Ratio and proportion
Section: Number 2 | Syllabus: Cambridge IGCSE Mathematics (0580)
Ratios in Simplest Form and Dividing a Quantity
A ratio compares two or more quantities by size. Writing a ratio in its simplest form, and splitting an amount according to a ratio, are two sides of the same idea.
- A ratio is simplified the same way as a fraction: divide every part by their highest common factor
- To divide a quantity in a given ratio:
- Step 1: Add the parts of the ratio to find the total number of shares
- Step 2: Divide the quantity by the total number of shares to find the value of one share
- Step 3: Multiply one share by each part of the ratio
Worked Example: Simplifying a Ratio
- Question: Three items cost 45, 27 and 63. Write the ratio of their costs in simplest form.
- Step 1: Write the ratio directly from the costs: 45 : 27 : 63
- Step 2: Divide every part by their HCF, 9: 45 ÷ 9 : 27 ÷ 9 : 63 ÷ 9
- Answer: 5 : 3 : 7
Worked Example: Dividing a Quantity in a Given Ratio
- Question: A school hall has 1500 seats. On sports day, 3/5 of the seats are filled. The ratio of adults to children in the filled seats is 7:2. Find the number of children.
- Step 1: Find the number of filled seats: (3/5) × 1500 = 900
- Step 2: The 900 filled seats split into 7 + 2 = 9 shares, so one share is 900 ÷ 9 = 100
- Step 3: Children's share is 2 parts: 2 × 100 = 200
- Answer: 200 children
Common Mistakes
MistakeSimplifying a ratio using a common factor that isn't the highest one, leaving it not fully simplified, e.g. writing 45:27:63 as 15:9:21
Fixalways find the highest common factor of all the parts - 15:9:21 can still be divided by 3 to reach 5:3:7
MistakeDividing the total by the number of parts written down (e.g. by 2, since a ratio has "2 numbers") instead of by the sum of the parts
Fixalways add the parts of the ratio together first - for a ratio of 7:2, divide by 7 + 2 = 9, not by 2
Map Scales
A map scale relates a distance on the map to the real distance it represents. Writing the scale in the form 1:n makes it easy to convert between the two, as long as both sides are in the same unit first.
- A scale such as "1 cm represents 500 m" can be written as a ratio 1 : n
- Before writing the ratio, convert both parts to the same unit - usually the smaller unit
- To find a real-world distance from a map distance, multiply the map distance by n
Worked Example: Writing a Scale in the Form 1:n
- Question: A map has a scale where 1 cm represents 500 m. Write this scale in the form 1:n.
- Step 1: Convert 500 m to cm: 500 × 100 = 50\,000 cm
- Answer: 1 : 50\,000
Worked Example: Using a Scale to Find a Real Distance
- Question: On a map with scale 1:50 000, the distance between two towns is measured as 8 cm. Find the real distance in km.
- Step 1: Multiply the map distance by n: 8 × 50\,000 = 400\,000 cm
- Step 2: Convert to km: 400\,000 cm = 4000 m = 4 km
- Answer: 4 km
Common Mistakes
MistakeForgetting to convert units before writing the scale, e.g. writing "1 cm represents 500 m" as the ratio 1:500
Fixboth parts of a 1:n scale must be in the same unit - convert 500 m to 50 000 cm first, giving 1:50 000
MistakeLeaving a calculated real-world distance in centimetres instead of converting to a sensible unit like metres or kilometres
Fixalways convert the final answer into a unit that suits the context - 400 000 cm is much clearer written as 4 km
Best Value and Recipe Scaling
Proportional reasoning shows up whenever quantities need to be compared fairly or scaled up and down - working out which shop item is genuinely cheaper, or adjusting a recipe for a different number of people.
- To find best value, compare the price per unit (e.g. price per kg or price per item) for each option - the smaller unit price is the better deal
- To scale a recipe (or any set of quantities in fixed proportion), find the scale factor by dividing the new amount by the original amount, then multiply every ingredient by that same factor
Worked Example: Determining Best Value
- Question: A shop sells rice in a 4 kg bag for 6.80, and in a 6 kg bag for 9.90. Which bag is better value?
- Step 1: Cost per kg for the 4 kg bag: 6.80 ÷ 4 = 1.70
- Step 2: Cost per kg for the 6 kg bag: 9.90 ÷ 6 = 1.65
- Answer: The 6 kg bag is better value, since 1.65 per kg is less than 1.70 per kg
Worked Example: Scaling a Recipe
- Question: A recipe for 8 pancakes uses 240 g of flour and 3 eggs. How much flour and how many eggs are needed for 20 pancakes?
- Step 1: Find the scale factor: 20 ÷ 8 = 2.5
- Step 2: Scale the flour: 240 × 2.5 = 600 g
- Step 3: Scale the eggs: 3 × 2.5 = 7.5, which rounds up to 8 eggs, since a whole number of eggs is needed and 7 would not be quite enough
- Answer: 600 g flour and 8 eggs
Common Mistakes
MistakeComparing total prices directly to judge value, e.g. saying the 6.80 bag is cheaper than the $9.90 bag without considering the different sizes
Fixalways compare the price per unit (per kg, per item) - the bag with the lower total price is not always the better value
MistakeScaling only one ingredient of a recipe and leaving the others unchanged
Fixevery ingredient must be multiplied by the same scale factor to keep the recipe in the correct proportion
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