Let's solve the problem step by step.

Problem:

The digits of a 2-digit number differ by 5. If the digits are interchanged and the resulting number is added to the original number, we get 99. Find the original number. Write the reversed number.

Solution:

Let the original 2-digit number be represented as $10x + y$, where:

• $x$ is the tens digit.
• $y$ is the units digit.

From the problem, we have the following conditions:

1. The digits differ by 5: $|x - y| = 5$

2. When the digits are interchanged, the resulting number $10y + x$ is added to the original number to give 99: $(10x + y) + (10y + x) = 99$

Let's simplify this equation:

$10x + y + 10y + x = 99$

$11x + 11y = 99$

$x + y = 9 \quad \text{(Equation 1)}$

Now, using the condition that the digits differ by 5:

$|x - y| = 5$

This gives us two cases:

Case 1: $x - y = 5$

From Equation 1:

$x + y = 9$

Adding the two equations:

$2x = 14 \implies x = 7$

Substituting $x = 7$ in Equation 1:

$7 + y = 9 \implies y = 2$

So, the original number is $72$, and the reversed number is $27$.

Case 2: $y - x = 5$

From Equation 1:

$x + y = 9$

Adding the two equations:

$2y = 14 \implies y = 7$

Substituting $y = 7$ in Equation 1:

$x + 7 = 9 \implies x = 2$

So, the original number is $27$, and the reversed number is $72$.

Answer:

• The original numbers are $72$ and $27$.
• If the original number is $72$, the reversed number is $27$. If the original number is $27$, the reversed number is $72$.

Let me know if you need further clarification!

Questions for Further Exploration:

1. How would the solution change if the digits differed by a different number, like 4?
2. What happens if the sum of the original and reversed number equals a different total, such as 108?
3. Can the original number be greater than 72?
4. What if we had a three-digit number with similar conditions?
5. How does changing the condition $|x - y| = 5$ to a product condition affect the outcome?

Tip:

When solving equations involving digits, remember to systematically set up relationships between the variables based on given conditions.