# The sum of the digits of a two digit number is 12. When the digits are reversed the new number is 18 less than the original number. How do you find the original number?

Sep 24, 2015

Express as two equations in the digits and solve to find original number $75$.

#### Explanation:

Suppose the digits are $a$ and $b$.

We are given:

$a + b = 12$

$10 a + b = 18 + 10 b + a$

Since $a + b = 12$ we know $b = 12 - a$

Substitute that into $10 a + b = 18 + 10 b + a$ to get:

$10 a + \left(12 - a\right) = 18 + 10 \left(12 - a\right) + a$

That is:

$9 a + 12 = 138 - 9 a$

Add $9 a - 12$ to both sides to get:

$18 a = 126$

Divide both sides by $18$ to get:

$a = \frac{126}{18} = 7$

Then:

$b = 12 - a = 12 - 7 = 5$

So the original number is $75$