Proportion with Powers & Roots

Grade 7

Beyond the straight line

Not all proportion is linear. Sometimes one quantity depends on the square (or cube, or square root) of another. The maths still works the same way — you just swap the equation.

For example, the area of a circle depends on the square of the radius. Double the radius and the area goes up by a factor of four, not two. That's proportion involving a power.

Common forms

y = kx² (y is directly proportional to x squared)

y = k√x (y is directly proportional to the square root of x)

y = k/x² (y is inversely proportional to x squared)

The method is always the same: substitute a known pair to find k, then use that k to answer the question.

Worked example — direct proportion with a square

EXAMPLE

y is directly proportional to x². When x = 3, y = 36. Find y when x = 5.

Write the equation: y = kx². Substitute the known pair: 36 = k × 9, so k = 4. Now find y when x = 5: y = 4 × 25 =100.

Worked example — inverse proportion with a square

EXAMPLE

y is inversely proportional to x². When x = 2, y = 50. Find y when x = 5.

Write: y = k/x². Find k: 50 = k/4, so k = 200. Find y when x = 5: y = 200/25 =8.

The multiplier shortcut

If x is doubled in a y = kx² relationship, y goes up by a factor of 2² = 4. If x is tripled, y goes up by 3² = 9. This multiplier trick can save a lot of time in exam questions that don't give you numbers at all.

Try this one

The cost, C pence, of producing a batch of sweets is directly proportional to the cube root of the number of sweets, n, in the batch.

When n = 8, C = 90.

(a) Find a formula for C in terms of n. [2]

(b) Find the cost when n = 27. [1]

Nova's hint:

This is direct proportion to a fractional power — specifically the cube root.

Ask yourself: 'How do I write cube root as a power, and what form does my equation take?' Try \( C = k \times n^{1/3} \) and substitute the known values to find k before doing any other part.

Try to Solve it with Nova?