The sum of the digits of a certain two digit number is 9. Reversing its digits decreases the number by 27. What is the number?

Question

i really don't understand this. 

Answer 1

Hi Sabrina,

The sum of the digits is 9.  If the tens digit is 'x', and the units digit is 9 - x.

 

The original number is ten times the tens digit plus the units digit, so the original number is 10x + 9 - x and that can be rearranged to 10x - x + 9, and simplified to 9x + 9.

 

When the number is reversed, the tens digit will now be ten time 9 - x plus the other digit 'x', which looks like this:

10(9 - x) + x   which can be simplified to 90 - 9x + x  which becomes 90 - 9x.

 

The second sentence in your problem says 'Original number minus the second number = 27'

So....     (9x + 9) - (90 - 9x) = 27     which simplifies to 9x + 9 - 90 + 9x = 27      which simplifies to 18x - 81 = 27, add 81 to each side, 18x = 108   and now divide by 18   to get x = 6.

Since x = 6    then 9 - x = 3

So the tens digit is 6 and units digit is 3.         The 'number' is 63.

 

To check, 63 - 36 = 27 

 

Hope this helps, Sabrina,

Mary Ann

Answer 2

Lets assume that there is a 2-digit number XY and write the system of two linear equations:

 

X + Y = 9

(10X+Y) - (10Y+X) = 27

 

Simplify

 

X + Y = 9

X - Y = 3

 

thus X=6, Y=3 and the number is 63