# There is a two digit number where the sum of the digits is 7. The actual value of the number is such that it equals 12 times the first digit minus 2. What is the number?

Feb 20, 2017

52

#### Explanation:

Let the first digit be $a \text{ "->" actual value is } 10 a$
Let the second digit be $b$

Then the number is $a b$ Note: this is not $a \times b$

Sum of the digits is:
$a + b = 7 \text{ " =>" } b = 7 - a$ Equation(1)

Considering the actual value

$\textcolor{g r e e n}{10 a + b} = 12 a - 2$ Equation(2)
$\text{ } \textcolor{g r e e n}{\uparrow}$
$\textcolor{g r e e n}{\text{The actual value}}$

Using Equation(1) substitute for $b$ in Equation(2)

$10 a + \left(7 - a\right) = 12 a - 2$

$11 a + 7 = 12 a - 2$

$\textcolor{red}{a = 5}$

From Equation(1)

$a + b = 7 \text{ } \implies \textcolor{red}{b = 2}$
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The number is $52$