If `a : b = 3/2 : 7/3` and `b : c= 1/5 : 1/7` , then find `a : b : c`.
(a)`14 : 9 : 10`
(b)`9 : 14 : 10`
(c)`10 : 9 : 14`
(d)`4 : 5 : 7`

Options

A. `14 : 9 : 10`
B. `9 : 14 : 10`
C. `10 : 9 : 14`
D. `4 " 5 : 7`

Answer

To find the ratio \( a : b : c \) given \( a : b = \frac{3}{2} : \frac{7}{3} \) and \( b : c = \frac{1}{5} : \frac{1}{7} \), we will follow these steps:

### Step 1: Convert \( a : b \) into a simple ratio
We have:
\[ a : b = \frac{3}{2} : \frac{7}{3} \]
To eliminate the fractions, we find the least common multiple (LCM) of the denominators (2 and 3), which is 6. We multiply both parts of the ratio by 6:
\[ a : b = 3 \times 3 : 7 \times 2 = 9 : 14 \]

### Step 2: Convert \( b : c \) into a simple ratio
Next, we have:
\[ b : c = \frac{1}{5} : \frac{1}{7} \]
Again, we find the LCM of the denominators (5 and 7), which is 35. We multiply both parts of the ratio by 35:
\[ b : c = 1 \times 7 : 1 \times 5 = 7 : 5 \]

### Step 3: Combine the ratios \( a : b \) and \( b : c \)
Now we have:
\[ a : b = 9 : 14 \quad \text{and} \quad b : c = 7 : 5 \]
To combine these ratios into \( a : b : c \), we need to make sure that \( b \) is the same in both ratios. The value of \( b \) in the first ratio is 14, and in the second ratio, it is 7. We can equalize \( b \) by finding a common multiple. The least common multiple of 14 and 7 is 14.

We can express \( b : c \) as:
\[ b : c = 14 : 10 \quad (\text{since } 7 \times 2 = 14 \text{ and } 5 \times 2 = 10) \]

### Step 4: Write the combined ratio
Now we can write:
\[ a : b : c = 9 : 14 : 10 \]

### Conclusion
Thus, the final ratio \( a : b : c \) is:
\[ \boxed{9 : 14 : 10} \]