5.3 Direct proportion — Learn

Prerequisites

You should be able to multiply and divide whole numbers and fractions. Check units before comparing quantities.

Key terms

Direct proportion means one quantity is a fixed multiple of another. The constant of proportionality is written $k$.

Objectives

• Find equations and values in direct proportion, including square and cube relationships.
• Check that the result matches the units and direction of change.

Learn the method

In direct proportion, one quantity is a fixed multiple of another: $y=kx$. For $y\propto x^2$, write $y=kx^2$. Substitute known values to find $k$, then find the new value.

Worked example. $y$ is proportional to $x^2$. When $x=3$, $y=45$. Find $y$ when $x=5$.

• Write the model: $y=kx^2$.
• Substitute the known pair: $45=9k$.
• Divide by 9: $k=5$.
• Substitute $x=5$: $y=5\times25=125$.
• Check: $45\times(25/9)=125$.

Worked example. A graph of direct proportion passes through $(0,0)$ and $(3,12)$. Find its gradient and equation.

• Find the gradient: $12\div3=4$.
• The graph passes through the origin, so its equation is $y=4x$.
• Check with $(3,12)$: $4\times3=12$.

Progress check: apply this multiplier idea to the garden path scale.

Watch out. For $y\propto x^2$, doubling $x$ multiplies $y$ by 4, not 2. Square the scale factor, then check with the original and new values.

Try it: direct and inverse proportion

Predict: as $x$ doubles, does $y$ double or halve? Decide for direct proportion, then for inverse proportion. Write your predictions down first.

Test: adjust $k$ and $x$, then tick the inverse box, and check each graph against your prediction.

Explain: what is the difference between $y = kx$ and $y = \frac{k}{x}$? Use the shape of each graph, and write your answer before reading on.

You should now be able to

• I can find $k$ in $y=kx$, $y=kx^2$ or $y=kx^3$.
• I can use the model to find a missing value.
• I can check the scaling factor and the answer.