A hiker walked for two days. On the second day the hiker walked 2 hours longer and at an average...

1. Translate the problem requirements: We need to find the hiker's speed on day 1. We know: day 2 was 2 hours longer than day 1, day 2 speed was 1 mph faster than day 1, total distance = 64 miles, total time = 18 hours.
2. Set up variables for the unknowns: Define variables for day 1 speed and time, then express day 2 conditions in terms of these base variables.
3. Create equations from the given constraints: Use the total time and total distance information to build two equations that capture all the relationships.
4. Solve the system strategically: Eliminate variables systematically to find the day 1 speed, checking our answer against the given choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday terms:

• The hiker walked on two different days
• On day 2, he walked for 2 more hours than on day 1
• On day 2, his speed was 1 mph faster than on day 1
• Total distance covered = 64 miles
• Total time spent walking = 18 hours
• We need to find his speed on day 1

Think of this like comparing two different workout sessions - one shorter and slower, one longer and faster, but we know the combined totals.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Set up variables for the unknowns

Let's use simple variables to represent what we don't know:

Let s = speed on day 1 (in mph) - this is what we're looking for
Let t = time walked on day 1 (in hours)

Now we can express day 2 in terms of day 1:

• Speed on day 2 = s + 1 (since it's 1 mph faster)
• Time on day 2 = t + 2 (since it's 2 hours longer)

This approach keeps everything connected to our original unknowns, making the math cleaner.

3. Create equations from the given constraints

Now let's translate our constraints into equations using the relationship: \(\mathrm{Distance = Speed × Time}\)

From the total time constraint:
Time on day 1 + Time on day 2 = 18 hours
\(\mathrm{t + (t + 2) = 18}\)
\(\mathrm{2t + 2 = 18}\)
\(\mathrm{2t = 16}\)
\(\mathrm{t = 8}\) hours

So day 1 was 8 hours, and day 2 was 10 hours.

From the total distance constraint:
Distance on day 1 + Distance on day 2 = 64 miles
(Speed on day 1 × Time on day 1) + (Speed on day 2 × Time on day 2) = 64
\(\mathrm{(s × 8) + ((s + 1) × 10) = 64}\)
\(\mathrm{8s + 10s + 10 = 64}\)
\(\mathrm{18s + 10 = 64}\)
\(\mathrm{18s = 54}\)
\(\mathrm{s = 3}\) mph

4. Solve the system strategically

Let's verify our answer makes sense:

• Day 1: Speed = 3 mph, Time = 8 hours, Distance = \(\mathrm{3 × 8 = 24}\) miles
• Day 2: Speed = 4 mph, Time = 10 hours, Distance = \(\mathrm{4 × 10 = 40}\) miles

Checking our constraints:
• Total time: \(\mathrm{8 + 10 = 18}\) hours ✓
• Total distance: \(\mathrm{24 + 40 = 64}\) miles ✓
• Day 2 is 2 hours longer: \(\mathrm{10 - 8 = 2}\) ✓
• Day 2 speed is 1 mph faster: \(\mathrm{4 - 3 = 1}\) ✓

Final Answer

The hiker's average speed on the first day was 3 mph, which corresponds to choice (B).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the relationship constraints:
Students often confuse which day has the longer time and higher speed. They might incorrectly assume day 1 had the longer time or higher speed, leading to setting up relationships like "day 1 time = day 2 time + 2" instead of "day 2 time = day 1 time + 2". This fundamental misreading derails the entire solution from the start.

2. Choosing overly complex variable assignments:
Many students try to define separate variables for all four unknowns (speed day 1, speed day 2, time day 1, time day 2) instead of recognizing that day 2 values can be expressed in terms of day 1 values. This leads to a more complicated system with four variables instead of the simpler two-variable approach shown in the solution.

3. Forgetting to use the distance = speed × time relationship:
Some students get overwhelmed by the multiple constraints and fail to recognize that they need to use the fundamental distance formula to connect the speed and time information with the total distance constraint. They might try to solve using only the time constraint, missing the crucial distance equation.

Errors while executing the approach

1. Arithmetic errors in equation setup and solving:
When expanding (s + 1) × 10, students commonly make errors like getting 10s + 1 instead of 10s + 10. Similarly, when solving 18s = 54, they might incorrectly calculate s = 4 instead of s = 3 due to hasty division.

2. Solving equations in the wrong order:
Students might attempt to solve the distance equation first before finding the time values, making the algebra unnecessarily complex. The solution becomes much cleaner when the simpler time constraint equation is solved first to find t = 8, then substituted into the distance equation.

3. Sign errors when setting up constraint equations:
When translating "2 hours longer" and "1 mph faster" into mathematical expressions, students may incorrectly use subtraction instead of addition, writing expressions like (s - 1) for day 2 speed or (t - 2) for day 2 time.

Errors while selecting the answer

1. Providing the speed for the wrong day:
After correctly calculating that day 1 speed is 3 mph and day 2 speed is 4 mph, students might mistakenly select the day 2 speed (4 mph) as their final answer since the question asks specifically for "his average speed on the first day."

2. Failing to verify the solution against original constraints:
Students might arrive at an answer but skip the verification step, missing computational errors that would be caught by checking whether their answer satisfies all the original conditions (total time = 18, total distance = 64, day 2 being 2 hours longer and 1 mph faster).