To learn Direct and Inverse Proportion quickly and correctly, begin by asking what happens to one quantity when the other changes. If two quantities grow or shrink together by the same factor, they may be in direct proportion. If one becomes twice as large while the other becomes half as large, they may be in inverse proportion. The key is to test a constant ratio or a constant product rather than guessing from the words “more” and “less”. These eduKateSG Mathematics Quick Notes teach both patterns through short explanations, tables, worked problems and simple checks.
- Direct proportion: y = kx, so y/x is constant when x is non-zero.
- Inverse proportion: y = k/x, so xy is constant when x is non-zero.
- Read the situation: identify what is fixed and which quantities can change.
- Test the pattern: use two table rows or values, not a guess.
- Find the constant k: substitute a known pair before finding a new value.
- Check the scale: doubling x doubles y in direct proportion, but halves y in inverse proportion.
First, picture a direct proportion situation. Imagine every notebook costs $3 and there is no extra fixed charge. One notebook costs $3, two cost $6 and five cost $15. The cost increases in the same ratio as the number of notebooks. The cost divided by the number of notebooks is always 3 dollars per notebook, so cost = 3 × number. If x is the number of notebooks and y is the cost, write y = 3x. A graph of a direct proportion relationship is a straight line through the origin when both axes use ordinary linear scales.
Second, picture inverse proportion through equal work. If six identical machines can complete a fixed job in eight hours, twelve identical machines might complete the same work in four hours, provided all machines work at the same steady rate and can work together without delays. The number of machines doubles while the time halves. The product stays fixed: 6 × 8 = 48 machine-hours and 12 × 4 = 48 machine-hours. For these assumptions, machines × hours = 48, or hours = 48 ÷ machines. It is the constant product, not just a downward trend, that defines inverse proportion.
Third, check the relationship before calculating. A table with x-values 2, 4, 8 and y-values 5, 10, 20 is direct proportion because every value of y/x equals 2.5. By contrast, if x = 2, 4, 8 and y = 20, 10, 5, every value of xy equals 40, so the relationship is inverse proportion. These tables look different: one rises with x while the other falls. But a relationship that merely rises or falls is not automatically proportional. Test the same factor or a constant ratio/product every time.
Worked example: direct proportion
A recipe uses 300 g of flour for 4 identical batches. How much flour is needed for 10 batches if each batch uses the same amount? One batch needs 300 ÷ 4 = 75 g. Ten batches need 10 × 75 = 750 g. Using algebra, let f be flour in grams and b the number of batches. Then f = 75b, so f = 75(10) = 750. Check that increasing the batches by a factor of 10/4 = 2.5 also multiplies the flour by 2.5: 300 × 2.5 = 750.
Worked example: inverse proportion
A team of 4 equally productive workers needs 15 hours for a fixed task. Under the assumption that work can be shared perfectly, how long would 6 such workers need? The number of worker-hours remains 4 × 15 = 60. New time = 60 ÷ 6 = 10 hours. Check: 6 workers × 10 hours also gives 60 worker-hours. The answer should be less than 15 hours, because more workers are helping with the same total work.
The easiest decision table
- Direct proportion: y/x = k, hence y = kx.
- Inverse proportion: xy = k, hence y = k/x.
- Direct doubling: x doubles and y doubles.
- Inverse doubling: x doubles and y halves.
- Graph clue: a direct proportion line passes through (0, 0); an inverse proportion graph is curved for positive x and y.
- Not proportional: if a fixed fee is added, y = 3x + 5 is not directly proportional to x, even though it is linear.
Why a fixed charge can break direct proportion
Suppose a taxi fare is $5 plus $2 per kilometre, simplified for this example. Travelling 1 km costs $7, while 2 km costs $9. Twice the distance does not mean twice the fare. The equation is fare = 5 + 2 × distance, which is linear, but it is not direct proportion because the ratio fare/distance changes. This is why “goes up with” is not enough to decide whether a relationship is directly proportional.
How to solve with the constant k
- Name the two changing quantities and keep their units clear.
- Decide whether y/x or xy should be constant based on the stated model.
- Use a known pair to calculate k.
- Write the equation y = kx or y = k/x.
- Insert the new x-value and calculate y.
- Check the direction and proportional change against the original pair.
Common mistakes and the correct repair
- Calling every increasing relationship direct proportion: test whether y/x is constant.
- Calling every decreasing relationship inverse proportion: test whether xy is constant.
- Mixing up the constant: direct uses a constant quotient; inverse uses a constant product.
- Forgetting conditions: worker and machine problems assume an unchanged job and constant rates.
- Dividing when you should multiply: use the model equation before arithmetic.
- Leaving out units: state dollars, grams, hours or the required measurement.
Mini-test: one relationship at a time
- If y is directly proportional to x and y = 18 when x = 4, find y when x = 10.
- If y is inversely proportional to x and y = 16 when x = 3, find y when x = 8.
- Five kilograms of rice cost $17.50 at a fixed unit price. Find the cost of 8 kg.
- Four identical taps fill a tank in 12 minutes. Assuming each tap provides the same steady flow and the tank is unchanged, how long will six taps take?
Answers with the reason
1. k = 18/4 = 4.5, so y = 4.5 × 10 = 45. 2. k = 3 × 16 = 48, so y = 48/8 = 6. 3. Unit price = $17.50 ÷ 5 = $3.50; 8 kg costs $28. 4. The fixed tap-minutes total is 4 × 12 = 48; with 6 taps, the time is 8 minutes.
A simple ten-minute practice routine
- Minutes 1–2: say what stays constant in each type of proportion.
- Minutes 3–4: use a two-row table to test a constant quotient and a constant product.
- Minutes 5–7: solve one direct and one inverse proportion problem.
- Minute 8: check whether doubling one variable changes the other in the expected way.
- Minutes 9–10: explain why an extra fixed charge can make a linear relationship non-proportional.
The best shortcut is to identify the structure before choosing the calculation. Once the constant is clear, a proportion question often becomes a single multiplication or division.
Keep learning: your next Quick Notes
Build this skill with Ratios · Percentages · Coordinate Graphs · Mathematics. Pick a different question and explain why your method works without looking at the example.
For the wider Secondary Mathematics learning pathway, read Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. The original reference page remains unchanged.