Choosing a proportion model
Choosing a proportion model
Decide what stays fixed. Direct proportion keeps the ratio \(\dfrac{y}{x}\) fixed; inverse proportion keeps the product \(xy\) fixed. If neither is fixed, the quantities are not in either proportion.
Direct: \(y=kx\), a straight line through \(O\).
Inverse: \(y=\dfrac{k}{x}\), a hyperbola.
Neither: for example \(y=x+3\), a line not through \(O\).
| Test | Direct proportion | Inverse proportion |
|---|---|---|
| Table of values | \(\dfrac{y}{x}\) is the same for every pair | \(xy\) is the same for every pair |
| Graph of \(y\) against \(x\) | straight line through \(O\) | hyperbola that never meets the axes |
| Doubling \(x\) | doubles \(y\) | halves \(y\) |
| Context | a fixed price or rate per unit | a fixed total job, distance or supply |
Why: each model is defined by the quantity that stays constant. Calculating that quantity for every pair is evidence; noticing that \(y\) rises or falls is not.
Method
- Calculate \(\dfrac{y}{x}\) and \(xy\) for every pair in the table.
- Decide which, if either, is constant.
- In a context, ask what stays fixed: a price per unit, a total job, or a fixed charge that is paid once.
- Write the equation and check it with one pair.
Example
| \(x\) | \(3\) | \(6\) | \(9\) |
|---|---|---|---|
| \(y\) | \(12\) | \(6\) | \(4\) |
The ratios \(\dfrac{y}{x}\) are \(4\), \(1\) and \(\dfrac{4}{9}\), which are not equal. The products \(xy\) are \(36\), \(36\) and \(36\). So \(y\) is inversely proportional to \(x\) and \(y=\dfrac{36}{x}\).
Check: a table that decreases steadily, such as \(y=20-2x\), is neither. Use every pair, because two pairs can agree by chance.
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