# The sum of the digits of a two-digit number is 14. If the numbers are reversed, the new number is 18 less than the original number. Find the original number.

I know Ana asked this question, but i don't understand how to get the equations.

## I thought Damon did a pretty good job of explaining this question before

Ok, here is my approach, perhaps it will make sense to you.

Let the unit digit of the original number be x
Let the tens digit by y
then the original number was 10y+x

We were told the sum of the digits is 14, so
x+y=14, this is your first equation

the number reversed would be 10x+y
but this is 18 less than the original number, so....

10x+y + 18 = 10y+x , (since it was 18 less, I added 18 to make them "equal")

9x - 9y = -18
x-y = -2 , this is your second equation.

I will leave it up to you to solve them

## would you agree that according to my definition, the original number is 10y+x and the number reversed is 10x+y ???

your problem stated "the new number is 18 less than the original number." which translates into

10x+y < 10y+x by 18, so I added 18 to the "smaller" side to make them "equal", thus

10x+y + 18 = 10y+x
surely you can see how that simplifies to x-y=-2

## Do the digits of 86 add up to 14?

Is 68 less than 86 by 18 ??

## To solve this problem, let's assume the two-digit number is represented as "10x + y," where x represents the tens digit, and y represents the units digit. We're given two pieces of information:

1. The sum of the digits is 14: x + y = 14
2. Reversing the digits results in a number 18 less than the original: 10y + x = 10x + y - 18

We can solve these two equations simultaneously to find the values of x and y, and ultimately determine the original number. Here's how:

This equation tells us that the sum of the tens and units digit is 14.

2. Simplify the second equation: 10y + x = 10x + y - 18
Rearrange the equation to bring all variables on one side:
10y - y = 10x - x - 18
Simplify both sides:
9y = 9x - 18
Divide both sides by 9 to isolate y:
y = x - 2

3. Substitute the value of y in terms of x into the first equation:
x + (x - 2) = 14
Simplify:
2x - 2 = 14
2x = 16
Divide both sides by 2:
x = 8

4. Substitute x = 8 back into the equation for y:
y = 8 - 2
y = 6

5. Now, we have the values of x and y. The original two-digit number is 10x + y:
Original number = 10(8) + 6 = 80 + 6 = 86

Therefore, the original number is 86.