To solve the problem, we need to follow these steps:
- Determine how much of the hoagie is left after Evan ate three-eighths of it.
- Calculate how much of what was left he ate for lunch.
Step 1: Calculate what is left after dinner.
Evan ate \( \frac{3}{8} \) of the hoagie. The amount left is: \[ 1 - \frac{3}{8} = \frac{8}{8} - \frac{3}{8} = \frac{5}{8} \]
Step 2: Calculate how much he ate for lunch.
Evan ate two-thirds of what was left, which is: \[ \frac{2}{3} \times \frac{5}{8} \]
Now we perform the multiplication: \[ \frac{2 \times 5}{3 \times 8} = \frac{10}{24} \]
This fraction can be simplified: \[ \frac{10}{24} = \frac{5}{12} \]
Thus, Evan ate \( \frac{5}{12} \) of the entire hoagie for lunch.
Options Analysis:
Looking at the choices provided:
- (a) \( \frac{3}{8} ÷ \frac{2}{3} \) does not apply, as it does not calculate the amount eaten for lunch.
- (b) \( \frac{3}{8} + \frac{2}{3} \) is incorrect, since we do not add but rather multiply the remaining amount.
- (c) \( \frac{3}{8} \times \frac{2}{3} \) is incorrect because it does not represent the calculation for lunch. Instead—it represents something different and is used incorrectly.
- (d) \( \frac{5}{8} \times \frac{2}{3} = \frac{10}{24} = \frac{5}{12} \) is the correct choice, as it correctly finds how much Evan ate for lunch.
- (e) \( \frac{5}{8} ÷ \frac{2}{3} \) is also incorrect, as it would not calculate the amount eaten, but a different operation.
Final Answers:
The correct process is (d) \( \frac{5}{8} \times \frac{2}{3} = \frac{5}{12} \).
Only choice (d) is appropriate in solving how much of the entire hoagie Evan ate for lunch.