If `(a)/(b) = (c )/(d ) = (e )/(f) = (3)/(4)`, what is the value of `(a+c +e )/( b+d +f)`?

Options

A. a) `9//16`
B. b) `3//4`
C. c) `2//3 `
D. d) `4//5`

Answer

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

\[ \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{3}{4} \]

### Step 1: Express variables in terms of a common variable
From the ratios, we can express \(a\), \(c\), and \(e\) in terms of a common variable \(x\):

- From \(\frac{a}{b} = \frac{3}{4}\), we can write:
\[ a = 3x \quad \text{and} \quad b = 4x \]

- From \(\frac{c}{d} = \frac{3}{4}\), we can write:
\[ c = 3y \quad \text{and} \quad d = 4y \]

- From \(\frac{e}{f} = \frac{3}{4}\), we can write:
\[ e = 3z \quad \text{and} \quad f = 4z \]

### Step 2: Set the common variable
Since all ratios are equal, we can set \(x = y = z\). Therefore, we can express all variables in terms of \(x\):

- \(a = 3x\)
- \(b = 4x\)
- \(c = 3x\)
- \(d = 4x\)
- \(e = 3x\)
- \(f = 4x\)

### Step 3: Calculate the numerator and denominator
Now we need to find the value of \(\frac{a+c+e}{b+d+f}\):

- The numerator:
\[ a + c + e = 3x + 3x + 3x = 9x \]

- The denominator:
\[ b + d + f = 4x + 4x + 4x = 12x \]

### Step 4: Form the final expression
Now we can substitute the values of the numerator and denominator into the fraction:

\[ \frac{a+c+e}{b+d+f} = \frac{9x}{12x} \]

### Step 5: Simplify the expression
We can simplify this fraction by canceling \(x\):

\[ \frac{9x}{12x} = \frac{9}{12} = \frac{3}{4} \]

### Final Answer
Thus, the value of \(\frac{a+c+e}{b+d+f}\) is:

\[ \frac{3}{4} \]