# One number is 2/3 of another number. The sum of the two numbers is 10. How do you find the two numbers?

Oct 26, 2016

The two numbers are $4$ and $6$.

#### Explanation:

Let one number be represented as $x$ and the other as $y$.

According to the problem:

$x = \frac{2}{3} y$ and $x + y = 10$

From the second equation we get:

$x + y = 10$

$\therefore \textcolor{red}{y = 10 - x}$ (subtracting $x$ from both sides)

Replacing the value of $y$ in the first equation we get:

$x = \frac{2}{3} \textcolor{red}{y}$

$x = \frac{2}{3} \textcolor{red}{\left(10 - x\right)}$

Multiplying both sides by $3$ we get:

$3 x = 2 \left(10 - x\right)$

Opening the brackets and simplifying we get:

$3 x = 20 - 2 x$

Add $2 x$ to both sides.

$5 x = 20$

Divide both sides by $5$.

$x = 4$

Since from the second equation we have:

$x + y = 10$

substituting $x$ with $4$ we get:

$4 + y = 10$

Subtract $4$ from both sides.

$y = 6$

Oct 26, 2016

The numbers are 4 and 6.

#### Explanation:

This question can also be done by using just one variable.
Define each variable and then form an equation.

Let the larger number be $x$.
The other number is $\frac{2}{3} x$

The sum of the numbers is 10.

$x + \frac{2}{3} x = 10 \text{ } \leftarrow$ multiply by 3

$3 x + \frac{3 \times 2 x}{3} = 30$

$3 x + 2 x = 30$

$5 x = 30$

$x = \frac{30}{5} = 6 \text{ } \leftarrow$this is the larger number

$\frac{2}{3} \left(6\right) = 4 \text{ } \leftarrow$ this is the smaller number.

The numbers are 4 and 6.