If a: b= 5:7 and c:d=2a:3b, then ac: bd is :

Options

A. `20:38`
B. `50:147`
C. `10:21`
D. `50:151`

Answer

To solve the problem, we need to find the ratio of \( ac : bd \) given the ratios \( a : b = 5 : 7 \) and \( c : d = 2a : 3b \).

### Step-by-Step Solution:

1. **Write down the given ratios:**
\[ \frac{a}{b} = \frac{5}{7} \]
This implies that \( a = 5k \) and \( b = 7k \) for some constant \( k \).

2. **Substitute \( a \) and \( b \) into the second ratio:**
We have:
\[ \frac{c}{d} = \frac{2a}{3b} \]
Substituting \( a \) and \( b \):
\[ \frac{c}{d} = \frac{2(5k)}{3(7k)} = \frac{10k}{21k} = \frac{10}{21} \]
This implies that \( c = 10m \) and \( d = 21m \) for some constant \( m \).

3. **Now, calculate \( ac \) and \( bd \):**
\[ ac = (5k)(10m) = 50km \]
\[ bd = (7k)(21m) = 147km \]

4. **Now, find the ratio \( ac : bd \):**
\[ ac : bd = 50km : 147km \]
Since \( km \) is common in both terms, we can simplify:
\[ ac : bd = 50 : 147 \]

5. **Final Result:**
The ratio \( ac : bd \) is:
\[ \boxed{50 : 147} \]