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 ratios in fraction form
We start by expressing the given ratios as fractions:
- From \( a : b = 3 : 7 \), we can write it as:
\[ \frac{a}{b} = \frac{3}{7} \]
- From \( b : c = 14 : 5 \), we can write it as:
\[ \frac{b}{c} = \frac{14}{5} \]

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

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

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

### Step 4: Eliminate \( b \) from the ratio
To simplify the ratio, we can eliminate \( b \) 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 fractions into a single ratio, we need a common denominator. The least common multiple of \( 7 \), \( 1 \), and \( 14 \) is \( 14 \):
- Convert \( \frac{3}{7} \) to have a denominator of \( 14 \):
\[ \frac{3}{7} = \frac{3 \times 2}{7 \times 2} = \frac{6}{14} \]
- The second term remains \( 1 = \frac{14}{14} \).
- Convert \( \frac{5}{14} \) to have a denominator of \( 14 \):
\[ \frac{5}{14} = \frac{5}{14} \]

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

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

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