If a : b= c : d = e:i = 1:2 , then find (3a + 5c + 7e): (36 + 5d + i7).

Options

A. `1 : 2`
B. `2 : 1`
C. `3 : 1`
D. `1 : 3`

Answer

To solve the problem, we start with the given ratios:

1. **Given Ratios**:
\( a : b = c : d = e : i = 1 : 2 \)

This means:
- \( \frac{a}{b} = \frac{1}{2} \)
- \( \frac{c}{d} = \frac{1}{2} \)
- \( \frac{e}{i} = \frac{1}{2} \)

2. **Assigning Variables**:
Let's assign values based on the ratio. We can let:
- \( a = x \)
- \( b = 2x \) (since \( b = 2a \))
- \( c = x \)
- \( d = 2x \) (since \( d = 2c \))
- \( e = x \)
- \( i = 2x \) (since \( i = 2e \))

3. **Calculating the Numerator**:
We need to find \( 3a + 5c + 7e \):
\[ 3a + 5c + 7e = 3(x) + 5(x) + 7(x) = 3x + 5x + 7x = 15x \]

4. **Calculating the Denominator**:
Now, we calculate \( 36 + 5d + 7i \):
\[ 5d = 5(2x) = 10x \]
\[ 7i = 7(2x) = 14x \]
Therefore:
\[ 36 + 5d + 7i = 36 + 10x + 14x = 36 + 24x \]

5. **Forming the Ratio**:
Now we can form the ratio:
\[ \frac{3a + 5c + 7e}{36 + 5d + 7i} = \frac{15x}{36 + 24x} \]

6. **Simplifying the Ratio**:
To simplify \( \frac{15x}{36 + 24x} \), we can factor out \( 3 \) from the denominator:
\[ = \frac{15x}{3(12 + 8x)} = \frac{5x}{12 + 8x} \]

7. **Finding the Value of the Ratio**:
To find the value of this ratio, we can set \( x = 1 \) (since the ratio is independent of the actual value of \( x \)):
\[ = \frac{5(1)}{12 + 8(1)} = \frac{5}{12 + 8} = \frac{5}{20} = \frac{1}{4} \]

Thus, the final answer is:
\[ \frac{1}{4} \]