Formula

Direct variation as a power: $y \propto x^{n}$

Arithmetic • Ratio and Proportion • Direct and Inverse Variation

Formula

y \propto x^{n} ⟺ y = c x^{n}

y

varies directly as

x^{n}

, then

y = c x^{n}

for a constant

c

that does not change across the problem. Consequently

\frac{y _{1}}{y _{2}} = (\frac{x _{1}}{x _{2}})^{n}

, and

c

need never be found when only a ratio is asked for.

How to use it

1Write the law as an equation with $c$ first; $c$ usually cancels, and when it does not, one given pair determines it.
2'Varies as the square' means $n = 2$ , so halving $x$ divides $y$ by $4$ — quantities that add do not have squares that add.
3For inverse variation use $y = \frac{c}{x ^{n}}$ instead.

The cost of a gem varies as the cube of its weight. A gem weighing

3

g costs Rs.

540

. What does a

5

g gem cost?

Step-by-step

1
$C = c w^{3} \Rightarrow 540 = c \times 27$
Why: Write the law as an equation and use the known pair.
2
$c = 20$
3
$C = 20 \times 5^{3} = 2500$

Answer

Rs.

2500

Watch out

For

n \neq = 1

the quantity is NOT additive: if a whole of weight

w

splits into pieces, the sum of the pieces' values is less than the whole's, which is exactly the loss such questions ask for.

Builds on

k

Also known as

direct variation, varies directly as, proportional to the square, constant of proportionality, varies as the cube