# The sum of the digits of a two-digit number is 7. When the digit are reversed,the numbers is increase by 27. How do you find the numbers?

May 16, 2018

The digits are $2$ and $5$

#### Explanation:

If we let the digits be $\textcolor{b l u e}{a}$ and $\textcolor{b l u e}{b}$
then the possible numbers composed of those two digits are
$\textcolor{b l u e}{10 a + b}$ and $\textcolor{b l u e}{10 b + a}$

We are told
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{a} + \textcolor{b l u e}{b} = 7$
and (assuming $10 a + b$ is the larger composite number)
$\textcolor{w h i t e}{\text{XXX}} \left(\textcolor{b l u e}{10 a + b}\right) - \left(\textcolor{b l u e}{10 b + a}\right) = 27$

 simplifies into
$\textcolor{w h i t e}{\text{XXX}} 9 a - 9 b = 27$
or
$\textcolor{w h i t e}{\text{XXX}} a - b = 3$

Adding  and , we get
$\textcolor{w h i t e}{\text{XXX}} 2 a = 10$
which implies
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{a} = 5$

Substituting $5$ for $\textcolor{b l u e}{a}$ back in 
we see that
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{b} = 2$

May 16, 2018

The two digit number is $25$

#### Explanation:

Let tens digit and units digit of the number be

x and y ; x+y=7; (1)

The two digit number is $10 x + y$, when reversed,

the two digit number becomes $10 y + x$, by given condition,

$10 y + x = 10 x + y + 27 \mathmr{and} 9 y - 9 x = 27$ or

$9 \left(y - x\right) = 27 \mathmr{and} \left(y - x\right) = 3 \mathmr{and} y = x + 3$ Putting

$y = x + 3$ in equation (1) we get,  x+x +3=7;  or

2 x= 4 :. x =2 ; y= 2+3=5

Hence the two digit number is $25$ [Ans]