simplified ratio is $5:2$.

5. Summary:

• Ratios can be written using "to," a colon, or as fractions.
• To simplify a ratio, divide all terms by their HCF.
• Convert units if necessary before writing a ratio.