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Cambridge IGCSE Mathematics — 0580 Extended

Topic 1.10: Number — Ratio & Proportion

Simplifying and dividing in ratio

A ratio compares parts of the same kind. Write it in its simplest form by dividing all parts by their HCF.

Example: \(12 : 18 = 2 : 3\).

Method — divide \(Q\) in ratio \(a : b\)

  1. Add the parts: total parts = \(a + b\).
  2. One part = \(Q \div (a + b)\).
  3. Multiply: first share = one part \(\times a\); second share = one part \(\times b\).

Divide \(240\) in the ratio \(2 : 3 : 5\).

Worked solution: 240 divided in the ratio 2 to 3 to 5 gives 48, 72 and 120
10 parts, so one part is 24: shares \(48\), \(72\), \(120\)

The ratio of red to blue paint is \(4 : 7\). How much blue paint is needed with \(20\) ml of red?

Worked solution: 20 ml of red in the ratio 4 to 7 needs 35 ml of blue
One part is \(5\) ml, so blue is \(7 \times 5 = 35\) ml

Proportional reasoning

If two quantities are directly proportional, they increase and decrease together. Multiply or divide both by the same factor.

Five identical pens cost \($12.50\). Find the cost of eight pens.

Worked solution: five pens cost 12.50 so eight pens cost 20.00
Unitary method: one pen is \($2.50\), so eight cost \($20.00\)

Ratio or proportion?

Sharing a fixed total into parts

Divide in ratio — find one part first.

One quantity changes and the other scales with it

Use proportion — find the unit value, then scale up or down.

Map or scale drawing

Write scale as a ratio; convert using consistent units.

Map scales and best value

A map scale \(1 : 50\,000\) means \(1\) cm on the map represents \(50\,000\) cm (\(= 0.5\) km) in real life.

On a map with scale \(1 : 25\,000\), two towns are \(4\) cm apart. Find the real distance in km.

Worked solution: 4 cm on a 1 to 25000 map is 1 km in real life
\(4 \times 25\,000 = 100\,000\) cm \(= 1\) km

Best value

  1. Find the cost per unit for each option (same units).
  2. Compare — the lower cost per unit is better value.
  3. Check the question: it may ask for cheapest total or best value per kg/litre/item.

Pack A: \(500\) g for \($3.50\). Pack B: \(750\) g for \($4.80\). Which is better value?

Worked solution: pack B at 0.0064 dollars per gram is better value than pack A
Pack B costs less per gram, so it is better value

Try this

Divide \($180\) in the ratio \(1 : 2 : 3\). A recipe uses flour and sugar in ratio \(5 : 2\). If you use \(300\) g flour, how much sugar?

Show answer
Answer

  1. Total parts \(1 + 2 + 3 = 6\); one part \(= 180 \div 6 = 30\)

    \[ 1 \times 30 = 30,\quad 2 \times 30 = 60,\quad 3 \times 30 = 90 \]

  2. Flour is 5 parts \(= 300\) g, so one part \(= 60\) g

    \[ \text{sugar} = 2 \times 60 = 120 \text{ g} \]

  3. Final answers

    \[ \$30,\; \$60,\; \$90 \quad\text{and}\quad 120\text{ g} \]

Exam Traps

  • Map scales need unit conversion — \(25\,000\) cm is \(250\) m, not \(25\) km. Convert cm → m → km in steps.
  • The cheaper pack is not always better value. Compare cost per gram (or per litre), not the sticker price.

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