If `x : y = 5:2`, then `(8x + 9y)/(8x + 2y)` = ?

Options

A. `22:29`
B. `26:61`
C. `29:22`
D. `61: 26`

Answer

To solve the problem where \( x : y = 5 : 2 \) and we need to find the value of \( \frac{8x + 9y}{8x + 2y} \), we can follow these steps:

### Step 1: Express x and y in terms of a common variable
Given the ratio \( x : y = 5 : 2 \), we can express \( x \) and \( y \) in terms of a common variable \( k \):
- Let \( x = 5k \)
- Let \( y = 2k \)

### Step 2: Substitute x and y into the expression
Now, substitute \( x \) and \( y \) into the expression \( \frac{8x + 9y}{8x + 2y} \):
\[ \frac{8(5k) + 9(2k)}{8(5k) + 2(2k)} \]

### Step 3: Simplify the numerator
Calculate the numerator:
\[ 8(5k) + 9(2k) = 40k + 18k = 58k \]

### Step 4: Simplify the denominator
Calculate the denominator:
\[ 8(5k) + 2(2k) = 40k + 4k = 44k \]

### Step 5: Form the fraction
Now we can form the fraction:
\[ \frac{58k}{44k} \]

### Step 6: Cancel out k
Since \( k \) is common in both the numerator and the denominator, we can cancel it out:
\[ \frac{58}{44} \]

### Step 7: Simplify the fraction
Now, simplify \( \frac{58}{44} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
\[ \frac{58 \div 2}{44 \div 2} = \frac{29}{22} \]

### Final Answer
Thus, the value of \( \frac{8x + 9y}{8x + 2y} \) is:
\[ \frac{29}{22} \]

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