# The sum of two numbers is 78. Their difference is 32. What are these numbers?

Feb 25, 2017

$\left(1\right) : x + y = 78 \text{ & } \left(2\right) : x - y = 32 , \left(x > y\right) .$

$\left(1\right) + \left(2\right) \Rightarrow 2 x = 110 \Rightarrow x = 55.$

& then, by $\left(1\right) , y = 78 - 55 = 23.$

Feb 25, 2017

$55$ and $23$

#### Explanation:

Given two numbers $a$ and $b$, take their sum and difference:

$a + b$

$a - b$

Then take the sum and difference of these two expressions:

$\left(a + b\right) + \left(a - b\right) = 2 a$

$\left(a + b\right) - \left(a - b\right) = 2 b$

Notice that we get back to the two numbers we started with, but doubled.

So with $78$ and $32$, form their sum and difference and halve the results to get the two original numbers:

$\frac{78 + 32}{2} = \frac{110}{2} = 55$

$\frac{78 - 32}{2} = \frac{46}{2} = 23$