Direct and Inverse Proportion: Practice

Find $k$ from the known pair by dividing.

$k = \frac{30}{5} = 6$

Then $y = 6x = 6 \times 8 = 48$.

Find the price per gallon $k$ from the known pair.

$k = \frac{27}{9} = 3$

Then the cost is $3 \times 14 = 42$ dollars.

Find the constant $k = xy$, the fixed amount of pumping work.

$k = 3 \times 8 = 24$

Then the time is $\dfrac{24}{4} = 6$ hours. More pumps means less time.

Inverse variation means $k = xy$.

$k = 12 \times 5 = 60$

Find how many km each cm represents.

$k = \frac{50}{2} = 25$

Then $7$ cm represents $25 \times 7 = 175$ km.

Find the weight per meter $k$.

$k = \frac{9}{6} = 1.5$

Then $10$ m weighs $1.5 \times 10 = 15$ kg.

Find the constant $k = xy$, the total person-days of food.

$k = 6 \times 20 = 120$

Then the supply lasts $\dfrac{120}{8} = 15$ days for $8$ people.

Inverse variation means $xy = k = 36$.

$9x = 36 \implies x = 4$

Substitute into $y = kx$ with $k = \dfrac{3}{4}$.

$y = \frac{3}{4} \times 20 = 15$

Find the constant product $k = xy$.

$k = 60 \times 12 = 720$

Then $y = \dfrac{720}{x} = \dfrac{720}{45} = 16$.

Direct variation is $y = kx$. Replacing $x$ with $5x$ gives $k(5x) = 5(kx)$.

$y_{\text{new}} = 5kx = 5y$

So $y$ is multiplied by $5$.

Find the constant product $k = xy$.

$k = 4 \times 7 = 28$

Then $y = \dfrac{28}{x} = \dfrac{28}{2} = 14$. Halving $x$ doubles $y$.