A number consists of 3 digits whose sum is 10. The middle digit is equal to the sum of the other two and the number will be increased by 99 if its digits are reversed. What is the number?

The sum of the three digits is 10: A + B + C = 10

The middle digit (B) is equal to the sum of the other two digits (A and C): B = A + C

The number will be increased by 99 if its digits are reversed. This means that if we reverse the order of the digits, we get a new number that is 99 greater than the original number.

Now, let's use this information to solve for the values of A, B, and C.

From equation 1, we know that A + B + C = 10.

From equation 2, we know that B = A + C.

Now, we can substitute the value of B from equation 2 into equation 1:

A + (A + C) + C = 10

Simplify the equation:

2A + 2C = 10

Divide both sides by 2:

A + C = 5

Now, we have two equations:

1. A + C = 5
2. B = A + C

Since B = A + C and A + C = 5, we can conclude that B = 5.

Now that we know B is 5, we can use the fact that the number will be increased by 99 if its digits are reversed. This means that the original number is 99 less than the reversed number.

Let's represent the original number as ABC and the reversed number as CBA. The difference between the reversed and original numbers is 99:

CBA - ABC = 99

Now, substitute the values of B (5) and A + C (5) into the equations:

(100C + 10B + A) - (100A + 10B + C) = 99

Now, simplify:

100C + 10B + A - 100A - 10B - C = 99

Combine like terms:

99C - 99A = 99

Now, divide both sides by 99:

C - A = 1

Now, we have two equations:

1. A + C = 5
2. C - A = 1

Solve this system of equations:

From equation 2, we can express C in terms of A: C = A + 1.

Now, substitute this into equation 1:

A + (A + 1) = 5

Combine like terms:

2A + 1 = 5

Subtract 1 from both sides:

2A = 4

Divide by 2:

A = 2

Now that we know A is 2, we can find C:

C = A + 1 = 2 + 1 = 3

So, A = 2, B = 5, and C = 3. Therefore, the number is ABC = 253.