If `a: b= c:d = e:f = 1:2`, then ( 3a + 5c +7e): (3b + 5d + 7 J) is equal to

Options

A. `8:7`
B. `2:1`
C. `1:4`
D. `1:2`

Answer

To solve the problem, we need to find the ratio \( (3a + 5c + 7e) : (3b + 5d + 7f) \) given that \( a : b = c : d = e : f = 1 : 2 \).

### Step-by-Step Solution:

1. **Understanding the Ratios:**
Given \( a : b = 1 : 2 \), we can express \( a \) and \( b \) in terms of a common variable. Let:
\[ a = 1x \quad \text{and} \quad b = 2x \]
Similarly, for \( c : d \) and \( e : f \):
\[ c = 1y \quad \text{and} \quad d = 2y \]
\[ e = 1z \quad \text{and} \quad f = 2z \]
Since all ratios are equal, we can set \( x = y = z = 1 \). Therefore:
\[ a = 1, \quad b = 2, \quad c = 1, \quad d = 2, \quad e = 1, \quad f = 2 \]

2. **Calculating the Numerator:**
Now, we calculate \( 3a + 5c + 7e \):
\[ 3a + 5c + 7e = 3(1) + 5(1) + 7(1) = 3 + 5 + 7 = 15 \]

3. **Calculating the Denominator:**
Next, we calculate \( 3b + 5d + 7f \):
\[ 3b + 5d + 7f = 3(2) + 5(2) + 7(2) = 6 + 10 + 14 = 30 \]

4. **Forming the Ratio:**
Now we can form the ratio:
\[ (3a + 5c + 7e) : (3b + 5d + 7f) = 15 : 30 \]

5. **Simplifying the Ratio:**
We can simplify \( 15 : 30 \) by dividing both sides by 15:
\[ 15 : 30 = 1 : 2 \]

### Final Answer:
Thus, the final answer is:
\[ (3a + 5c + 7e) : (3b + 5d + 7f) = 1 : 2 \]