# The sum of the digits of a two-digit number is 10. If the digits are reversed, a new number is formed. The new number is one less than twice the original number. How do you find the original number?

May 21, 2018

Original number was $37$

#### Explanation:

Let $m \mathmr{and} n$ be the first and second digits respectively of the original number.

We are told that: $m + n = 10$
$\to n = 10 - m$ [A]

Now. to form the new number we must reverse the digits. Since we can assume both numbers to be decimal, the value of the original number is $10 \times m + n$ [B]

and the new number is: $10 \times n + m$ [C]

We are also told that the new number is twice the original number minus 1.

Combining [B] and [C] $\to 10 n + m = 2 \left(10 m + n\right) - 1$ [D]

Replacing [A] in [D]

$\to 10 \left(10 - m\right) + m = 20 m + 2 \left(10 - m\right) - 1$

$100 - 10 m + m = 20 m + 20 - 2 m - 1$

$100 - 9 m = 18 m + 19$

$27 m = 81$

$m = 3$

Since $m + n = 10 \to n = 7$

Hence the original number was: $37$

Check: New number $= 73$

$73 = 2 \times 37 - 1$