If y is directly proportional to x, then y ÷ x is the same number for every pair, and you can write y = kx. This lesson shows how to test that from a table and then use the rule, which links ratio work in ratio and proportional reasoning to graphs and formulas later.
What makes a table proportional?
Direct proportion means that if x doubles, y doubles, and if x triples, y triples. Both values grow in the same steps.
The test that never fails is to divide. If y ÷ x gives the same value in every column, that value is k, the constant of proportionality. A table where y just rises is not enough.
How to model it, step by step
- Divide y by x for every pair in the table.
- Compare the results. If they are all equal, the relationship is direct proportion.
- Write the rule y = kx with the value of k.
- Use the rule to find a missing y (multiply) or a missing x (divide).
Worked example
| Items (x) | 3 | 5 | 8 |
|---|---|---|---|
| Cost in RM (y) | 7.50 | 12.50 | 20.00 |
Step 1, divide: 7.5 ÷ 3 = 2.5, 12.5 ÷ 5 = 2.5 and 20 ÷ 8 = 2.5.
Step 2, compare: all three are 2.5, so cost is directly proportional to the number of items.
Step 3, rule: y = 2.5x.
Step 4, use it: for 14 items, y = 2.5 × 14 = RM35. For a cost of RM27.50, x = 27.5 ÷ 2.5 = 11 items.
Check: 2.5 × 11 = 27.5, and 2.5 × 14 = 35.
The mistake to watch for
A common slip is to call a table proportional because both rows increase, or to use one pair to extend a pattern.
Mistaken working: x = 2, 4, 6 and y = 5, 9, 13. The first pair gives 5 ÷ 2 = 2.5, so when x = 10, y = 25.
The other pairs give 9 ÷ 4 = 2.25 and 13 ÷ 6 = 2.17, so the ratio is not constant.
The table goes up by 4 each time x goes up by 2, so y = 2x + 1.
That is a linear rule with a starting value. When x = 10, y = 21, not 25. Always test every pair before you trust the ratio.
Check yourself
1. Is y directly proportional to x? If so, find y when x = 12.
| x | 2 | 5 | 9 |
|---|---|---|---|
| y | 7 | 17.5 | 31.5 |
Show answer
7 ÷ 2 = 3.5, 17.5 ÷ 5 = 3.5 and 31.5 ÷ 9 = 3.5. All equal, so yes, y = 3.5x.
When x = 12, y = 3.5 × 12 = 42.
2. Is y directly proportional to x?
| x | 1 | 2 | 3 |
|---|---|---|---|
| y | 4 | 7 | 10 |
Show answer
4 ÷ 1 = 4, 7 ÷ 2 = 3.5 and 10 ÷ 3 = 3.33. The ratios differ, so no. The rule is y = 3x + 1.
3. Distance is directly proportional to time. A car travels 150 km in 2 hours. How far does it go in 5 hours, and how long does it need for 525 km?
Show answer
k = 150 ÷ 2 = 75 km per hour, so distance = 75 × time.
5 hours: 75 × 5 = 375 km. For 525 km: 525 ÷ 75 = 7 hours.
Where this leads next
Compare this lesson with recognising inverse proportion from a changing product, where x × y is constant instead. The non-calculator working trainer helps you check each division.
If you can test a table but struggle to explain why the result matters, online one-to-one Mathematics tuition can help you practise writing the reason in the wording an exam expects.