If y varies directly as x, write y = kx, use one known pair to find k, then use the equation for the rest. The constant k never changes within one problem.
This lesson opens the variation section. The next lesson, solving inverse variation problems, reverses the relationship.
How do I solve a direct variation problem?
Use the same three steps every time.
- Write the relationship with k, for example m = kL.
- Substitute the known pair and solve for k.
- Write the full equation with the value of k, then answer the question.
Worked example: a steel rod
The mass m of a steel rod varies directly as its length L. A rod of length 40 cm has a mass of 320 g. Find the mass of a rod 55 cm long.
- Write m = kL.
- Substitute 320 = k × 40, so k = 8.
- The equation is m = 8L. For L = 55, m = 8 × 55 = 440 g.
Check: 55 ÷ 40 = 1.375 and 320 × 1.375 = 440. The ratio method agrees.
When a power is involved
Suppose the cost C of a square tile varies directly as the square of its side s. A tile with s = 10 cm costs RM5.
- Write C = ks².
- Substitute 5 = k × 100, so k = 0.05.
- For s = 14: C = 0.05 × 196 = RM9.80.
Raising s from 10 to 14 multiplies the cost by 1.96, not by 1.4. The power of s belongs inside the equation from the first line.
The mistake that costs marks
The common slip is to treat a squared relationship as a simple one. A student assumes that a tile with a side 1.4 times longer costs 1.4 times as much, giving RM7.
| Step | Wrong | Right |
|---|---|---|
| Equation | C = ks | C = ks² |
| Find k | 5 = k × 10, so k = 0.5 | 5 = k × 100, so k = 0.05 |
| Cost at s = 14 | 0.5 × 14 = RM7.00 | 0.05 × 196 = RM9.80 |
The wording “square of” is the signal to write the power. Underline it before you write the equation.
Check yourself
A varies directly as the square root of B. A = 12 when B = 16. Find A when B = 25.
Answer
Write A = k√B. Substitute 12 = k × √16 = 4k, so k = 3.
For B = 25: A = 3 × √25 = 3 × 5 = 15.
Check: 15 ÷ 12 = 1.25 and √25 ÷ √16 = 5 ÷ 4 = 1.25.
What to study next
Continue with solving inverse variation problems, then test the chapter with the variation practice set. If the algebra of solving for k slows you down, the algebra step repair trainer targets that step.
For a teacher to go through these with you, see online one-to-one Mathematics tuition.