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

Given the equation \(ABC - CAB = 243\), we need to find the value of \(\left(\frac{A}{B}\right) \times C\).

First, let's express \(ABC\) and \(CAB\) mathematically. Assume \(A\), \(B\), and \(C\) are digits, and the numbers \(ABC\) and \(CAB\) can be written as:

\[
ABC = 100A + 10B + C
\]

\[
CAB = 100C + 10A + B
\]

The equation given is:

\[
ABC - CAB = 243
\]

Substitute the expressions for \(ABC\) and \(CAB\) into the equation:

\[
(100A + 10B + C) - (100C + 10A + B) = 243
\]

Simplify the equation:

\[
100A + 10B + C - 100C - 10A - B = 243
\]

Combine like terms:

\[
(100A - 10A) + (10B - B) + (C - 100C) = 243
\]

\[
90A + 9B - 99C = 243
\]

Now, simplify further by dividing the entire equation by 9:

\[
10A + B - 11C = 27
\]

So, the equation simplifies to:

\[
10A + B = 27 + 11C
\]

Now, we can test possible values of \(C\) (since \(C\) is a digit from 0 to 9) to find integer values for \(A\) and \(B\).

Let's test different values of \(C\):

- If \(C = 1\):

\[
10A + B = 27 + 11(1) = 38
\]

Possible values for \(A\) and \(B\) are \(A = 3\) and \(B = 8\).

Now we check if \(A = 3\), \(B = 8\), and \(C = 1\) satisfy the original equation \(ABC - CAB = 243\):

\[
ABC = 381 \quad \text{and} \quad CAB = 138
\]

\[
381 - 138 = 243
\]

The values satisfy the equation.

Now, calculate \(\left(\frac{A}{B}\right) \times C\):

\[
\left(\frac{A}{B}\right) \times C = \left(\frac{3}{8}\right) \times 1 = \frac{3}{8}
\]

Thus, the value is \(\frac{3}{8}\).

Would you like to see more details or have any questions?

Here are 5 related questions:
1. What happens if \(C\) is chosen differently?
2. Can \(A\), \(B\), and \(C\) be zero in any case?
3. What if the given difference was negative?
4. How would the solution change if the difference was 100 or 500?
5. How can this problem be solved using modular arithmetic?

**Tip:** When solving problems like this, always check the possible values for digits systematically.

Learn how to solve the problem where ABC - CAB = 243 and find the value of (A/B) * C step-by-step. This problem involves equations with digits, requiring understanding of basic mathematical operations. Suitable for students in Grades 5-7.

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