Variation
How to Solve a joint variation
Use this when one quantity varies with two or more others at the same time.
Before you start
- Single direct and inverse variation
- Solving for the constant k
- Substituting values including powers like z²
When to use it
Use this when one quantity varies with two or more others at the same time.
The steps
- Read the wording to see which variables are direct and which are inverse.
- Write one combined relationship, e.g. y = kxz or y = kx/z.
- Substitute the first full set of values to find k.
- Write the complete equation with k in place.
- Substitute the new values to find the unknown.
Worked example
y varies directly as x and z. When x = 2 and z = 3, y = 24.
Find y when x = 4 and z = 5.
- Read the wording: y is direct with both x and z.
- Write one combined relationship: y = kxz.
- Substitute the first full set x = 2, z = 3, y = 24: 24 = k × 2 × 3 = 6k, so k = 4.
- Write the complete equation: y = 4xz.
- Substitute the new values x = 4, z = 5: y = 4 × 4 × 5 = 80.
A second example, with a twist
One variable is inverse and squared (z²), so the combined relation mixes direct and inverse with a power. y varies directly as x and inversely as the square of z.
When x = 6 and z = 2, y = 9. Find y when x = 8 and z = 4.
- Read the wording: y is direct with x and inverse with z, which is squared.
- Write one combined relationship: y = kx/z².
- Substitute the first full set x = 6, z = 2, y = 9: 9 = k × 6 / 2² = 6k/4, so 6k = 36 and k = 6.
- Write the complete equation: y = 6x/z².
- Substitute the new values x = 8, z = 4: y = 6 × 8 / 4² = 48/16 = 3.
Formula pages
Practise this in a KBAT problem
Frequently asked questions
How do I know which variables go on top and which on the bottom?
Direct variables go in the top of the fraction, multiplied together; inverse variables go on the bottom. If y varies directly as x and inversely as z, write y = kx/z.
Reading each variable as 'directly' or 'inversely' tells you exactly where it belongs.
Do I need all the values to find k?
Yes, you need one complete set, a value for every variable in the relationship, to find k. Substitute them all at once into y = kxz or y = kx/z and solve for k.
Missing even one value means you cannot pin down the constant.
What happens to the square when I substitute z?
Square the value of z first, then divide by it. For z = 2 the term z² is 4, not 2, so you divide by 4.
Doing the power before the division keeps the arithmetic correct; squaring after dividing is a common and costly mistake.