Prabhu reversed the digit of the two-digit number ‘x’ and labelled as ‘y’. The difference between x and y is 72. Find the sum of possible values of x which is greater than 50.

Given that \( x \) is a two-digit number and \( y \) is the number obtained by reversing the digits of \( x \), we need to find the values of \( x \) where the difference \( x - y = 72 \). Additionally, we are interested in the sum of possible values of \( x \) that are greater than 50.

Let \( x = 10a + b \) where \( a \) and \( b \) are the digits of \( x \), with \( a \) being the tens digit and \( b \) being the units digit. Then, the reversed number \( y \) is represented as \( y = 10b + a \).

The given condition is:
\[
x - y = 72
\]

Substitute the expressions for \( x \) and \( y \):
\[
(10a + b) - (10b + a) = 72
\]

Simplify the equation:
\[
10a + b - 10b - a = 72
\]
\[
9a - 9b = 72
\]
\[
a - b = 8
\]

Since \( a \) and \( b \) are digits (0-9), and \( x \) is a two-digit number, \( a \) must be between 1 and 9. \( b \) is calculated as:
\[
b = a - 8
\]

For \( b \) to be a valid digit (0-9):
\[
0 \leq a - 8 \leq 9
\]
\[
8 \leq a \leq 9
\]

Thus, \( a \) can be either 8 or 9. Let's find the corresponding \( x \) values for these cases.

1. If \( a = 8 \):
\[
b = 8 - 8 = 0
\]
\[
x = 10a + b = 10 \cdot 8 + 0 = 80
\]

2. If \( a = 9 \):
\[
b = 9 - 8 = 1
\]
\[
x = 10a + b = 10 \cdot 9 + 1 = 91
\]

Both \( x = 80 \) and \( x = 91 \) are greater than 50.

Finally, we sum these values:
\[
80 + 91 = 171
\]

Therefore, the sum of possible values of \( x \) which are greater than 50 is:
\[
\boxed{171}
\]

Would you like details on any step or have any further questions?

---

Here are 8 related questions to consider:
1. What are other possible conditions for digit manipulation in two-digit numbers?
2. How would the solution change if \( x - y \) were a different value?
3. Can this problem be extended to three-digit numbers? How?
4. What if \( x + y \) were given instead of \( x - y \)?
5. How can we determine the validity of other digit manipulations in different number ranges?
6. What would be the steps if \( x - y \) resulted in a negative value?
7. How can we verify the solution for \( x - y = 72 \) programmatically?
8. What are the properties of digit manipulation in base 10 that affect such problems?

**Tip:** When solving digit manipulation problems, carefully define the relationship between the original number and its manipulated form using algebraic expressions to streamline the process.

Learn how to solve the problem of reversing the digits of a two-digit number and finding the sum of possible values greater than 50. This problem involves concepts from number theory and algebra, including digit manipulation and equation solving. Suitable for students in Grades 7-9.

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