# If y varies directly as x, and y = 9 when x= 5, find y when x = 10.

## Question:

If y varies directly as x, and y = 9 when x= 5, find y when x = 10.

## Direct proportionality

Direct proportionality means that two quantities are changing, either increasing or decreasing, with the same ratio. It is given by the notation, {eq}y \propto x {/eq} and to make the proportionality an equality we introduce the idea of proportionality constant, {eq}k {/eq}. So we have, {eq}y \propto x \to y = kx. {/eq}

If y varies directly with x, then we can say that {eq}y \propto x {/eq} which means that there is a direct proportionality between variables x and y. To make the relation an equality, we introduce the constant of proportionality k so we have {eq}y = kx {/eq}.

We need to calculate the constant k to solve this problem.

{eq}\displaystyle \begin{align*} y &= kx \\ y &= 9,\ \&\ x = 5 \\ 9 &= k(5) \\ k &= \frac{9}{5} \end{align*} {/eq}.

Now we can find the y, when x = 10.

{eq}\displaystyle \begin{align*} y &= kx \\ &= \frac{9}{5}(10)\\ \end{align*} {/eq}.

{eq}\boxed{ y = 18} {/eq}