If `x : y = 3 : 4` then find the value of `(2x+3y)/(3x+4y)`

Options

A. `9/25`
B. `18/25`
C. `7/18`
D. `17/25`

Answer

To solve the problem where \( x : y = 3 : 4 \) and we need to find the value of \( \frac{2x + 3y}{3x + 4y} \), we can follow these steps:

### Step 1: Express \( x \) in terms of \( y \)
From the ratio \( x : y = 3 : 4 \), we can express \( x \) in terms of \( y \):
\[ x = \frac{3}{4}y \]

### Step 2: Substitute \( x \) in the expression
Now, substitute \( x \) in the expression \( \frac{2x + 3y}{3x + 4y} \):
\[ \frac{2\left(\frac{3}{4}y\right) + 3y}{3\left(\frac{3}{4}y\right) + 4y} \]

### Step 3: Simplify the numerator
Calculate the numerator:
\[ 2\left(\frac{3}{4}y\right) + 3y = \frac{6}{4}y + 3y = \frac{6}{4}y + \frac{12}{4}y = \frac{18}{4}y = \frac{9}{2}y \]

### Step 4: Simplify the denominator
Now, calculate the denominator:
\[ 3\left(\frac{3}{4}y\right) + 4y = \frac{9}{4}y + 4y = \frac{9}{4}y + \frac{16}{4}y = \frac{25}{4}y \]

### Step 5: Combine the results
Now we can combine the simplified numerator and denominator:
\[ \frac{\frac{9}{2}y}{\frac{25}{4}y} \]

### Step 6: Cancel \( y \) and simplify the fraction
The \( y \) in the numerator and denominator cancels out:
\[ \frac{\frac{9}{2}}{\frac{25}{4}} = \frac{9}{2} \times \frac{4}{25} = \frac{36}{50} \]

### Step 7: Further simplify the fraction
Now simplify \( \frac{36}{50} \):
\[ \frac{36}{50} = \frac{18}{25} \]

### Final Answer
Thus, the value of \( \frac{2x + 3y}{3x + 4y} \) is:
\[ \frac{18}{25} \]