The average gas mileage for n vehicles on one car lot is 21 miles per gallon. The average gas mileage for p vehicles on another car lot is 25 miles per gallon. When the vehicles on both lots are combined, the average gas mileage is 22.5 miles per gallon. What is the value of `(p)/(n)`?

Options

A. `(3)/(8)`
B. `(1)/(2)`
C. `(3)/(5)`
D. `1`

Explanation

To solve the problem, we need to find the ratio \( \frac{p}{n} \) where \( n \) is the number of vehicles on one car lot with an average gas mileage of 21 miles per gallon, and \( p \) is the number of vehicles on another car lot with an average gas mileage of 25 miles per gallon. The combined average gas mileage of both lots is 22.5 miles per gallon.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( n \) = number of vehicles on lot A (average mileage = 21 mpg)
- Let \( p \) = number of vehicles on lot B (average mileage = 25 mpg)

2. **Write the Equations for the Sums**:
- The total mileage for lot A: \( \text{Sum}_A = 21n \)
- The total mileage for lot B: \( \text{Sum}_B = 25p \)

3. **Write the Equation for the Combined Average**:
The combined average mileage for both lots can be expressed as:
\[ \frac{\text{Sum}_A + \text{Sum}_B}{n + p} = 22.5 \]
Substituting the sums:
\[ \frac{21n + 25p}{n + p} = 22.5 \]

4. **Cross-Multiply to Eliminate the Fraction**:
Multiply both sides by \( n + p \):
\[ 21n + 25p = 22.5(n + p) \]

5. **Distribute on the Right Side**:
\[ 21n + 25p = 22.5n + 22.5p \]

6. **Rearrange the Equation**:
Move all terms involving \( n \) to one side and all terms involving \( p \) to the other side:
\[ 21n - 22.5n = 22.5p - 25p \]
This simplifies to:
\[ -1.5n = -2.5p \]

7. **Divide Both Sides by -1**:
\[ 1.5n = 2.5p \]

8. **Rearranging to Find the Ratio**:
Divide both sides by \( n \) and \( p \):
\[ \frac{p}{n} = \frac{1.5}{2.5} \]

9. **Simplify the Fraction**:
To simplify \( \frac{1.5}{2.5} \), multiply the numerator and denominator by 2 to eliminate the decimal:
\[ \frac{1.5 \times 2}{2.5 \times 2} = \frac{3}{5} \]

10. **Final Answer**:
Thus, the value of \( \frac{p}{n} \) is \( \frac{3}{5} \).