If 6(A's capital) = 8(B's capital) = 10(C 's capital) , then the ratio of their capital is

Options

A. 3:4:5
B. 12:15:20
C. 20:15:12
D. 6:8:10

Answer

To find the ratio of the capitals of A, B, and C given the equation \(6(A's \, capital) = 8(B's \, capital) = 10(C's \, capital)\), we can follow these steps:

### Step-by-Step Solution:

1. **Set the Common Value**:
Since \(6A = 8B = 10C\), we can set this common value equal to \(N\). Thus, we have:
\[ 6A = N, \quad 8B = N, \quad 10C = N \]

2. **Express A, B, and C in Terms of N**:
From the equations above, we can express A, B, and C in terms of \(N\):
\[ A = \frac{N}{6}, \quad B = \frac{N}{8}, \quad C = \frac{N}{10} \]

3. **Write the Ratio**:
Now, we can write the ratio of A, B, and C:
\[ A : B : C = \frac{N}{6} : \frac{N}{8} : \frac{N}{10} \]

4. **Simplify the Ratio**:
We can simplify this ratio by removing \(N\) from each term:
\[ A : B : C = \frac{1}{6} : \frac{1}{8} : \frac{1}{10} \]

5. **Find the LCM of the Denominators**:
To express the ratio in whole numbers, we need to find the least common multiple (LCM) of the denominators (6, 8, and 10). The LCM of 6, 8, and 10 is 120.

6. **Multiply Each Term by the LCM**:
Now, we multiply each term of the ratio by 120 to eliminate the fractions:
\[ A : B : C = 120 \cdot \frac{1}{6} : 120 \cdot \frac{1}{8} : 120 \cdot \frac{1}{10} \]

7. **Calculate Each Term**:
Now, we calculate each term:
\[ A = \frac{120}{6} = 20, \quad B = \frac{120}{8} = 15, \quad C = \frac{120}{10} = 12 \]

8. **Final Ratio**:
Thus, the final ratio of A, B, and C is:
\[ A : B : C = 20 : 15 : 12 \]

### Final Answer:
The ratio of their capitals is \(20 : 15 : 12\).