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From simplifying ratios to solving proportion equations, master every ratio question type on the IGCSE paper.
A ratio compares quantities in the same units (e.g. 3:5). Proportion describes how quantities change together. Direct proportion means as one increases, the other increases at the same rate (y = kx). Inverse proportion means as one increases, the other decreases (y = k/x). Ratio and proportion carry 6-10 marks on the IGCSE paper and often appear as multi-step real-world problems.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
A ratio compares two or more quantities using the same units. To simplify a ratio, divide every part by the highest common factor (HCF). For example, 12:18 simplifies to 2:3 because the HCF of 12 and 18 is 6.
When a ratio involves fractions or decimals, multiply through to make whole numbers first. For example, 1.5 : 2.5 becomes 3 : 5 (multiply both by 2), and 1/3 : 1/4 becomes 4 : 3 (multiply both by 12, the LCM of the denominators).
Two quantities are in direct proportion if they increase and decrease at the same rate. If you double one, the other doubles. The relationship is written as y ∝ x, which means y = kx where k is the constant of proportionality.
The graph of a direct proportion relationship is a straight line through the origin. If the line does not pass through (0, 0), the relationship is linear but not directly proportional.
Two quantities are in inverse proportion if one increases as the other decreases. If you double one, the other halves. The relationship is y ∝ 1/x, which gives y = k/x.
The solving method is identical to direct proportion: substitute to find k, then use the equation to find the unknown.
On the Higher tier, proportion questions may involve squared, cubed, or square root relationships. The method is the same — only the equation changes:
The key is translating the proportionality statement into the correct equation. Once you have the equation, substitute to find k and then solve for the unknown — exactly as before.
Share 360 in the ratio 2 : 3 : 4.
Step 1 (Total parts): 2 + 3 + 4 = 9 parts
Step 2 (Value of one part): 360 / 9 = 40
Step 3 (Multiply): 2 x 40 = 80, 3 x 40 = 120, 4 x 40 = 160
Step 4 (Check): 80 + 120 + 160 = 360 ✓
Answer: 80, 120, 160
y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
Step 1 (Write equation): y = kx
Step 2 (Find k): 20 = k x 4, so k = 5
Step 3 (Complete equation): y = 5x
Step 4 (Substitute): y = 5 x 7 = 35
Answer: y = 35
y is inversely proportional to x. When x = 3, y = 12. Find y when x = 9.
Step 1 (Write equation): y = k/x
Step 2 (Find k): 12 = k/3, so k = 36
Step 3 (Complete equation): y = 36/x
Step 4 (Substitute): y = 36/9 = 4
Answer: y = 4
y is directly proportional to x². When x = 3, y = 45. Find x when y = 125.
Step 1 (Write equation): y = kx²
Step 2 (Find k): 45 = k x 3² = 9k, so k = 5
Step 3 (Complete equation): y = 5x²
Step 4 (Substitute and solve): 125 = 5x², so x² = 25, therefore x = 5
Answer: x = 5
Dividing by the wrong number when sharing: Students divide by one of the ratio numbers instead of the total parts. For 3:5, divide by 8, not by 3 or 5.
Confusing direct and inverse proportion: If doubling x halves y, the relationship is inverse (y = k/x), not direct. Read the question carefully to identify which type applies.
Forgetting to square x in y ∝ x² problems: When the proportion involves a power, you must apply the power to x before multiplying by k. If y = kx² and x = 3, use 9 not 3.
Not simplifying ratios with different units: Before writing a ratio, convert quantities to the same unit. A ratio of 2 m to 50 cm is 200:50 = 4:1, not 2:50.
Write the proportionality equation first: Translate "y is directly proportional to x squared" into y = kx² immediately. This earns the first method mark.
Always find k and state the full equation: Even if the question does not ask for k, writing it out shows clear working and earns marks.
Check your ratio answer adds up: When sharing in a ratio, the sum of all parts must equal the original quantity. A quick addition check catches arithmetic errors.
Use a table for complex proportion: Organise x and y values in a two-column table to keep your working tidy, especially when multiple substitutions are required.
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