# y varies inversely with x, and x=4.5 when y=2.4. What is the value of x when the value of y=4.32?

Jun 6, 2018

$\textcolor{b l u e}{x = 2.5}$

#### Explanation:

Inverse variation is given by:

$y \propto \frac{k}{x} ^ n$

Where $\boldsymbol{k}$ is the constant of variation.

To find $\boldsymbol{k}$ we substitute $x = 4.5$ and $y = 2.4$

$2.4 = \frac{k}{4.5}$

$k = 2.4 \cdot 4.5 = 10.8$

When $y = 4.32$

$4.32 = \frac{10.8}{x}$

$x = \frac{10.8}{4.32} = 2.5$

Jun 6, 2018

$x = 2.5$

#### Explanation:

Direct variation uses the equation $y = k x$

Inverse variation uses the equation $y = \frac{k}{x}$

Where $k$ represents the constant of variation.

In order to solve this problem we need to use the numbers provided for the first scenario to solve for the constant of
variation $k$ and then use $k$ to solve for the second set of numbers.

${x}_{1} = 4.5$
${y}_{1} = 2.4$
k=?

$y = \frac{k}{x}$ Equation of Inverse Variation

$2.4 = \frac{k}{4.5}$
Use the multiplicative inverse to isolate $k$

$4.5 \cdot 2.4 = \frac{k}{\cancel{4.5}} \cdot \cancel{4.5}$

$k = 10.8$

x_2=?
${y}_{2} = 4.32$
$k = 10.8$

$y = \frac{k}{x}$ Equation of Inverse Variation

$4.32 = \frac{10.8}{x}$
Use the multiplicative inverse to bring $x$ out of the denominator

$x \cdot 4.32 = \frac{10.8}{\cancel{x}} \cdot \cancel{x}$

$4.32 x = 10.8$

Divide both sides by $4.32$ to isolate $x$

$\frac{\cancel{4.32} x}{\cancel{4.32}} = \frac{10.8}{4.32}$

$x = 2.5$

Jun 6, 2018

$x = 2.5$

#### Explanation:

$y \propto \frac{1}{x} \to \text{Inverse variation}$

$y = \frac{k}{x}$, where $k$ is constant

When;

$x = 4.5 \mathmr{and} y = 2.4$

Substituting the values of $x \mathmr{and} y$ into the equation..

$2.4 = \frac{k}{4.5}$

$\frac{2.4}{1} = \frac{k}{4.5}$

Cross multiplying;

$2.4 \times 4.5 = k \times 1$

$10.8 = k$

Therefore;

$k = 10.8$

Now the relationship between the two unknowns becomes;

$y = \frac{10.8}{x}$

What is $x$ when $y = 4.32$

Substituting the value of $y$ into the relationship equation..

$4.32 = \frac{10.8}{x}$

$\frac{4.32}{1} = \frac{10.8}{x}$

Cross multiplying;

$4.32 \times x = 10.8 \times 1$

$4.32 x = 10.8$

$x = \frac{10.8}{4.32}$

$x = 2.5$