gsm8k-0008

# GSM8K Problem 9

John drives for 3 hours at a speed of 60 mph and then turns around because he realizes he forgot something very important at home.  He tries to get home in 4 hours but spends the first 2 hours in standstill traffic.  He spends the next half-hour driving at a speed of 30mph, before being able to drive the remaining time of the 4 hours going at 80 mph.  How far is he from home at the end of those 4 hours?

Reason step by step, then give ONLY the final answer on the last line as \boxed{ANSWER}.

<!-- imported from openai/gsm8k (test) -->

First, we calculate the distance John traveled away from home.
He drove for 3 hours at a speed of 60 mph.
$$ \text{Distance}_{\text{out}} = 60 \text{ mph} \times 3 \text{ hours} = 180 \text{ miles} $$

Next, we analyze the return trip, which takes a total of 4 hours. We need to calculate the distance he covered during this time to determine how close he got to home. The return trip consists of three segments:

1.  **First 2 hours (Standstill traffic):**
    Speed = 0 mph.
    $$ \text{Distance}_1 = 0 \text{ mph} \times 2 \text{ hours} = 0 \text{ miles} $$

2.  **Next half-hour (0.5 hours):**
    Speed = 30 mph.
    $$ \text{Distance}_2 = 30 \text{ mph} \times 0.5 \text{ hours} = 15 \text{ miles} $$

3.  **Remaining time:**
    The total time for the return trip is 4 hours. We subtract the time spent in the first two segments to find the time for the last segment.
    $$ \text{Time}_3 = 4 \text{ hours} - 2 \text{ hours} - 0.5 \text{ hours} = 1.5 \text{ hours} $$
    Speed = 80 mph.
    $$ \text{Distance}_3 = 80 \text{ mph} \times 1.5 \text{ hours} = 120 \text{ miles} $$

Now, we sum the distances traveled during the return trip:
$$ \text{Distance}_{\text{return}} = 0 + 15 + 120 = 135 \text{ miles} $$

Finally, we calculate how far John is from home by subtracting the distance he returned from the distance he originally traveled away.
$$ \text{Distance from home} = \text{Distance}_{\text{out}} - \text{Distance}_{\text{return}} $$
$$ \text{Distance from home} = 180 \text{ miles} - 135 \text{ miles} = 45 \text{ miles} $$

\boxed{45}