Two quantities are in direct proportion only if the ratio between them never changes. If a fixed amount is added, the relationship is straight-line but not proportional, and scaling breaks.
What is the small gap?
Most early examples (price of pens, kilograms of rice) are proportional, so students learn “double one, double the other” as a law. It is a property of proportion, not of every pair of quantities.
How do you test a situation?
- Write the pairs in a table.
- Divide the second number by the first for each pair.
- Same answer every time means direct proportion. Different answers mean it is not.
A scaffolded example
An original taxi fare: RM4 to start, then RM2 per km.
| Distance (km) | Fare (RM) | Fare ÷ distance |
|---|---|---|
| 2 | 8 | 4 |
| 4 | 12 | 3 |
| 8 | 20 | 2.5 |
The ratios differ, so the fare is not proportional to distance. Doubling the distance from 4 km to 8 km does not double the fare. Doubling RM12 gives RM24, but the real fare is RM20.
Each extra kilometre adds a constant RM2 to the fare, which is a constant rate of change. That is the signature of a constant-difference relationship, written as fare = 2 × distance + 4.
What does the common mistake look like?
A student reasons: 4 km costs RM12, so 8 km costs RM24. The starting charge of RM4 is counted twice when scaling. The repair is to ask first whether the ratio is constant, before any scaling.
Compare a proportional case: 4 kg of rice at RM12 means 8 kg at RM24, because there is no starting charge.
Self-check
A plumber charges RM30 to come and RM25 per hour. Is the charge proportional to hours? What does a 3-hour job cost?
Answer
One hour costs 30 + 25 = RM55, and two hours cost RM80. The ratios 55 ÷ 1 = 55 and 80 ÷ 2 = 40 differ, so it is not proportional.
Three hours: 30 + 25 × 3 = 30 + 75 = RM105.
Where does this show up in SPM?
In SPM Mathematics, it appears in linear functions and variation. In Physics, a proportional graph passes through the origin, and a line with an intercept, read with reading an intercept in context, tells a different story.
The companion skill is scaling a recipe-like model, which assumes proportion. See the hub for the order.
One-to-one Mathematics tuition starts with a one-hour trial class (from RM50).