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# The sum of the digits of a certain two-digit number is 7. Reversing its digits increases the number by 9. What is the number?

Jan 22, 2016

b=4 a=3

$\textcolor{b l u e}{\text{The first digit is 3 and the second 4 so the original number is 34}}$

To be honest! It would be much quicker to solve by trial and error.

#### Explanation:

$\textcolor{m a \ge n t a}{\text{Building the equations}}$

Let the first digit be $a$
Let the second digit be $b$

$\textcolor{b l u e}{\text{The first condition}}$

$a + b = 7$ ...............................(1)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{The Second condition}}$

$\textcolor{g r e e n}{\text{The first order value: }}$
$\textcolor{w h i t e}{\times \times} a$ is a counting in tens. So actual value is $10 \times a$
$\textcolor{w h i t e}{\times \times} b$ is counting in units. So actual value is $1 \times b$

$\textcolor{g r e e n}{\text{The first Order Value} = 10 a + b}$...............................(2)
'-----------------------------------------------------------------------'
$\textcolor{p u r p \le}{\text{The second order value:}}$

$\textcolor{w h i t e}{\times \times} b$ is a counting in tens. So actual value is $10 \times b$
$\textcolor{w h i t e}{\times \times} a$ is counting in units. So actual value is $1 \times a$

$\textcolor{p u r p \le}{\text{The second Order Value} = 10 b + a}$.........................(3)
'----------------------------------------------------------------------'

From the question
$\textcolor{red}{\text{Equation (3)" - "Equation (2)} = 9}$.................................(4)

$\textcolor{m a \ge n t a}{\text{|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||}}$

$\textcolor{b l u e}{\text{Putting it all together}}$

$\text{Equation 4 becomes} \to \left(10 b + a\right) - \left(10 a + b\right) = 9$

$9 b - 9 a = 9 \textcolor{w h i t e}{. .} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left({4}_{a}\right)$
$a + b = 7 \textcolor{w h i t e}{. .} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(1\right)$

From equation (1)
$a = 7 - b$

Substitute in $\left({4}_{a}\right)$ giving:
$9 b - 9 \left(7 - b\right) = 9$

$9 b + 9 b - 63 = 9$

$18 b = 72$

$\textcolor{b l u e}{b = \frac{72}{18} = 4}$

Substitute in Equation (1) giving
$a + b = 7 \to a + 4 = 7$

$\textcolor{b l u e}{a = 3}$