If `a/b=c/d=e/f=9/5` then `(2a + 3c + 5e)/( 2b + 3d + 5f) = ?`

Options

A. `3/4`
B. `2/5`
C. `9/5`
D. `3/7`

Answer

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

Given:
\[ \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{9}{5} \]

We can express \(a\), \(c\), and \(e\) in terms of \(b\), \(d\), and \(f\) respectively.

### Step 1: Express \(a\), \(c\), and \(e\) in terms of \(b\), \(d\), and \(f\)

From the ratios, we can write:
\[ a = \frac{9}{5}b \]
\[ c = \frac{9}{5}d \]
\[ e = \frac{9}{5}f \]

### Step 2: Substitute \(a\), \(c\), and \(e\) into the expression \((2a + 3c + 5e)\)

Now, we substitute these expressions into the numerator:
\[ 2a + 3c + 5e = 2\left(\frac{9}{5}b\right) + 3\left(\frac{9}{5}d\right) + 5\left(\frac{9}{5}f\right) \]
\[ = \frac{18}{5}b + \frac{27}{5}d + \frac{45}{5}f \]
\[ = \frac{18b + 27d + 45f}{5} \]

### Step 3: Substitute \(b\), \(d\), and \(f\) into the expression \((2b + 3d + 5f)\)

Next, we calculate the denominator:
\[ 2b + 3d + 5f \]

### Step 4: Form the complete expression

Now we can form the complete expression:
\[ \frac{2a + 3c + 5e}{2b + 3d + 5f} = \frac{\frac{18b + 27d + 45f}{5}}{2b + 3d + 5f} \]

### Step 5: Simplify the expression

To simplify this, we can multiply both the numerator and the denominator by 5:
\[ = \frac{18b + 27d + 45f}{5(2b + 3d + 5f)} \]

### Step 6: Factor out the common terms

Notice that \(18b + 27d + 45f\) can be factored:
\[ = \frac{9(2b + 3d + 5f)}{5(2b + 3d + 5f)} \]

### Step 7: Cancel the common terms

Now, we can cancel \(2b + 3d + 5f\) from the numerator and denominator (assuming \(2b + 3d + 5f \neq 0\)):
\[ = \frac{9}{5} \]

### Final Answer

Thus, the value of \(\frac{2a + 3c + 5e}{2b + 3d + 5f}\) is:
\[ \frac{9}{5} \]