If a : b = 2 : 3 , b: c = 2 : 4, then a : b : c = ?

Options

A. ` 4 : 6 : 12`
B. ` 6 : 1 : 5`
C. ` 5 : 6 : 2`
D. ` 7 : 1 : 2`

Answer

To solve the problem of finding the combined ratio \( a : b : c \) given the ratios \( a : b = 2 : 3 \) and \( b : c = 2 : 4 \), we can follow these steps:

### Step-by-Step Solution:

1. **Write the given ratios in fractional form**:
- From \( a : b = 2 : 3 \), we can express this as:
\[ \frac{a}{b} = \frac{2}{3} \]
- From \( b : c = 2 : 4 \), we can express this as:
\[ \frac{b}{c} = \frac{2}{4} = \frac{1}{2} \]

2. **Express \( b \) in terms of \( c \)**:
- From \( \frac{b}{c} = \frac{1}{2} \), we can rearrange this to find \( b \):
\[ b = \frac{1}{2}c \]

3. **Substitute \( b \) into the first ratio**:
- We know \( \frac{a}{b} = \frac{2}{3} \). Substituting \( b = \frac{1}{2}c \) into this ratio gives:
\[ \frac{a}{\frac{1}{2}c} = \frac{2}{3} \]
- To eliminate the fraction, we can multiply both sides by \( \frac{1}{2}c \):
\[ a = \frac{2}{3} \cdot \frac{1}{2}c = \frac{2c}{6} = \frac{1}{3}c \]

4. **Now express \( a \), \( b \), and \( c \) in terms of a common variable**:
- Let \( c = 6k \) for some variable \( k \):
- Then \( b = \frac{1}{2}c = \frac{1}{2}(6k) = 3k \)
- And \( a = \frac{1}{3}c = \frac{1}{3}(6k) = 2k \)

5. **Write the combined ratio \( a : b : c \)**:
- Now we have:
- \( a = 2k \)
- \( b = 3k \)
- \( c = 6k \)
- Therefore, the combined ratio \( a : b : c \) is:
\[ a : b : c = 2k : 3k : 6k \]
- We can simplify this by dividing each term by \( k \):
\[ a : b : c = 2 : 3 : 6 \]

### Final Answer:
Thus, the combined ratio \( a : b : c \) is \( 2 : 3 : 6 \).