Ratio and Proportion: Sharing, Rates and Worked Examples

Comparing Quantities

A recipe uses two cups of rice for every cup of beans. Concrete is mixed one part cement to two parts sand to four parts gravel. A map shows 1 cm for every 50,000 cm on the ground. Business partners share profit according to how much each invested. All of these compare quantities using ratios, and many everyday calculations, such as costs, recipes, speeds and work rates, depend on proportion. This tutorial covers ratio and proportion as tested in WAEC, NECO, BECE and JAMB mathematics: simplifying ratios, sharing in a given ratio, combining ratios, increasing and decreasing in a ratio, direct and inverse proportion, the unitary method, rates such as speed and unit price, and map scales, with worked examples.

What Is a Ratio?

A ratio compares two or more quantities of the same kind, measured in the same units. The ratio of a to b is written a : b, or as the fraction a/b.

A ratio has no units. If a class has 15 boys and 20 girls, the ratio of boys to girls is 15 : 20, which simplifies to 3 : 4. The order matters: the ratio of girls to boys is 4 : 3.

Simplifying ratios

Divide all the parts by their highest common factor, exactly as for fractions. Convert to the same units first, and clear fractions or decimals by multiplying.

Worked example 1

Simplify (a) 45 : 60 (b) 75 cm : 2 m (c) 1½ : 2¼.

(a) Divide by 15: 3 : 4.

(b) Convert to centimetres: 75 : 200. Divide by 25: 3 : 8.

Ratios in the form 1 : n

To compare ratios easily, divide by the first part. For example, 4 : 10 = 1 : 2.5. Map scales and gear ratios are often written this way.

Sharing in a Given Ratio

To share an amount in the ratio a : b : c:

Add the parts to find the total number of shares: a + b + c.
Find the value of one share: amount ÷ total shares.
Multiply by each part.

Worked example 2

Ada and Bayo share ₦240,000 in the ratio 3 : 5. How much does each receive?

Total shares = 3 + 5 = 8, so one share = ₦240,000 ÷ 8 = ₦30,000.

Ada receives 3 × ₦30,000 = ₦90,000, and Bayo receives 5 × ₦30,000 = ₦150,000. Check: 90,000 + 150,000 = 240,000.

Worked example 3: concrete

Concrete is mixed with cement, sand and gravel in the ratio 1 : 2 : 4. How much of each is needed for 3.5 tonnes of concrete?

Total shares = 7, so one share = 0.5 tonnes. Cement = 0.5 t, sand = 1 t, gravel = 2 t.

Worked example 4: working back from one share

Money is shared between Chidi and Dami in the ratio 4 : 7. Dami receives ₦18,000 more than Chidi. How much was shared?

The difference is 7 − 4 = 3 shares, so 3 shares = ₦18,000, and one share = ₦6,000.

The total is 11 shares = ₦66,000.

Combining Ratios

If a : b and b : c are given separately, make the common term b the same in both.

Worked example 5

If a : b = 2 : 3 and b : c = 4 : 5, find a : b : c.

The values of b are 3 and 4, whose lowest common multiple is 12. Multiply the first ratio by 4 and the second by 3: a : b = 8 : 12, and b : c = 12 : 15.

So a : b : c = 8 : 12 : 15.

Increasing or Decreasing in a Ratio

To change a quantity in the ratio new : old, multiply it by the fraction new/old.

Worked example 6

(a) Increase ₦3,600 in the ratio 5 : 4. (b) A school's enrolment of 1,200 falls in the ratio 7 : 8. Find the new enrolment.

(a) 3,600 × 5/4 = ₦4,500.

(b) 1,200 × 7/8 = 1,050 students.

A ratio greater than 1 (like 5 : 4) increases the quantity, and a ratio less than 1 (like 7 : 8) decreases it.

Direct Proportion

Two quantities are in direct proportion if they increase or decrease together at the same rate: doubling one doubles the other, and their ratio stays constant. The cost of identical items and the number bought, or the distance travelled at a steady speed and the time taken, are examples.

The unitary method

First find the value for one unit, then multiply.

Worked example 7

Five textbooks cost ₦7,500. How much do eight cost?

One textbook costs ₦7,500 ÷ 5 = ₦1,500, so eight cost 8 × ₦1,500 = ₦12,000.

Alternatively, by ratio: 8/5 × ₦7,500 = ₦12,000.

Inverse Proportion

Two quantities are in inverse proportion if one increases as the other decreases, so that their product stays constant: doubling one halves the other. The number of workers and the time to finish a job, or speed and the time taken for a fixed journey, are examples.

Worked example 8

Six workers can build a fence in 10 days. How long would four workers take, working at the same rate?

The total work is 6 × 10 = 60 worker-days. Four workers need 60 ÷ 4 = 15 days.

Fewer workers take longer, which is the check that the proportion is inverse.

Worked example 9

Food in a camp lasts 30 days for 120 students. If 30 more students arrive, how long will the food last?

Total = 30 × 120 = 3,600 student-days. With 150 students: 3,600 ÷ 150 = 24 days.

Before solving any proportion problem, ask: "If one quantity doubles, does the other double (direct) or halve (inverse)?"

Rates

A rate compares two quantities of different kinds, so it has units, such as km/h, ₦ per litre or litres per minute.

Worked example 10: speed

A bus travels 120 km in 1 hour 30 minutes. Find its average speed in km/h and in m/s.

Speed = 120 ÷ 1.5 = 80 km/h.

To convert km/h to m/s, multiply by 1000/3600, which is the same as dividing by 3.6: 80 ÷ 3.6 ≈ 22.2 m/s.

Worked example 11: best buy

A 500 g tin of milk powder costs ₦2,400, and a 900 g tin costs ₦4,140. Which is better value?

500 g: ₦2,400 ÷ 500 = ₦4.80 per gram. 900 g: ₦4,140 ÷ 900 = ₦4.60 per gram.

The 900 g tin is better value.

Worked example 12: currency

Suppose the exchange rate is $1 = ₦1,500. How many naira is $250, and how many dollars is ₦60,000?

$250 = 250 × 1,500 = ₦375,000. ₦60,000 = 60,000 ÷ 1,500 = $40.

Map Scales

A map scale of 1 : 50,000 means that 1 cm on the map represents 50,000 cm (0.5 km) on the ground.

Worked example 13

On a map with scale 1 : 50,000, two towns are 4.2 cm apart. Find the actual distance in kilometres.

4.2 × 50,000 = 210,000 cm. Since 1 km = 100,000 cm, this is 2.1 km.

For areas, the scale factor is squared, as covered in the tutorial on similar triangles.

Common Mistakes with Ratio and Proportion

Comparing quantities in different units, such as 75 cm : 2 m written as 75 : 2.
Reversing the order of the ratio. "Boys to girls" means boys first.
Dividing by one part instead of the total when sharing. For 3 : 5, divide by 8.
Treating inverse proportion as direct, for example thinking fewer workers finish sooner.
Forgetting the units of a rate, such as km/h or ₦ per gram.

Ratio and Proportion: Practice Questions

Simplify 1.2 kg : 800 g.
Share ₦45,000 among three sisters in the ratio 2 : 3 : 4.
If x : y = 3 : 4 and y : z = 6 : 5, find x : y : z.
Seven pens cost ₦1,050. Find the cost of twelve pens.
A journey takes 4 hours at 60 km/h. How long does it take at 80 km/h?
Convert 72 km/h to m/s.
On a map with scale 1 : 25,000, a road is 8 cm long. Find its actual length in km.

Answers: 1. 1200 : 800 = 3 : 2. 2. One share = ₦45,000 ÷ 9 = ₦5,000, so ₦10,000, ₦15,000 and ₦20,000. 3. Make y equal to 12: x : y = 9 : 12 and y : z = 12 : 10, so x : y : z = 9 : 12 : 10. 4. One pen costs ₦150, so twelve cost ₦1,800. 5. This is inverse proportion: 4 × 60 ÷ 80 = 3 hours. 6. 72 ÷ 3.6 = 20 m/s. 7. 8 × 25,000 = 200,000 cm = 2 km.

Conclusion

A ratio compares quantities of the same kind in the same units, and it is simplified like a fraction. To share in a ratio, divide by the total number of shares, then multiply. Combine ratios by matching the common term, and change a quantity in a ratio by multiplying by new/old. In direct proportion the ratio stays constant, so use the unitary method. In inverse proportion the product stays constant. Rates such as speed and unit price compare different kinds of quantity, and map scales turn map distances into real distances.