Direct proportion
A direct model has a constant output-to-input ratio at nonzero inputs and a zero output at zero input
\[y=kx\]
- Test matching nonzero inputs Divide each output by its matching nonzero input. Equal ratios support a direct model for the displayed data \(\displaystyle k=\frac{y}{x}\quad(x\ne0)\)
- Check zero and the graph A direct rule has no fixed added amount. Its straight-line graph passes through the origin \(\displaystyle y=kx,\quad y(0)=0\)
- Keep units consistent The constant has output units per input unit. Convert the input before applying a rate
- Interpret the model Check the allowed inputs and whether the constant-rate assumption is reasonable. A pattern in recorded data does not prove the model outside those conditions
Worked example
A ribbon costs $8.40 for 2.4 metres, with no fixed fee and a constant price per metre. Find the cost of 3.6 metres
\[C=kl\]
- \(\displaystyle k=\frac{8.40}{2.4}=3.5\)The price is $3.50 per metre
- \(\displaystyle C=3.5\times3.6=12.60\)The required ribbon costs $12.60 if the same price per metre applies
Common mistakes
Confusing linear with directly proportional
✕\(\displaystyle y=3x+4\text{ is direct}\)✓\(\displaystyle y=3x\text{ is direct}\)
A fixed added value prevents direct proportionUsing seconds with an hourly rate
✕\(\displaystyle 150\times90=13500\)✓\(\displaystyle 150\times\frac{90}{3600}=3.75\)
For 90 seconds at 150 L/hour, convert seconds to hours before multiplyingDifficulty levels
introRecognise a direct-proportion rule or find k from a simple exact pair
standardFind k, use y = kx in both directions and complete a table of values
challengeConvert a rate between units or recognise direct proportion from a graph
examSelect the direct-proportion rule, constant and units from diagnostic alternatives
Questions about this topic
Can the ratio of output to input be calculated at zero?+
No. Use nonzero inputs to determine the constant, then check that the rule gives zero output at zero input
Can measurement error affect the prediction?+
Yes. A range for the measured rate gives a range for the predicted output. The range also depends on assumptions such as the rate staying constant
Does a straight line always show direct proportion?+
No. It must also pass through the origin. A nonzero fixed amount gives a more general linear model
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 6: Linear graphs
In this chapter
6A Reading linear graphs: Read and interpret a straight-line graphInterpret and sketch distance-time graphsConstruct a table of values from a formula6C Plotting lines from tables: Recognise a line or find points for its graph6C Plotting lines from tables: Read and interpret a straight-line graph6C Plotting lines from tables: Construct a linear model from a descriptionMidpoint and distance of a coordinate segment6E Gradient from coordinates: Classify and compare gradients from graphsFind the gradient of a line through two pointsCalculate a rate from a straight-line graphRead the vertical-axis intercept and gradient from an equation6G Gradient-intercept form: Recognise a line or find points for its graph6G Gradient-intercept form: Classify and compare gradients from graphsForm an equation of a parallel or perpendicular lineFind the axis intercepts of a line6I Sketching lines from intercepts: Recognise a line or find points for its graph6J Linear modelling: Construct a linear model from a descriptionInterpret the graph of a linear modelRead the point of intersection of two linesDirect proportion and the constant of proportionalityIdentify direct or inverse variation from tables and graphsFind the constant of variation and complete the table