One week a certain truck rental lot had a total of 20 trucks, all of which were on the lot...

1. Translate the problem requirements: We start with 20 trucks on Monday. Some get rented during the week. Of those rented, 50% return by Saturday. We need at least 12 trucks on Saturday morning. Find the maximum number that could have been rented.
2. Set up the constraint equation: Express the Saturday morning truck count in terms of trucks rented, using the return rate to create our main constraint.
3. Work backwards from the minimum requirement: Use the "at least 12 trucks" condition to find the maximum number that could have been rented out.
4. Verify the solution makes logical sense: Check that our answer satisfies all given conditions and represents a realistic scenario.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening with these trucks in plain English:

• We start Monday morning with 20 trucks sitting on the lot
• During the week, some trucks get rented out (this is what we want to maximize)
• Of the trucks that were rented, exactly \(\mathrm{50\%}\) (or half) come back by Saturday morning
• By Saturday morning, we need at least 12 trucks on the lot

The key insight is understanding what trucks are on the lot Saturday morning. These trucks come from two sources:

1. Trucks that were never rented during the week (stayed on the lot the whole time)
2. Trucks that were rented but returned by Saturday morning (\(\mathrm{50\%}\) of those rented)

Let's call the number of trucks rented during the week "R". Then:

• Trucks that stayed on lot all week = \(\mathrm{20 - R}\)
• Trucks that returned by Saturday = \(\mathrm{0.5 \times R}\)
• Total trucks on lot Saturday morning = \(\mathrm{(20 - R) + 0.5R = 20 - 0.5R}\)

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

2. Set up the constraint equation

Now we can express our main constraint mathematically. We know that:

Trucks on Saturday morning \(\mathrm{\geq 12}\)

Substituting our expression from step 1:

\(\mathrm{20 - 0.5R \geq 12}\)

This inequality tells us the mathematical relationship between the number rented (R) and the minimum requirement for Saturday morning.

3. Work backwards from the minimum requirement

To find the maximum number of trucks that could be rented, we need to solve our inequality:

\(\mathrm{20 - 0.5R \geq 12}\)

Subtract 20 from both sides:

\(\mathrm{-0.5R \geq 12 - 20}\)

\(\mathrm{-0.5R \geq -8}\)

Divide both sides by -0.5 (remember to flip the inequality sign when dividing by a negative):

\(\mathrm{R \leq 16}\)

This means the maximum number of trucks that could be rented is 16.

Process Skill: APPLY CONSTRAINTS - Using the minimum requirement to find the maximum rental

4. Verify the solution makes logical sense

Let's check our answer by plugging R = 16 back into our scenario:

• Start with 20 trucks on Monday
• Rent out 16 trucks during the week
• Trucks remaining on lot all week: \(\mathrm{20 - 16 = 4}\) trucks
• Trucks that return by Saturday: \(\mathrm{50\% \text{ of } 16 = 8}\) trucks
• Total trucks on lot Saturday morning: \(\mathrm{4 + 8 = 12}\) trucks ✓

This exactly meets our minimum requirement of "at least 12 trucks" on Saturday morning.

If we tried to rent out 17 trucks:

• Trucks remaining: \(\mathrm{20 - 17 = 3}\)
• Trucks returning: \(\mathrm{50\% \text{ of } 17 = 8.5}\)
• Total Saturday: \(\mathrm{3 + 8.5 = 11.5}\) trucks

But we can't have half a truck, and this would give us fewer than 12 trucks, violating our constraint.

5. Final Answer

The greatest number of different trucks that could have been rented out during the week is 16.

The answer is (B) 16.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "50% of trucks rented were returned" means

Students often confuse this to mean that \(\mathrm{50\%}\) of ALL trucks (10 trucks) were returned, rather than understanding it means \(\mathrm{50\%}\) of only the rented trucks came back. This fundamental misinterpretation leads to setting up completely wrong equations from the start.

2. Incorrectly identifying what trucks are on the lot Saturday morning

Many students fail to recognize that Saturday morning's truck count comes from TWO sources: trucks that were never rented (stayed on lot all week) AND trucks that were rented but returned by Saturday. They might only count one source, leading to an incomplete constraint equation.

3. Setting up the wrong optimization objective

Students may try to minimize the number of trucks rented instead of maximizing it, or they might try to find the exact number rather than the maximum possible number. The phrase "greatest number" clearly indicates we need to maximize, but this optimization aspect is sometimes missed.

Errors while executing the approach

1. Sign errors when solving the inequality

When solving \(\mathrm{-0.5R \geq -8}\), students frequently forget to flip the inequality sign when dividing both sides by the negative number -0.5. This leads to \(\mathrm{R \geq 16}\) instead of the correct \(\mathrm{R \leq 16}\), giving a completely wrong interpretation of the constraint.

2. Arithmetic mistakes in the constraint equation

Students may make basic calculation errors when simplifying \(\mathrm{20 - 0.5R \geq 12}\), such as incorrectly calculating \(\mathrm{12 - 20 = 8}\) instead of -8, or making errors when handling the 0.5 coefficient.

3. Mishandling fractional truck returns

When testing their answer or working with odd numbers of rented trucks, students might not properly handle the fact that \(\mathrm{50\%}\) of an odd number gives a fractional result (like 8.5 trucks), forgetting that we can't have half a truck in reality.

Errors while selecting the answer

1. Selecting the minimum required trucks instead of maximum rented trucks

After correctly calculating that \(\mathrm{R \leq 16}\), some students get confused about what the question is asking for and might select 12 (the minimum trucks required on Saturday) instead of 16 (the maximum trucks that can be rented).

2. Off-by-one errors when interpreting "at least 12"

Students might interpret "at least 12 trucks" to mean "more than 12 trucks" and use \(\mathrm{> 12}\) instead of \(\mathrm{\geq 12}\) in their constraint. This would lead them to conclude that 11.99... trucks are needed, pushing them to select 15 trucks rented instead of the correct 16.