The digits of a two-digit number sum to 10. If the digits swap places, the number is 36 more than the original number. What is the original number?

Question

The answer must be calculated through the use of simultaneous equations.

Answer 1

If the original number is xy

x + y = 10   {equation 1}

 

By basic definition of base 10 numbers we have,

xy = 10x + y

As in 23 = 2(10) + 3 and so forth.

By the same logic,

yx = 10y + x

 

If the digits swap places, the number is 36 more than the original number:

10y + x = 10x + y + 36

subtract (10x+y) from both sides

-9x + 9y = 36

Divide by -9

x - y = -4    {equation 2}

 

We now have 2 simple simultaneous equations with 2 variables

 

x + y = 10

x - y = -4    add equations

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

2x = 6

x = 3

 

x-y = -4

3-y = -4

-y = -7

y = 7

 

Original number = 37

New number = 73

 

Check:  37 + 36 = 73   yes

             3 + 7 = 10     yes