Joe, a fruit vendor , sold 3 apples, 4 oranges and 6 bananas to a customer for $25. He sold 5 apples , 3 oranges and 11 bananas to another custmore for $36. If a third customer wants to buy one apple, 5 oranges and one banana, how much would be he have to pay ?

Options

A. `$9`
B. `$14`
C. `$18`
D. `$22`

Answer

To solve the problem, we need to set up equations based on the information given about the sales of Joe, the fruit vendor.

### Step-by-Step Solution:

1. **Define Variables**:
Let:
- \( X \) = price of one apple
- \( Y \) = price of one orange
- \( Z \) = price of one banana

2. **Set Up Equations**:
From the information provided:
- For the first customer:
\[ 3X + 4Y + 6Z = 25 \quad \text{(Equation 1)} \]
- For the second customer:
\[ 5X + 3Y + 11Z = 36 \quad \text{(Equation 2)} \]

3. **Multiply Equations**:
To eliminate one variable, we can multiply the equations:
- Multiply Equation 1 by 5:
\[ 15X + 20Y + 30Z = 125 \quad \text{(Equation 3)} \]
- Multiply Equation 2 by 3:
\[ 15X + 9Y + 33Z = 108 \quad \text{(Equation 4)} \]

4. **Subtract Equations**:
Now, subtract Equation 4 from Equation 3:
\[ (15X + 20Y + 30Z) - (15X + 9Y + 33Z) = 125 - 108 \]
This simplifies to:
\[ 11Y - 3Z = 17 \quad \text{(Equation 5)} \]

5. **Express Z in terms of Y**:
Rearranging Equation 5 gives:
\[ 3Z = 11Y - 17 \implies Z = \frac{11Y - 17}{3} \quad \text{(Equation 6)} \]

6. **Substitute Z back into one of the original equations**:
Substitute Equation 6 into Equation 1:
\[ 3X + 4Y + 6\left(\frac{11Y - 17}{3}\right) = 25 \]
Simplifying this:
\[ 3X + 4Y + 2(11Y - 17) = 25 \]
\[ 3X + 4Y + 22Y - 34 = 25 \]
\[ 3X + 26Y - 34 = 25 \]
\[ 3X + 26Y = 59 \quad \text{(Equation 7)} \]

7. **Express X in terms of Y**:
Rearranging Equation 7 gives:
\[ 3X = 59 - 26Y \implies X = \frac{59 - 26Y}{3} \quad \text{(Equation 8)} \]

8. **Find the cost for the third customer**:
The third customer wants to buy:
- 1 apple, 5 oranges, and 1 banana, which translates to:
\[ X + 5Y + Z \]
Substituting Equations 6 and 8 into this gives:
\[ \left(\frac{59 - 26Y}{3}\right) + 5Y + \left(\frac{11Y - 17}{3}\right) \]
Combining these:
\[ = \frac{59 - 26Y + 15Y + 11Y - 17}{3} \]
\[ = \frac{59 - 17}{3} = \frac{42}{3} = 14 \]

9. **Final Answer**:
The total cost for the third customer is **$14**.