Explain the concept of direct proportion.

Direct proportion means that as one quantity increases, the other quantity increases at a constant rate.

In more detail, when two quantities are in direct proportion, they increase or decrease in such a way that their ratio remains constant. For example, if you double one quantity, the other quantity also doubles. This relationship can be expressed mathematically as \( y = kx \), where \( y \) and \( x \) are the two quantities, and \( k \) is the constant of proportionality.

Imagine you are buying apples, and each apple costs 50p. If you buy 1 apple, it costs 50p. If you buy 2 apples, it costs £1, and if you buy 3 apples, it costs £1.50. Here, the cost is directly proportional to the number of apples you buy. The constant of proportionality \( k \) is 50p per apple.

Graphs of direct proportion relationships are straight lines that pass through the origin (0,0). This is because when one quantity is zero, the other must also be zero, maintaining the constant ratio. For instance, if you plot the number of apples on the x-axis and the total cost on the y-axis, you will get a straight line starting from the origin.

Understanding direct proportion is crucial in various real-life situations, such as converting currencies, calculating speed, or working out recipes. It helps you predict one quantity if you know the other, making problem-solving more straightforward.