Harry started a 6-mile hike with a full 10-cup canteen of water and finished the hike in 2 hours with...

1. Translate the problem requirements: Harry starts with 10 cups, ends with 1 cup after 2 hours. Water is lost in two ways: leaking (\(1 \text{ cup/hour}\)) and drinking (3 cups in last mile, unknown amount in first 5 miles). We need to find cups consumed per mile during the first 5 miles.
2. Calculate total water loss: Determine how much water Harry lost overall by comparing starting amount to ending amount.
3. Account for leakage loss: Calculate how much water was lost due to the canteen leaking over the 2-hour period.
4. Determine total consumption: Subtract leakage loss from total loss to find how much Harry actually drank.
5. Calculate consumption in first 5 miles: Subtract the known consumption in the last mile from total consumption, then divide by 5 miles to get per-mile rate.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know and what we're looking for in everyday terms.

Harry begins his hike with a canteen containing 10 cups of water. After completing his 6-mile hike in 2 hours, he has 1 cup left. During this journey, water disappears from his canteen in two ways: first, the canteen has a leak that causes him to lose 1 cup every hour, and second, Harry drinks water as he hikes.

We know specifically that Harry drank 3 cups during the last mile of his hike. What we need to figure out is how many cups he drank per mile during the first 5 miles.

Process Skill: TRANSLATE - Converting the story into clear mathematical understanding

2. Calculate total water loss

Let's figure out how much water Harry lost overall during his hike.

He started with 10 cups and ended with 1 cup, so the total amount of water that left his canteen is:
\(\text{Total water loss} = \text{Starting amount} - \text{Ending amount}\)
\(\text{Total water loss} = 10 \text{ cups} - 1 \text{ cup} = 9 \text{ cups}\)

So 9 cups of water disappeared from his canteen during the 2-hour hike.

3. Account for leakage loss

Now let's figure out how much of that 9-cup loss was due to the leak versus how much Harry actually drank.

The canteen leaked at 1 cup per hour, and the hike took 2 hours, so:
\(\text{Water lost to leakage} = 1 \text{ cup/hour} \times 2 \text{ hours} = 2 \text{ cups}\)

This means 2 cups of the 9-cup total loss was due to the leak.

4. Determine total consumption

Since 2 cups were lost to leakage, the remaining water loss must be what Harry actually drank.

\(\text{Total amount Harry drank} = \text{Total water loss} - \text{Water lost to leakage}\)
\(\text{Total amount Harry drank} = 9 \text{ cups} - 2 \text{ cups} = 7 \text{ cups}\)

So Harry consumed 7 cups of water during his entire hike.

5. Calculate consumption in first 5 miles

We know Harry drank 7 cups total, and 3 of those cups were consumed during the last mile. This means during the first 5 miles, he drank:

\(\text{Water consumed in first 5 miles} = \text{Total consumption} - \text{Consumption in last mile}\)
\(\text{Water consumed in first 5 miles} = 7 \text{ cups} - 3 \text{ cups} = 4 \text{ cups}\)

To find his consumption rate per mile during the first 5 miles:
\(\text{Consumption per mile} = \text{Water consumed in first 5 miles} ÷ \text{Number of miles}\)
\(\text{Consumption per mile} = 4 \text{ cups} ÷ 5 \text{ miles} = \frac{4}{5} \text{ cups per mile}\)

6. Final Answer

Harry drank \(\frac{4}{5}\) cups of water per mile during the first 5 miles of his hike.

This matches answer choice (A) \(\frac{4}{5}\).

Common Faltering Points

Errors while devising the approach

1. Forgetting to account for the canteen leak
Many students focus only on Harry's drinking and forget that water also leaves the canteen due to the 1 cup per hour leak. They incorrectly assume that all 9 cups of water loss (10 cups start - 1 cup end) represents what Harry drank, leading them to calculate 6 cups consumed in the first 5 miles instead of 4 cups.

2. Misinterpreting the time constraint
Students might not clearly connect that the 2-hour total hike time is crucial for calculating leakage loss. They may try to break down drinking rates by time periods rather than focusing on the mile-based consumption pattern that the problem actually requires.

3. Confusing what needs to be calculated
Some students may set up equations to find Harry's speed or try to determine how long he spent on each mile, when the problem simply requires finding his drinking rate per mile during the first 5 miles. This leads to unnecessary complexity and potential errors.

Errors while executing the approach

1. Arithmetic errors in the subtraction sequence
The solution requires multiple sequential subtractions: \(10-1=9\) (total loss), \(9-2=7\) (actual consumption), and \(7-3=4\) (consumption in first 5 miles). Students often make calculation mistakes in this chain, particularly when working under time pressure, leading to incorrect intermediate results.

2. Incorrect leakage calculation
Students may miscalculate the leakage as 1 cup total instead of \(1 \text{ cup per hour} \times 2 \text{ hours} = 2 \text{ cups}\) total. This error makes their subsequent calculations incorrect, even if their approach is sound.

Errors while selecting the answer

1. Selecting total consumption rate instead of first 5 miles rate
Students who correctly calculate that Harry drank 7 cups total during 6 miles might mistakenly select an answer representing \(\frac{7}{6}\) cups per mile (which isn't offered), or they might calculate the overall average incorrectly. They lose sight that the question specifically asks for the rate during the first 5 miles only.