Question

In each example, find the constant of proportionally and write the equation of variation.

(1) The quantity y varies directly as x. When y is 20, x is 4.
(2) p α q. When p is 12, q is 18.
(3) c α d. When c is 28, d is 21.
(4) m α n. When m is 7.5, n is 10.

Answer 1

A set of numbers is said to be in proportion if their fractions in the simplest form are equal.
(1) There is a direct variation between y and x; when y is 20, x is 4.
$$\begin{gathered} \therefore \frac{y}{x}=k \\ \Rightarrow \frac{20}{4}=k \\ \Rightarrow k=5 \end{gathered}$$

So ,the constant of proportionality is 5 and equation of variation is y = 5x .

(2) p and q are in direct variation; when p is 12, q is 18.
$$\begin{gathered} \therefore \frac{p}{q}=k \\ \Rightarrow \frac{12}{18}=k \\ \Rightarrow k=\frac{2}{3} \end{gathered}$$
So, the constant of proportionality is $\frac{2}{3}$ and equation of variation is p = $\frac{2}{3}q$ .

(3) c and d are in direct variation; when c is 28, d is 21.
$$\begin{gathered} \therefore \frac{c}{d}=k \\ \Rightarrow \frac{28}{21}=k \\ \Rightarrow k=\frac{4}{3} \end{gathered}$$
So, the constant of proportionality is $\frac{4}{3}$ and equation of the variation is $c=\frac{4}{3}d$ .

(4) m and n are in direct variation; when m is 7.5, n is 10.
$$\begin{gathered} \therefore \frac{m}{n}=k \\ \Rightarrow \frac{7.5}{10}=k \\ \Rightarrow k=\frac{3}{4} \end{gathered}$$
So, the constant of proportionality is $\frac{3}{4}$ and equation of the variation is m = $\frac{3}{4}n$ .