Direct and inverse proportion (Higher Tier)

Direct Proportion

A variable is in direct proportion to another when they increase or decrease together at the same rate.
When two quantities are related by direct proportion, as one quantity doubles, the other also doubles.
You can represent direct proportion with the formula y = kx where k is the constant of proportionality.
If y is directly proportional to x (written as y ∝ x), it means that y = kx where k is a non-zero constant.
For example, if you’re buying apples and you know that 5 apples cost £1.50, you can say the cost (y) is directly proportional to the number of apples (x). If you double the number of apples to 10, the cost would also double to £3.

A variable is in inverse proportion to another if it increases as the other decreases.
When two quantities x and y are inversely proportional, as x increases, y decreases or vice versa.
You can represent inverse proportion with the formula y = k/x or xy = k, where k is the constant of proportionality.
If y is inversely proportional to x (written as y ∝ 1/x), it means y = k/x or xy = k where k is a non-zero constant.
For example, in a journey, if your speed is constant, the time taken to complete the journey is inversely proportional to the speed. The faster you go (increase in speed), the less time it takes (decrease in time).

To solve problems involving direct or inverse proportion, you first need to find the value of the constant k.
To find the constant of proportionality (k), you insert the known x and y values into the formula and solve for k.
Once you have found k, you can use it to solve for any unknown values.
Remember to check that your answers make sense in context. For example, it wouldn’t make sense to have a negative time or speed.

Graphs of direct proportion are straight lines through the origin (0, 0).
The gradient of the line in a graph of direct proportion is equal to the constant of proportionality k.
In an inverse proportion, the graph is a hyperbola and will never cross the axes.
The product of corresponding x and y values on a graph of inverse proportion remains constant.