If a : b = 3 : 7 and b : c = 14 : 5, then find a : b : c.

Options

A. `6:14:5`
B. `6:7:5`
C. `3:7:5`
D. `3:14:5`

Answer

To solve the problem of finding the ratio \( a : b : c \) given \( a : b = 3 : 7 \) and \( b : c = 14 : 5 \), we can follow these steps:

### Step 1: Write the given ratios in fraction form
We start with the given ratios:
- \( a : b = 3 : 7 \) can be written as \( \frac{a}{b} = \frac{3}{7} \)
- \( b : c = 14 : 5 \) can be written as \( \frac{b}{c} = \frac{14}{5} \)

### Step 2: Express \( a \) and \( c \) in terms of \( b \)
From the first ratio \( \frac{a}{b} = \frac{3}{7} \), we can express \( a \) in terms of \( b \):
\[ a = \frac{3}{7}b \]

From the second ratio \( \frac{b}{c} = \frac{14}{5} \), we can express \( c \) in terms of \( b \):
\[ c = \frac{5}{14}b \]

### Step 3: Substitute the expressions of \( a \) and \( c \) into the ratio \( a : b : c \)
Now we can express the entire ratio \( a : b : c \) in terms of \( b \):
\[ a : b : c = \frac{3}{7}b : b : \frac{5}{14}b \]

### Step 4: Eliminate \( b \) from the ratio
We can eliminate \( b \) from the ratio by dividing each term by \( b \):
\[ a : b : c = \frac{3}{7} : 1 : \frac{5}{14} \]

### Step 5: Find a common denominator
To combine these ratios, we need a common denominator. The denominators are 7, 1, and 14. The least common multiple of these is 14. We can express each term with this common denominator:
- \( \frac{3}{7} = \frac{3 \times 2}{7 \times 2} = \frac{6}{14} \)
- \( 1 = \frac{1 \times 14}{1 \times 14} = \frac{14}{14} \)
- \( \frac{5}{14} = \frac{5}{14} \)

### Step 6: Write the final ratio
Now we can write the ratio as:
\[ a : b : c = 6 : 14 : 5 \]

### Conclusion
Thus, the final ratio \( a : b : c \) is \( 6 : 14 : 5 \).

---