Ratio: Dividing and Applying

What is a Ratio?

A ratio compares quantities of the same kind. The ratio $a : b$ means for every $a$ parts of one quantity, there are $b$ parts of another.

Simplifying ratios: Divide all parts by the HCF (highest common factor).

• $12 : 18 = 2 : 3$ (HCF = 6)
• $0.5 : 1.5 = 1 : 3$ (multiply both by 2 first to clear decimals)

Dividing a Quantity in a Given Ratio

Method:

1. Add the parts of the ratio to find the total number of shares.
2. Divide the quantity by the total to find the value of one share.
3. Multiply by each part.

Example: Divide £240 in the ratio $3 : 5$.

• Total shares: $3 + 5 = 8$
• One share: $£240 \div 8 = £30$
• First part: $3 \times £30 = £90$
• Second part: $5 \times £30 = £150$
• Check: $£90 + £150 = £240$ ✓

Example (three-way): Share 360 g in the ratio $1 : 2 : 3$.

• Total: $1 + 2 + 3 = 6$ shares
• One share: $360 \div 6 = 60$ g
• Parts: 60 g, 120 g, 180 g

Finding a Ratio Given One Part

If you know one share and the total quantity, you can find the ratio.

Example: Ali gets £120 and Beth gets £80 from a prize. Write the ratio of Ali's share to Beth's. $$120 : 80 = 3 : 2 \text{ (divide by HCF 40)}$$

Ratio and Fractions

The ratio $a : b$ means the first quantity is $\frac{a}{a+b}$ of the total.

Example: In the ratio $2 : 3$, the first quantity is $\frac{2}{5}$ of the total, and the second is $\frac{3}{5}$.

Recipes and Scaling (Proportion Context)

Ratios appear in recipe and mixing problems: if a recipe uses 200 g flour for 4 people, for 10 people you need $200 \times \frac{10}{4} = 500$ g.

Finding an Unknown from a Ratio

Example: The ratio of red to blue sweets is $3 : 7$. There are 18 red sweets. How many blue sweets are there?

$$\frac{\text{blue}}{\text{red}} = \frac{7}{3}$$ $$\text{blue} = 18 \times \frac{7}{3} = 42$$

Or: $3$ parts = 18, so 1 part = 6, so 7 parts = 42.

WJEC Exam Tips

• Always check your answer by adding the parts — they should sum to the original quantity.
• Ratios must be in the same units (convert first if needed).
• For 3-part ratios, the method is identical — just add all three parts.
• "Best value" problems: find unit cost (cost per gram, cost per litre) and compare.