Some money is divided among A,B and C in such a way that 5 times A'S share 3 times B'S share and 2 times C'S share are all equal the ratio between the shares of A,B,C is

Options

A. 5 : 3 : 2
B. 2 : 2 : 5
C. 15 : 10 : 6
D. 6 : 10 : 15

Answer

To solve the problem, we need to express the shares of A, B, and C in terms of a common variable based on the information given.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that:
- \( 5 \times \text{A's share} = 3 \times \text{B's share} = 2 \times \text{C's share} \)
Let's denote A's share as \( a \), B's share as \( b \), and C's share as \( c \).

2. **Setting Up the Equations**:
From the given information, we can write:
- \( 5a = 3b \) (1)
- \( 5a = 2c \) (2)

3. **Expressing B and C in terms of A**:
From equation (1):
\[ b = \frac{5a}{3} \]
From equation (2):
\[ c = \frac{5a}{2} \]

4. **Finding a Common Ratio**:
Now we have:
- A's share: \( a \)
- B's share: \( \frac{5a}{3} \)
- C's share: \( \frac{5a}{2} \)

5. **Finding a Common Denominator**:
To express all shares in a common ratio, we need to find a common denominator for \( \frac{5a}{3} \) and \( \frac{5a}{2} \). The least common multiple of 3 and 2 is 6.

- A's share: \( a = \frac{6a}{6} \)
- B's share: \( b = \frac{5a}{3} = \frac{10a}{6} \)
- C's share: \( c = \frac{5a}{2} = \frac{15a}{6} \)

6. **Writing the Final Ratio**:
Now we can express the shares as:
- A : B : C = \( 6a : 10a : 15a \)

Removing \( a \) from the ratio, we get:
- A : B : C = \( 6 : 10 : 15 \)

7. **Simplifying the Ratio**:
To simplify this ratio, we can divide each term by the greatest common divisor (GCD), which is 1 in this case:
- A : B : C = \( 6 : 10 : 15 \)

This can also be simplified further:
- Dividing by 5 gives us \( 2 : 2 : 3 \)

8. **Final Answer**:
The final ratio of A, B, and C is:
- A : B : C = \( 2 : 3 : 5 \)