The sum of the digits in a two-digit number is 15. When the digits are reversed, the new number is 9 more than the original number. What is the original number?

Question

I know that the answer is 78 because it is provided on the answer key, but I need help with finding out HOW to get to that..

Answer 1

THX for wanting to know HOW?; also, the question WHY? is very important.  Getting the right answer is only a very little achievement; learning the process is the main goal.

This problem makes us think about the decimal number system (and you will use others, like binary, as you use math).  Decimal is a positional number system (with units, tens, hundreds, … positions).  That means that a value of digit d is different in the numbers represented by d, d4, 3d5, 99d3, and so on.  In the number  d, d is worth d, but in the numbers d4, 3d5, and 99d3, it is worth 10*d.

See if you  can match these math equations to the words of the problem:
       Let 'xy' be a two-digit number.    (with 0<=x<=9 and 0<=y<=9;
                                                                                         and 'xy' means two digits, not multiply)
      x + y = 15
      'yx' = 9 + 'xy'        (to make this easier to understand, we will rewrite it:
                                  10y + x = 9 + (10x + y)
      The problem asks us to find xy.  Well, really, 10x + y, in regular math terms.  So, let’s use those while we do the easy math:

      From x + y = 15, we solve for one of them, say   x = 15-y,   then substitute that into the other equation:
      10y + (15-y) = 9 + 10(15-y) + y                    (now it’s all y, so we can solve for y)
        9y + 15   =   159 - 9y
             18y = 144
               y = 8

The original number was 78.

 Now, plug that value back into either equation (x+y=15 is the easiest):
             x + 8 = 15
             x = 7

Checking (very important):
        Is 7 + 8 = 15  ?    yes.
        Is 87 = 9 + 78   ?    yes.

Answer 2

Let's call the two digits x and y. We know x + y = 15.

 

Sonce it is a two digit number, one of the digits is in the tens place and the other is in the ones place. So we can call the first number 10x + y.

 

The second number has the digits reversed, so that's 10y + x.

 

The second number is 9 more than the first. That means we can add 9 to the first number, 10x + y, to equal the second, 10y + x.

 

In other words, 10x + y + 9 = 10y + x.

 

Use inverse operations to rewrite this equation. I will subtract the 9 from both sides as well as the 10y and the x. I get  9x - 9y = -9.

 

Of course, this entire equation is divisible by 9, so I can divide out 9 from all terms to get this beautifully simple equation.

x - y = -1

 

Going back to what I knew originally, that the digits make a sum of 15, I can now write a system of equations:

 

x + y = 15

x - y = -1

 

Using the elimination method, I add this system together.

 

    x + y = 15

+  x -  y = -1

    2x      = 14

 

That's an easy solve to see that x = 7.

 

Once we know that, finding y is simple.

 

x + y = 15

7 + y = 15

 

y = 8

 

So the initial number was 78.