Variation
How to Solve a variation problem
Use this to solve a problem where one quantity varies directly or inversely with another.
Before you start
- The idea of two quantities in proportion
- Solving a simple equation to find the constant k
- Substituting values, including squared terms
When to use it
Use this to solve a problem where one quantity varies directly or inversely with another.
The steps
- Read the wording to decide the type: direct (y = kx) or inverse (y = k/x).
- Write the relationship in symbols, including any power (like x²).
- Substitute the first pair of values to find the constant k.
- Write the full equation with k filled in.
- Substitute the new value to find the unknown.
Worked example
y varies directly as x. When x = 4, y = 20.
Find y when x = 7.
- The words say 'varies directly', so use y = kx.
- There is no power here, so the relationship in symbols is y = kx.
- Substitute the first pair x = 4, y = 20: 20 = k × 4, so k = 5.
- Write the full equation: y = 5x.
- Substitute the new value x = 7: y = 5 × 7 = 35.
A second example, with a twist
This is inverse variation and involves a square, so k is found by multiplying rather than a simple divide. y varies inversely as the square of x.
When x = 2, y = 9. Find y when x = 3.
- The words say 'varies inversely', so use y = k/x.
- With the square, the relationship in symbols is y = k/x².
- Substitute the first pair x = 2, y = 9: 9 = k / 2² = k / 4, so k = 36.
- Write the full equation: y = 36/x².
- Substitute the new value x = 3: y = 36 / 3² = 36 / 9 = 4.
Formula pages
Practise this in a KBAT problem
Frequently asked questions
How do I tell direct from inverse variation?
Read what happens as one quantity grows. In direct variation, when x increases y increases too, and you use y = kx.
In inverse variation, when x increases y decreases, and you use y = k/x. Words like 'inversely' or 'the more, the less' point to the inverse form.
Does the constant k change between the two parts?
No. The whole point is that k stays the same throughout one problem.
You use the first pair of values only to find k, and then that same k is used for the new value. If you accidentally recalculate k for the second part, something has gone wrong.
What do I do with the power, like x²?
Keep the power in the relationship from the start: write y = k/x² or y = kx², not just x. When you substitute, square the value first, then divide or multiply.
Forgetting to square is the most common slip, so square before you do anything else with that value.