Ratio, proportion & rates of change · Direct & inverse proportion

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Proportion with squares, cubes and roots

Why y ∝ x², y ∝ √x and y ∝ 1/x² all follow the same three-step method — write the equation, find k, use it — and how the power changes the scaling.

Ratio, proportion & rates of change · Direct & inverse proportion

Proportion with squares, cubes and roots

Why y ∝ x², y ∝ √x and y ∝ 1/x² all follow the same three-step method — write the equation, find k, use it — and how the power changes the scaling.

Why it works

Proportion doesn't have to be to

x

itself —

y

can be proportional to

x^2

, to

x^3

, to

\sqrt{x}

, or inversely to any of them. The wording maps directly to an equation:

words	equation
$y$ is proportional to the square of $x$	$y = kx^2$
$y$ is proportional to the square root of $x$	$y = k\sqrt{x}$
$y$ is inversely proportional to the square of $x$	$y = \dfrac{k}{x^2}$

and every question then has the same three steps:

Write the equation with an unknown $k$ .
Substitute the given pair of values to find $k$ .
Use the completed formula to answer whatever is asked.

The only place the power enters is that you substitute the powered value. If

y = kx^2

and

y = 75

when

x = 5

, then

75 = k \times 25

(not

k \times 5

), so

k = 3

and

y = 3x^2

.

The power changes how scaling works. In plain direct proportion doubling

x

doubles

y

. But if

y \propto x^2

, doubling

x

multiplies

y

2^2 = 4

; if

y \propto x^3

, by

8

. And for inverse square,

y \propto \frac{1}{x^2}

, doubling

x

divides

y

4

— not by

2

. The rule: whatever factor

x

changes by,

y

changes by that factor pushed through the power. This is worth internalising because exam questions ask it directly ("when

x

is doubled, what happens to

y

?") with no numbers at all.

Coming back the other way needs the root. If

y = 3x^2

and you're told

y = 300

, then

x^2 = 100

— and

x^2

is not the answer. Undo the square:

x = 10

(taking the positive root, since these quantities are positive). Forgetting this final un-powering step is the classic way to lose the last mark.

Chains of proportion combine into one. If

a \propto b^2

and

b \propto \sqrt{c}

, then

b^2 \propto c

, so

a \propto c

: substitute one formula into the other and the two constants merge into a single new one.