This article covers the essential skills of rate, ratio, direct proportion, inverse proportion, and estimation – a recurring topic in DSE Paper 1 Section A(1) worth 3–5 marks. You will learn how to simplify ratios, solve direct and inverse proportion problems, convert between rates and ratios, and apply estimation techniques to real-world problems. The article includes step-by-step worked examples, DSE exam techniques, practice questions, and a full set of homework with solutions. These are straightforward marks that every candidate should secure.
By the end of this article, you should be able to:
Rate, ratio, and proportion problems appear every year in DSE Paper 1 Section A(1). Questions may ask you to:
These are straightforward marks if you know the rules. Don't lose them!
Questions on rate, ratio, and proportion often appear in Section A(1) as short-answer questions worth 2–3 marks each. They are designed to test your basic numeracy and algebraic skills.
A ratio compares two or more quantities of the same kind. It can be written as \( a:b \) or \( a:b:c \).
To simplify a ratio, divide all terms by their highest common factor (HCF).
| Ratio | Simplified Form | Method |
|---|---|---|
| \( 12:18 \) | \( 2:3 \) | Divide by HCF = 6 |
| \( 0.5:1.5 \) | \( 1:3 \) | Multiply by 10, then divide by HCF = 5 |
| \( \frac{1}{2}:\frac{1}{3} \) | \( 3:2 \) | Multiply by LCM of denominators (6) |
| \( 2.4:3.6:4.8 \) | \( 2:3:4 \) | Multiply by 10, divide by HCF = 12 |
When simplifying ratios, always divide by the HCF – just like simplifying fractions!
If \( a:b = c:d \), then \( ad = bc \) (cross-multiplication).
If \( 3:5 = x:15 \), find \( x \).
\( 3 \times 15 = 5 \times x \) → \( 45 = 5x \) → \( x = 9 \)
A rate compares two quantities of different kinds. For example, speed (km/h), density (g/cm³), or population density (people/km²).
| Rate | Conversion |
|---|---|
| Speed: \( \text{km/h} \to \text{m/s} \) | Multiply by \( \frac{5}{18} \) |
| Speed: \( \text{m/s} \to \text{km/h} \) | Multiply by \( \frac{18}{5} \) |
| Density: \( \text{g/cm}^3 \to \text{kg/m}^3 \) | Multiply by 1000 |
Convert \( 72 \text{ km/h} \) to \( \text{m/s} \).
\( 72 \times \frac{5}{18} = 20 \text{ m/s} \)
Two quantities \( x \) and \( y \) are in direct proportion if:
where \( k \) is the constant of proportionality.
As \( x \) increases, \( y \) increases at the same rate.
Question: \( y \) is directly proportional to \( x \). When \( x = 4 \), \( y = 20 \). Find \( y \) when \( x = 7 \).
\( y = kx \) → \( 20 = k \times 4 \) → \( k = 5 \)
\( y = 5 \times 7 = 35 \)
Answer: \( y = 35 \)
Two quantities \( x \) and \( y \) are in inverse proportion if:
where \( k \) is the constant of proportionality.
As \( x \) increases, \( y \) decreases proportionally.
Question: \( y \) is inversely proportional to \( x \). When \( x = 3 \), \( y = 12 \). Find \( y \) when \( x = 6 \).
\( y = \frac{k}{x} \) → \( 12 = \frac{k}{3} \) → \( k = 36 \)
\( y = \frac{36}{6} = 6 \)
Answer: \( y = 6 \)
Do not confuse direct and inverse proportion. In direct proportion, \( \frac{y}{x} \) is constant. In inverse proportion, \( xy \) is constant.
Estimation involves approximating a value using rounding or other techniques. In DSE, you may be asked to estimate a calculation by rounding to 1 significant figure.
Estimate \( \frac{48.3 \times 19.8}{5.02} \).
Round: \( 48.3 \approx 50 \), \( 19.8 \approx 20 \), \( 5.02 \approx 5 \)
\( \frac{50 \times 20}{5} = \frac{1000}{5} = 200 \)
Estimated answer: 200
Question: Simplify \( 0.25:0.75:1.25 \).
Multiply by 100: \( 25:75:125 \)
Divide by HCF = 25: \( 1:3:5 \)
Answer: \( 1:3:5 \)
Question: \( y \) is directly proportional to the square of \( x \). When \( x = 2 \), \( y = 12 \). Find \( y \) when \( x = 5 \).
\( y = kx^2 \) → \( 12 = k \times 4 \) → \( k = 3 \)
\( y = 3 \times 25 = 75 \)
Answer: \( y = 75 \)
Question: It takes 6 workers 8 days to complete a job. How many days would it take 4 workers to complete the same job? (Assume all workers work at the same rate.)
Workers and days are inversely proportional.
\( \text{Workers} \times \text{Days} = k \)
\( 6 \times 8 = 48 \) → \( k = 48 \)
\( 4 \times \text{Days} = 48 \) → \( \text{Days} = 12 \)
Answer: 12 days
Question 1 MC
Simplify \( 0.6:1.2:2.4 \).
A. \( 1:2:4 \) B. \( 2:4:8 \) C. \( 3:6:12 \) D. \( 1:2:3 \)
Question 2 MC
If \( y \) is directly proportional to \( x \) and \( y = 18 \) when \( x = 3 \), find \( y \) when \( x = 7 \).
A. \( 30 \) B. \( 36 \) C. \( 42 \) D. \( 54 \)
Question 3 Short Answer
If \( x:y = 5:7 \) and \( x = 20 \), find \( y \).
Question 4 Short Answer
Estimate \( \frac{38.7 \times 21.3}{4.98} \) by rounding to 1 significant figure.
Question 5 MC
If \( y \) is inversely proportional to \( x \) and \( y = 10 \) when \( x = 4 \), find \( y \) when \( x = 8 \).
A. \( 5 \) B. \( 10 \) C. \( 20 \) D. \( 40 \)
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An Exercise on Rate, Ratio & Estimation | Direct & Inverse Proportion