Ratio, proportion and scale · Year 10
In Ratio, proportion and scale (Year 10) you practise 5 skills: A ratio and its meaning · Proportion and rates · Dividing in a ratio · Scale · Areas under a scale. A ratio compares two amounts of the same kind. Before writing a ratio, write both amounts in the same unit: 250 g of sugar to a kilogram of flour is 250 to 1,000, which simplifies to one to four. A ratio simplifies like a fraction, by dividing both parts by the same number.
A proportion says that two ratios are equal. When one part is missing, call it x and cross multiply: the product of the outer parts equals the product of the inner parts. A rate, such as kilometres per litre or people per square kilometre, compares amounts of different kinds; multiply or divide by it. To divide a whole in a ratio, add the parts of the ratio to find how many equal parts there are. Divide the whole by that number, then multiply by each part. The same idea gives the sides of a rectangle from their ratio and its perimeter or area. A scale is the ratio of a length in a drawing or on a map to the real length, and its first part is one. At a scale of one to a hundred, each centimetre in the drawing is a hundred centimetres in real life. From the drawing to real life, multiply; from real life to the drawing, divide, once both lengths are in the same unit. At a scale of one to a hundred every length grows a hundred times, but an area grows a hundred times a hundred, that is ten thousand times. So find the real lengths first, and only then multiply them to get the area.
Tip: Before any working, write the amounts in the same unit.
What you learn
A ratio and its meaning
For example: A recipe uses 750 g of sugar and 1 kg of flour. What is the ratio of the sugar to the flour? Write the ratio in its simplest form.
Answer and solution
The answer: 3 : 4
In the same unit, 1 kg is 1,000 g. The ratio is: 750 : 1,000 Simplify by dividing both parts by 250: 750 : 1,000 = 3 : 4
Proportion and rates
For example: In a recipe, 10 eggs are enough for 14 biscuits. How many eggs are needed for 21 biscuits?
Answer and solution
The answer: 15
The proportion, with the unknown amount as x: 10 : 14 = x : 21 Cross multiply: 14 × x = 10 × 21 x = 210 ÷ 14 = 15
Dividing in a ratio
For example: A sum of £2,990 is shared between three partners in the ratio 2 : 4 : 7 . How much does the first partner get?
Answer and solution
The answer: £460
The number of equal parts: 2 + 4 + 7 = 13 One part: 2,990 ÷ 13 = 230 The first partner has 2 parts: 2 × 230 = 460 That is £460.
Scale
For example: A drawing of a wall has a scale of 1 : 25 . The wall is 20 cm long in the drawing. How long is the real wall, in metres?
Answer and solution
The answer: 5 m
Each centimetre is 25 cm in real life: 20 × 25 = 500 That is 500 cm, which is 5 m.
Areas under a scale
For example: A plan of a flat is drawn at a scale of 1 : 50 . The flat is a rectangle 12 cm by 4 cm. How long is the long side of the real flat, in metres?
Answer and solution
The answer: 6 m
Each centimetre on the plan is 50 cm in real life, so: 12 × 50 = 600 That is 600 cm, which is 6 m.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).