If a/b = c/d = e/f = 2/5, then what is (5a + 3c + 2e) : (5b + 3d + 2f)?

Options

A. `2:5`
B. `3:5`
C. `16:25`
D. `20:33`

Answer

To solve the problem, we need to find the ratio \( (5a + 3c + 2e) : (5b + 3d + 2f) \) given that \( \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{2}{5} \).

### Step-by-Step Solution:

1. **Express Variables in Terms of a Common Variable**:
Since \( \frac{a}{b} = \frac{2}{5} \), we can express \( a \) and \( b \) in terms of a common variable \( k \):
\[ a = 2k, \quad b = 5k \]

Similarly, for \( c \) and \( d \):
\[ c = 2m, \quad d = 5m \]

And for \( e \) and \( f \):
\[ e = 2n, \quad f = 5n \]

2. **Substituting Values into the Expression**:
Now we substitute \( a, b, c, d, e, \) and \( f \) into the expression \( 5a + 3c + 2e \):
\[ 5a + 3c + 2e = 5(2k) + 3(2m) + 2(2n) = 10k + 6m + 4n \]

For the denominator \( 5b + 3d + 2f \):
\[ 5b + 3d + 2f = 5(5k) + 3(5m) + 2(5n) = 25k + 15m + 10n \]

3. **Forming the Ratio**:
Now we have:
\[ \text{Ratio} = \frac{10k + 6m + 4n}{25k + 15m + 10n} \]

4. **Factoring Out Common Terms**:
We can factor out a common factor from both the numerator and the denominator:
\[ = \frac{2(5k + 3m + 2n)}{5(5k + 3m + 2n)} \]

5. **Canceling Common Terms**:
The terms \( (5k + 3m + 2n) \) cancel out:
\[ = \frac{2}{5} \]

6. **Final Ratio**:
Thus, the ratio \( (5a + 3c + 2e) : (5b + 3d + 2f) \) is:
\[ 2 : 5 \]