If ABC - CAB =243,Find the value of (A÷B)xC 

Let's solve the equation \( ABC - CAB = 243 \).

First, let's express the numbers \( ABC \) and \( CAB \) in terms of \( A \), \( B \), and \( C \):
- \( ABC = 100A + 10B + C \)
- \( CAB = 100C + 10A + B \)

Given:
\[
ABC - CAB = 243
\]
Substituting the values:
\[
(100A + 10B + C) - (100C + 10A + B) = 243
\]
Simplifying the equation:
\[
100A + 10B + C - 100C - 10A - B = 243
\]
\[
(100A - 10A) + (10B - B) + (C - 100C) = 243
\]
\[
90A + 9B - 99C = 243
\]
Now, simplify the equation by dividing everything by 9:
\[
10A + B - 11C = 27
\]
This equation relates \( A \), \( B \), and \( C \). 

We need to find the value of \( \frac{A}{B} \times C \). To solve for \( A \), \( B \), and \( C \), let's test different digits that satisfy the equation \( 10A + B - 11C = 27 \).

### Case 1: Assume \( C = 1 \)
\[
10A + B - 11(1) = 27 \implies 10A + B = 38
\]
This is valid when \( A = 3 \) and \( B = 8 \).

### Calculation
Now that \( A = 3 \), \( B = 8 \), and \( C = 1 \), let's calculate \( \frac{A}{B} \times C \):
\[
\frac{A}{B} \times C = \frac{3}{8} \times 1 = \frac{3}{8} = 0.375
\]

Therefore, the value of \( \frac{A}{B} \times C \) is \( 0.375 \).

Would you like more details on how I arrived at this result, or do you have any other questions? 

Here are five related questions to further your understanding:
1. What would the value of \( \frac{A}{B} \times C \) be if \( CAB - ABC = 243 \) instead?
2. Can you determine a general formula for similar problems where digits are rearranged?
3. How would the solution change if \( C = 2 \) instead?
4. What other combinations of \( A \), \( B \), and \( C \) could satisfy the equation \( 10A + B - 11C = 27 \)?
5. How can this problem be generalized for numbers with more than three digits?

**Tip:** When dealing with digit-based puzzles, carefully express the numbers in terms of their positional values (units, tens, hundreds, etc.) to establish the relationship between the digits.

Learn how to solve the equation ABC - CAB = 243 with step-by-step explanations and calculations. Explore the relationship between digits and positional notation in this mathematical problem.

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