Ratios
Section: Number and Calculation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What is a Ratio?
A ratio is a way of comparing two or more quantities of the same type, written as a : b (read as "a to b").
- Ratios compare quantities of the same unit (e.g. metres to metres, kg to kg).
- The order matters: 3:5 is not the same as 5:3.
- Ratios can be simplified, just like fractions.
- Ratios have no units - they are pure numbers.
Example: if a class has 3 boys and 5 girls, the ratio of boys to girls is 3:5.
Writing Ratios
To write a ratio from a real situation, put the quantities in the order the question asks for, separated by a colon.
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Question: In a class of 30 students, there are 18 girls and 12 boys. Write the ratio of: (a) girls to boys (b) boys to girls (c) girls to total students.
- (a) Girls to boys = 18:12
- (b) Boys to girls = 12:18
- (c) Girls to total students = 18:30
Simplifying Ratios
Just like fractions, a ratio can be simplified by dividing all parts by their Highest Common Factor (HCF).
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Question: Simplify 18:12.
- Step 1: HCF of 18 and 12 = 6
- Step 2: 18 ÷ 6 = 3, and 12 ÷ 6 = 2
- Answer: 3:2
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Question: Simplify 24:36:60.
- Step 1: HCF of 24, 36 and 60 = 12
- Step 2: 24 ÷ 12 = 2, 36 ÷ 12 = 3, 60 ÷ 12 = 5
- Answer: 2:3:5
Ratios with Different Units
Before writing a ratio, both quantities must be converted to the same units.
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Question: Write the ratio of 2 m to 50 cm in simplest form.
- Step 1: Convert to the same units: 2 m = 200 cm
- Step 2: Ratio: 200:50
- Step 3: Simplify (÷ 50): 4:1
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Question: Write the ratio of 1.5 kg to 250 g in simplest form.
- Step 1: Convert: 1.5 kg = 1500 g
- Step 2: Ratio: 1500:250
- Step 3: Simplify (÷ 250): 6:1
Sharing in a Given Ratio
To share an amount in a given ratio, use the "parts" method:
- Add up all parts of the ratio to find the total number of parts.
- Divide the total amount by the total parts to find the value of one part.
- Multiply each ratio part by the value of one part.
£120 shared in the ratio 2:3, using the bar model
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Question: Share £120 in the ratio 2:3.
- Step 1: Total parts = 2 + 3 = 5
- Step 2: One part = £120 ÷ 5 = £24
- Step 3: First share = 2 × £24 = £48. Second share = 3 × £24 = £72
- Check: £48 + £72 = £120 ✓
- Answer: £48 and £72
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Question: Divide 500 g of sweets between three children in the ratio 2:3:5.
- Step 1: Total parts = 2 + 3 + 5 = 10
- Step 2: One part = 500 ÷ 10 = 50 g
- Step 3: Child 1 = 100 g, Child 2 = 150 g, Child 3 = 250 g
- Answer: 100 g, 150 g, 250 g
Finding Quantities from Ratios
If you know the ratio and one of the actual quantities, find the value of one part first, then scale up to find the others.
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Question: The ratio of cats to dogs is 3:5. If there are 15 cats, how many dogs are there?
- Step 1: 3 parts = 15 cats, so 1 part = 15 ÷ 3 = 5
- Step 2: Dogs = 5 parts = 5 × 5 = 25
- Answer: 25 dogs
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Question: In a recipe, the ratio of flour to sugar is 4:1. If you use 300 g of flour, how much sugar do you need?
- Step 1: 4 parts = 300 g, so 1 part = 300 ÷ 4 = 75 g
- Step 2: Sugar = 1 part = 75 g
- Answer: 75 g
Finding a Total from One Part
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Question: Boys and girls in a class are in the ratio 3:5. There are 18 boys. Find (a) the number of girls (b) the total number of students.
- Step 1: 3 parts = 18 boys, so 1 part = 18 ÷ 3 = 6
- Step 2 (a): Girls = 5 parts = 5 × 6 = 30
- Step 3 (b): Total = 8 parts = 8 × 6 = 48
- Answer: (a) 30 girls (b) 48 students
Ratio and Fraction Connection
A ratio a:b can be written as the fraction a/(a+b) for the first quantity and b/(a+b) for the second, since a+b represents the whole.
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Question: Money is shared in the ratio 2:3. What fraction does each person get?
- Step 1: Total parts = 2 + 3 = 5
- Step 2: First person gets 2/5, second person gets 3/5
- Answer: 2/5 and 3/5
Map Scales and Ratios
A map scale of 1:50 000 means 1 cm on the map represents 50 000 cm in real life.
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Question: A map has scale 1:25 000. Two towns are 8 cm apart on the map. What is the actual distance?
- Step 1: 1 cm on the map = 25 000 cm in reality
- Step 2: 8 cm on the map = 8 × 25 000 = 200 000 cm
- Step 3: Convert to km: 200 000 cm = 2000 m = 2 km
- Answer: 2 km
Comparing Ratios
To compare two ratios, convert each to a fraction of its own total and compare those fractions.
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Question: Which mixture has a stronger orange flavour? Mixture A - orange to water = 2:5. Mixture B - orange to water = 3:7.
- Step 1: Mixture A: orange fraction = 2/(2+5) = 2/7 ≈ 0.286
- Step 2: Mixture B: orange fraction = 3/(3+7) = 3/10 = 0.3
- Step 3: 0.3 > 0.286
- Answer: Mixture B has the stronger orange flavour
Direct and Inverse Proportion
Two quantities are in direct proportion when they increase or decrease at the same rate - if one doubles, so does the other. Two quantities are in inverse proportion when one increases as the other decreases, in such a way that their product stays constant.
Direct Proportion
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Question: A car uses 8 litres of fuel to travel 100 km. Assuming fuel use is directly proportional to distance, how much fuel is needed to travel 250 km?
- Step 1: Fuel per km = 8 ÷ 100 = 0.08 litres/km
- Step 2: Fuel for 250 km = 250 × 0.08 = 20 litres
- Answer: 20 litres
Inverse Proportion
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