A boy's age if formed by reversing the two digits in his grandfather 's ages. If the sum of the digits used 9 and the grandfather is 63 years older than his grandson. Find their present ages?

please solve by process

81 and 18

Let's assume the boy's age is represented by the digits "ab" where 'a' is the tens digit and 'b' is the unit digit.

The grandfather's age is then represented by the digits "ba".

We also know that the sum of these two digits is 9.

So, the equation we can set up is: a + b = 9 --- (1)

Additionally, we are given that the grandfather is 63 years older than the grandson.

So, the equation we can set up is: 10b + a = 10a + b + 63 --- (2)

To solve these equations, let's simplify equation (2):

10b + a = 10a + b + 63
10b - b = 10a - a + 63
9b = 9a + 63
b = a + 7 --- (3)

Now, substitute equation (3) into equation (1):

a + (a + 7) = 9
2a + 7 = 9
2a = 9 - 7
2a = 2
a = 1

Substitute the value of 'a' back into equation (3):

b = a + 7
b = 1 + 7
b = 8

Therefore, the boy's age (ab) is 18 and the grandfather's age (ba) is 81.

To solve this problem, let's denote the grandson's age as "ab" (where "a" represents the tens digit and "b" represents the units digit) and the grandfather's age as "ba" (where "b" represents the tens digit and "a" represents the units digit).

According to the given information, the sum of the digits used is 9. So, we have two possibilities: a + b = 9 or b + a = 9.

Now, it is also mentioned that the grandfather is 63 years older than his grandson. Therefore, we can set up the following equation:

10b + a = 10a + b + 63

Simplifying, we get:
10b + a - 10a - b = 63
9b - 9a = 63
b - a = 7

Now let's consider the two possibilities for a + b = 9:
1. If a = 1 and b = 8, then b - a = 8 - 1 = 7, which satisfies our equation.
2. If a = 2 and b = 7, then b - a = 7 - 2 = 5, which does not satisfy our equation.

Therefore, a = 1 and b = 8.

Hence, the grandson's age is 18 years and the grandfather's age is 81 years.