Question

The sum of the digits of a two-digit number is 7. When the digit are reversed,the numbers is increase by 27. How do you find the numbers?

Answer 1

The digits are `2` and `5`

Explanation:

If we let the digits be `color(blue)a` and `color(blue)b`
then the possible numbers composed of those two digits are
`color(blue)(10a+b)` and `color(blue)(10b+a)`

We are told
[1]`color(white)("XXX")color(blue)a+color(blue)b=7`
and (assuming `10a+b` is the larger composite number)
[2]`color(white)("XXX")(color(blue)(10a+b))-(color(blue)(10b+a))=27`

[2] simplifies into
[3]`color(white)("XXX")9a-9b=27`
or
[4]`color(white)("XXX")a-b=3`

Adding [1] and [4], we get
[5]`color(white)("XXX")2a=10`
which implies
[6]`color(white)("XXX")color(blue)a=5`

Substituting `5` for `color(blue)a` back in [1]
we see that
[7]`color(white)("XXX")color(blue)b=2`

Answer 2

The two digit number is `25`

Explanation:

Let tens digit and units digit of the number be

`x and y ; x+y=7; (1)`

The two digit number is `10 x+y`, when reversed,

the two digit number becomes `10 y+x`, by given condition,

`10 y+x =10 x+y+27 or 9 y -9 x =27` or

`9 (y - x) =27 or (y-x)=3 or y = x+3` Putting

`y=x+3` in equation (1) we get, `x+x +3=7;` or

`2 x= 4 :. x =2 ; y= 2+3=5`

Hence the two digit number is `25` [Ans]