The sum of the digits...

Question

The sum of the digits of a two-digit counting number is 15. When the digits are reversed, the new number is 27 more than the original number. What was the original number?

Answer 1

x + y = 15

 

Our original number is xy  which, as a sum is actually 10x + y.

When we reverse the two digit number from xy to yx

we actually have 10y + x  ... That's how numbers work in base 10.

 

Our equation for the new number is 27 more than the original

looks like:

 

10y + x = 10x + y + 27

 

Let's rearrange this to look like   ax + by = c

 

9x - 9y = -27      divide through by 9

x - y = -3

 

So we now have two equations:

 

x + y = 15

x - y = -3          add the two equations to eliminate y

-------------

2x = 12

x = 6

 

x+y=15

6+y = 15

y = 9

 

The original number is 69

 

check:  x + y = 15    6+9=15    yes

 

           96 = 69 + 27    yes

 

Hope this helps.  Good luck with your class.

 

Answer 2

So we know two things:

 

1]  x + y = 15

 

2]  yx = xy + 27  (note, yx and xy are not being multiplied here)

 

As a consequence of 2],  we also know that yx > xy, which means y > x.

 

Now you can solve this one intuitively by thinking about what two single digits, the smaller one first, add up to 15.

 

7 + 8 doesn't work, because, 78 + 27 ≠ 87

6 + 9 is the next logical choice, and 69 + 27 does equal 96.

 

The original number is 69.

 

Hope this helps Rachel!