Direct and Inverse Proportion: Practice

Substitute the known values into $y = \dfrac{kx}{z}$.

$12 = \frac{k \times 8}{2} = 4k \implies k = 3$

Joint variation with $k = 2$ means $y = 2xz$.

$y = 2 \times 5 \times 7 = 70$

Test the product: $2 \times 24 = 48$, $4 \times 12 = 48$, and so on, so the product is constant and the variation is inverse with $k = 48$.

$y = \frac{48}{12} = 4$

Test the ratio: $\dfrac{12}{3} = \dfrac{20}{5} = \dfrac{28}{7} = 4$, so the ratio is constant and the variation is direct with $k = 4$.

$y = 4 \times 10 = 40$

Find $k$ from the known pair.

$k = \frac{15}{6} = 2.5$

Then $y = 2.5x$, so $40 = 2.5x$ gives $x = 16$.

Substitute the values directly.

$y = \frac{5 \times 2 \times 6}{3} = \frac{60}{3} = 20$

Find $k$ from the first reading.

$8 = \frac{k \times 12}{3} = 4k \implies k = 2$

Then $I = \dfrac{2 \times 20}{5} = 8$.

The product $xy$ is $36$ in every row ($1 \times 36$, $2 \times 18$, and so on), so the variation is inverse.

$xy = 36 \implies y = \frac{36}{x}$

Find $k$ from $y = kxz$.

$40 = k \times 4 \times 5 = 20k \implies k = 2$

Then $y = 2 \times 6 \times 10 = 120$.

Find the constant product $k = st$, the fixed distance.

$k = 60 \times 4 = 240 \text{ miles}$

Then $3 = \dfrac{240}{s}$ gives $s = 80$ mph.

Direct variation is $y = kx$. Increasing $x$ by $50\%$ replaces $x$ with $1.5x$.

$y_{\text{new}} = k(1.5x) = 1.5(kx) = 1.5y$

That is a $50\%$ increase in $y$, since the ratio is fixed.

The product $xy$ is $18$ in every row ($2 \times 9$, $3 \times 6$, $6 \times 3$), so the variation is inverse with $k = 18$.

$y = \frac{18}{9} = 2$