# Question #f7758

Dec 18, 2017

$y = 11.5$
We can know if something is a direct variation, if both the quantities increase or decrease simultaneously.

#### Explanation:

$y$ varies directly with $x$

Direct variation means: When the value of $x$increases, the value of $y$ also increases, and vice versa, i.e. the values of both the variables will increase or decrease simultaneously.

Writing with proportionality sign: $x \propto y$

When the proportionality sign is replaced by equal- to sign, a proportionality constant is to be multiplied:

$x = a \times y$ .... where $a$ is proportionality constant.

Given : y=5 when x=160,

$\implies x = a y$
OR
$y = \frac{x}{a}$

We will first calculate the value of proportionality constant from above equation:
$\implies a = \frac{x}{y}$

$\implies a = \frac{160}{5} = 32$

Asked :what is $y$ when $x = 368$ ?

$y = \frac{x}{a} = \frac{368}{32} = 11.5$

We can see that as the value of $x$ increased, the value of $y$ also increased.
Hence we can know if something is a direct variation by observing the variation proportionality.

When the value of one variable decreases with increase in value of other (and vice versa), it is inverse variation. Ex: If the speed of a vehicle is more, the time taken to cover a certain distance is less. Here Speed and time are the variables which are inversely proportional.

Dec 18, 2017

$y = 11.5$

#### Explanation:

$\text{the initial statement is } y \propto x$

$\text{to convert to an equation multiply by k the constant}$
$\text{of variation}$

$\Rightarrow y = k x$

$\text{to find k use the given condition}$

$y = 5 \text{ when } x = 160$

$y = k x \Rightarrow k = \frac{y}{x} = \frac{5}{160} = \frac{1}{32}$

$\text{equation is } \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = \frac{x}{32}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{when } x = 368$

$y = \frac{368}{32} = 11.5$