Sophie's favorite number is a two-digit number. If she reverses the digits, the result is 45 less than her favorite number. Also, one digit is one less than double the other digit. What is Sophie's favorite number?

Guest May 7, 2023

#1**+1 **

*Sophie's favorite number is a two-digit number. If she reverses the digits, the result is 45 less than her favorite number. Also, one digit is one less than double the other digit. What is Sophie's favorite number?*

Let's call the digits A and B,

and arbitarily select A to be in the 10's position of Sophie's favorite number.

So (10A + B) = (10B + A) + 45 Remember this; we'll use it later.

Also, one digit is one less than double the other. But we don't know which is which.

So it's either A = 2B – 1 or B = 2A – 1

Now all I know what to do is brute force ... that is, try everything.

IF A = 2B – 1 (10 * (2B – 1)) + B = (10 * B) + (2B – 1) + 45

20B – 10 + B = 10B + 2B – 1 + 45

21B – 10 = 12B + 44

9B = 54

B = 6 which makes A = 11 not a digit

IF B = 2A – 1 (10 * A) + (2A – 1)) = (10 * (2A – 1) ) + A + 45

10A + 2A – 1 = 20A – 10 + A + 45

12A – 1 = 21A + 35

9A = –36

A = –4 which makes B = –9

So if Sophie's favorite number is –49 then the digits reversed makes –94.

The difference is 45, and it's in the negative direction, so –94 is less than –49.

So –49 does satisfy the stipulation. Sophie's favorite number is **–49**.

_{.}

Guest May 7, 2023