Ratio, proportion and rates of change: understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; interpret equations that describe direct and inverse proportion (Foundation tier)

The idea

The equation tells you which kind of proportion you are looking at. If y = 4x then y is a fixed multiple of x: double x and y doubles, and where x is zero y is zero too. If y = 24/x the two quantities multiply to a fixed number, so doubling x halves y. Saying y is inversely proportional to x is the same as saying y is proportional to 1/x, the reciprocal of x. To read an equation, substitute a value and watch what happens: in y = 24/x, x = 3 gives y = 8, and x = 6 gives y = 4. In a table of values, a constant product points to an inverse relationship and a constant ratio between the two rows points to a direct one.

What earns the marks

• Identifying the type of proportion from the equation or the table, supported by a reason such as a constant product.
• A correct substitution into the given equation, even if the arithmetic that follows is wrong.
• The value found, given to the accuracy the question asks for.
• A correct statement of the effect of a change, such as saying that in an inverse relationship trebling one quantity divides the other by three.

Common mistakes and how to put them right

• Reading y = 24/x as direct proportion because both quantities are changing, then multiplying where a division was needed.

• Assuming any equation with the variable underneath is inverse proportion, including ones such as y = 2/x + 5 where no product stays constant.

• Swapping the roles of the two quantities, so working out x when the question asked for y.

• Calling a graph direct proportion because it passes through the origin, even when it is a curve.

Exam tips

• Test the equation with an easy value such as one, then double it and see what happens to the other quantity; that identifies the relationship quickly.
• For an inverse relationship, multiply each pair in the table together; if every product agrees, you have found the constant.
• Answer using the names of the quantities in the question, not only the letters.