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This page sets out the method for direct & inverse proportion in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Direct & inverse proportion topic page.
When you see a price or amount for several items and need it for a different number, find one first.
When the question says ' is directly (or inversely) proportional to ' and gives one pair of values.
When more workers, taps or machines make a job quicker, the total work is fixed.
More workers, less time. Multiply workers by hours, then divide by the new team.
y is directly proportional to x, so y = kx. If y = 10 when x = 2, then k = 10 ÷ 2 = 5. When x = 5, y = 5 × 5 = 25.
y is inversely proportional to x. The product xy stays fixed.
For direct proportion, k = y/x. For inverse proportion, k = xy.
This lesson uses y against x. Questions can also use x², but the method is the same: find k first.
Direct: both grow together. Inverse: one grows, the other falls.
If y is directly proportional to x, there is one multiplier that turns every x into its y. Knowing y = 18 when x = 6 tells you the multiplier is 18 ÷ 6 = 3. The equation is y = 3x. A new x is multiplied by 3, not by 18.
The reason is scaling. Six units of x produced 18. One unit produces 3, so eleven units produce 33. Adding 11 to 18 would treat the link as a shift, not a proportion. Direct proportion has no added constant. When x is 0, y is 0.
Method: 1. Divide the known y by the known x to find k. 2. Write y = kx with that k. 3. Multiply k by the new x. 4. Check that doubling x would double y.
The known pair is the only information about k. Use it once, write k down, and do not return to the old y as if it were the multiplier. For direct proportion the division is y ÷ x. For inverse proportion the multiplication is x × y. Mixing those two formulas swaps the type of proportion.
A sensible check is the size. If x grows and the proportion is direct, the new y should be larger than the old y. If the proportion is inverse, the new y should be smaller. When your answer moves the wrong way, the formula for k was probably the other one.
Method: 1. Decide whether the link is direct or inverse. 2. Calculate k from the known pair. 3. Substitute the new x into y = kx or y = k/x. 4. Compare the new y with the old y to see if the direction is right.
Inverse proportion keeps a product constant. If y = 24 when x = 5, then k = 5 × 24 = 120, and y = 120/x. When x = 8, y = 120 ÷ 8 = 15. The new y is smaller because the new x is larger. That is the whole point of the model.
Multiplying 24 by 8 ignores the constant product and treats the old y as a direct multiplier. The old y already includes the old x inside the product. You have to build k first, then divide by the new x. The division is what makes y fall.
Method: 1. Multiply the known x by the known y to find k. 2. Write y = k/x. 3. Divide k by the new x. 4. Check that the new product equals the old product.
Five workers taking 12 hours do 5 × 12 = 60 worker-hours of painting. The fence does not get bigger when more people arrive. Ten workers share those same 60 worker-hours, so each hour of the clock covers more of the job. The time is 60 ÷ 10 = 6 hours.
Halving the team doubles the time, for the same reason. The product is fixed, so one factor falling forces the other to rise. Do not add the workers to the hours. Workers and hours are different units, and the job is their product.
Method: 1. Multiply the known number of workers by the known time. 2. Record the product as the size of the job. 3. Divide the product by the new number of workers. 4. Check that more workers give fewer hours.
Exam questions go beyond y propto x and y propto 1/x. You will see statements such as "y is directly proportional to the square of x", which means y = kx^2, or "y is inversely proportional to the square root of x", which means y = dfrack√(x). The word "directly" puts the power on the top and "inversely" puts it underneath. The constant k is a fixed number that does not change.
Read the statement slowly and write the formula straight away, before you use any numbers. Square means power 2, cube means power 3, and square root means power tfrac12. The formula is always y = k × (the power of x) for direct, and y = k ÷ (the power of x) for inverse. Getting the formula right earns the first mark, even if later arithmetic slips.
Method: 1. Underline whether the proportion is direct or inverse, and what is done to x. 2. Write the formula with k, such as y = kx^2 or y = k/(x^2). 3. Substitute the pair of values you were given. 4. Solve for k and write the full formula.
Tip: Substitute the number for the power too. If x = 4 and the rule uses x^2, you must use 16, not 4.
The standard exam pattern has two stages. Part (a) asks you to "work out an equation connecting" the two letters, for 3 marks: one for the correct form, one for finding k, and one for the final equation. Part (b) uses the equation for a new value. Do not skip the equation and use ratios between values, as the marks are tied to the formula and a clear k.
When you are given y and need x in a power relationship, you must undo the power. For y = 3x^2 and y = 48, divide first to get x^2 = 16, then take the square root. Write both roots if the question does not restrict the sign, then keep the correct one. Read the question for words such as "positive" or "negative" that tell you which root to give.
Method: 1. Write the general formula with k and use the given pair to find k. 2. Write the equation with the value of k in it. 3. Substitute the new value, and solve, undoing any power step by step. 4. Check the answer by substituting back into the formula.
Some questions give two statements: "y is proportional to the square root of t" and "t is inversely proportional to the square of x". Find each constant on its own, then substitute one formula into the other to get y in terms of x. Work in steps: first y = a√(t), then t = b/(x^2), then y = a√(b/(x^2)). Simplify the root at the end, for example √(x^2) = x.
Percentage change questions use multipliers. If y is proportional to x^2 and x increases by 20%, the multiplier on x is 1.2, so the multiplier on y is 1.2^2 = 1.44, an increase of 44%. If p is proportional to √(q) and q increases by 69%, then √(1.69) = 1.3 and p increases by 30%. Do not just copy the percentage across.
Method: 1. For two linked statements, find each constant separately. 2. Substitute one formula into the other and simplify. 3. For percentage change, turn the percentage into a multiplier. 4. Apply the same power or root to the multiplier, then convert back to a percentage.
Tip: The question sometimes asks for the answer "in its simplest form". Cancel any numbers and write the formula neatly.
More workers means less time, so workers and time are in inverse proportion. The quantity that stays the same is the total work, the product of workers and time. For 15 workers taking 8 days, the total work is 120 worker-days. With 20 workers the time is 120 ÷ 20 = 6 days. State the assumption, "all workers work at the same rate", when the question tells you to.
Graphs show the shape of each type. Direct proportion y = kx is a straight line through the origin. y = kx^2 is a curve from the origin that rises more steeply. Inverse proportion y = k/x is a curve that falls and never touches either axis. If a question asks you to sketch or match, check where the graph starts and whether it goes through the origin.
Method: 1. For workers and time, find the total work by multiplying them. 2. Divide the total work by the new number of workers to get the new time. 3. Compare the time with the target in the question, and give a clear conclusion. 4. For graph questions, use the origin and the overall shape to choose.
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