If `(A)/(3) = (B)/(2) = (C )/(5)`, then what is the value of ratio `(C +A)^(2): (A+B)^(2) : (B+C)^(2)`?

Options

A. `9: 4: 25`
B. `25: 4: 9`
C. `64: 25: 49`
D. `49: 25: 64`

Answer

To solve the problem, we start with the given equation:

\[ \frac{A}{3} = \frac{B}{2} = \frac{C}{5} = k \]

Here, \( k \) is a constant. From this, we can express \( A \), \( B \), and \( C \) in terms of \( k \):

1. **Finding values of A, B, and C:**
- From \(\frac{A}{3} = k\), we have:
\[ A = 3k \]
- From \(\frac{B}{2} = k\), we have:
\[ B = 2k \]
- From \(\frac{C}{5} = k\), we have:
\[ C = 5k \]

2. **Calculating \( C + A \), \( A + B \), and \( B + C \):**
- \( C + A = 5k + 3k = 8k \)
- \( A + B = 3k + 2k = 5k \)
- \( B + C = 2k + 5k = 7k \)

3. **Finding the squares of these sums:**
- \( (C + A)^2 = (8k)^2 = 64k^2 \)
- \( (A + B)^2 = (5k)^2 = 25k^2 \)
- \( (B + C)^2 = (7k)^2 = 49k^2 \)

4. **Setting up the ratio:**
Now we need to find the ratio:
\[ (C + A)^2 : (A + B)^2 : (B + C)^2 = 64k^2 : 25k^2 : 49k^2 \]

5. **Simplifying the ratio:**
Since \( k^2 \) is common in all terms, we can simplify the ratio:
\[ 64 : 25 : 49 \]

Thus, the final answer is:
\[ \text{The value of the ratio } (C + A)^2 : (A + B)^2 : (B + C)^2 \text{ is } 64 : 25 : 49. \]