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.
Nova
Try this one

A printing machine prints pages at a constant rate. It prints 360 pages in 6 minutes.

(a) How many pages does it print in 10 minutes? [2]

(b) A document has 480 pages. How many minutes does it take to print this document? [2]

(c) Two identical machines work together. How long does it take them to print 480 pages? [2]

Nova's hint:

Parts (a) and (b) use direct proportion; part (c) uses inverse proportion (more machines, less time).

Ask yourself: 'How many pages per minute does one machine print, and what happens to the time when two machines share the work?'

For example, if 1 machine takes 10 minutes, then 2 machines take 10 ÷ 2 = 5 minutes.

Try to Solve it with Nova?