Applying Ratio

Applying Ratio

Ratio problems appear across a wide range of real-world contexts — sharing quantities, scaling recipes, mixing paints, comparing prices, and converting currencies. The key technique is to find the value of one part and then multiply to find any number of parts. This "unitary method" works reliably regardless of the context.

Sharing in a Given Ratio

  1. Add the ratio parts to find the total number of parts.
  2. Divide the total quantity by the total number of parts to find the value of one part.
  3. Multiply by the relevant ratio number for each share.
\[ \text{Share } £360 \text{ in ratio } 3:5: \quad 3+5=8 \text{ parts} \implies 1 \text{ part} = £45 \implies 3 \text{ parts} = £135,\ 5 \text{ parts} = £225 \]

Scaling (Recipes and Mixing)

To scale a recipe or mixture, find the scale factor (ratio of desired to given) and multiply every ingredient by it.

\[ \text{If a recipe for 4 serves uses 300 g flour, for 10 serves: scale factor} = \frac{10}{4} = 2.5 \implies 300 \times 2.5 = 750 \text{ g} \]

Finding a Quantity Given the Ratio and One Part

If you know the value of one part, multiply by the ratio number to find other parts, or by the total to find the whole.

Example: in ratio 2 : 5, if the first share is 14 kg, then one part = 7 kg, so the second share = 35 kg and the total = 49 kg.

Ratio and Proportion Connections

Many proportion problems are disguised ratio problems. "If 6 items cost £9, what do 10 cost?" — this is a ratio comparison resolved by finding the unit cost: £9 ÷ 6 = £1.50 per item, so 10 × £1.50 = £15.

Worked Examples

Share £480 between Alice and Ben in the ratio 5 : 3.
\[ 5+3=8 \text{ parts}. \quad 1 \text{ part} = \frac{480}{8} = £60. \] \[ \text{Alice} = 5 \times 60 = £300. \quad \text{Ben} = 3 \times 60 = £180. \]

Check: \( 300+180=480 \). ✓

A paint mix uses red : white = 2 : 5. How much red paint is needed to make 21 litres of mixed paint?
\[ 2+5=7 \text{ parts}. \quad 1 \text{ part} = \frac{21}{7} = 3 \text{ litres}. \quad \text{Red} = 2 \times 3 = 6 \text{ litres}. \]
A mortar mix needs cement : sand = 1 : 4. A builder has 12 kg of cement. How much sand is needed?
\[ \text{Sand} = 4 \times 12 = 48 \text{ kg}. \]
Two quantities are in the ratio 3 : 7. The smaller is 15 kg. Find the larger quantity and the total.
\[ 1 \text{ part} = \frac{15}{3} = 5 \text{ kg}. \quad \text{Larger} = 7 \times 5 = 35 \text{ kg}. \quad \text{Total} = 15+35 = 50 \text{ kg}. \]

 Key Takeaways

  • Sharing in ratio: (1) total parts = sum of ratio; (2) one part = total ÷ total parts; (3) each share = ratio number × one part.
  • Always verify by checking all shares sum to the original total.
  • Scaling problems: find the scale factor (new ÷ original) and multiply every ingredient/component by it.
  • If one part is given: find the value of 1 ratio unit, then multiply up for other parts or for the total.
  • The unitary method — finding the value of one unit — is the most reliable approach to all ratio and proportion problems.