Five pieces of wood have an average (arithmetic mean) length of 124 centimeters and a median length of 140 centimeters....

1. Translate the problem requirements: We have 5 pieces of wood with average length 124 cm and median 140 cm. We need to find the maximum possible length of the shortest piece. This means arranging 5 numbers where the middle value is 140, they sum to 620 (\(5 \times 124\)), and we want the smallest number as large as possible.
2. Establish the structural constraints: Since median is 140 cm, when arranged in order, the middle (3rd) piece must be 140 cm. The sum of all 5 pieces must equal 620 cm (from the average condition).
3. Set up the optimization framework: To maximize the shortest piece, we need to minimize the four longer pieces. The two pieces shorter than median should be as large as possible (equal to the shortest piece we're maximizing), and the two pieces longer than median should be as small as possible (equal to the median).
4. Apply the optimization and solve: Set up the equation where shortest piece = x, so we have pieces of lengths x, x, 140, 140, 140. Their sum equals 620, giving us \(2x + 3(140) = 620\), which solves to \(x = 100\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday language first. We have 5 pieces of wood, and we're told two important facts about them:

First, their average length is 124 centimeters. Think of this like the "balanced" length - if we could redistribute all the wood equally among the 5 pieces, each would be 124 cm long. This means the total amount of wood is \(5 \times 124 = 620\) centimeters.

Second, the median length is 140 centimeters. The median is the "middle" value when we line up all 5 pieces from shortest to longest. So the 3rd piece in this lineup is exactly 140 cm.

Our goal is to find the maximum possible length of the shortest piece. In other words, we want to arrange our 5 pieces so that even the smallest one is as long as possible.

Process Skill: TRANSLATE - Converting the problem's language about average and median into concrete mathematical constraints

2. Establish the structural constraints

Now let's think about what these conditions mean for how we can arrange our 5 pieces.

Since the median is 140 cm, when we arrange our pieces from shortest to longest, the middle piece (the 3rd one) must be exactly 140 cm. We can visualize this as:

[Piece 1] ≤ [Piece 2] ≤ [Piece 3 = 140] ≤ [Piece 4] ≤ [Piece 5]

Also, because the average is 124 cm, we know that:
Piece 1 + Piece 2 + Piece 3 + Piece 4 + Piece 5 = 620 cm

Since Piece 3 = 140 cm, we can rewrite this as:
Piece 1 + Piece 2 + 140 + Piece 4 + Piece 5 = 620

This means: Piece 1 + Piece 2 + Piece 4 + Piece 5 = 480 cm

3. Set up the optimization framework

Here's the key insight: to make the shortest piece as long as possible, we need to make all the other pieces as short as possible while still following our rules.

Let's call the length of the shortest piece "x". Since we want to maximize x, we should:

Make Piece 1 = x (this is our shortest piece)
Make Piece 2 as small as possible. Since Piece 2 can't be shorter than Piece 1, the smallest it can be is also x.

For the pieces longer than the median (Pieces 4 and 5), to minimize their lengths, we should make them as close to the median as possible. The smallest they can be is 140 cm each (equal to the median).

So our arrangement becomes:
Piece 1 = x, Piece 2 = x, Piece 3 = 140, Piece 4 = 140, Piece 5 = 140

Process Skill: APPLY CONSTRAINTS - Using the ordering requirements strategically to minimize all pieces except the one we're maximizing

4. Apply the optimization and solve

Now we can substitute our optimized arrangement into our total length equation:

x + x + 140 + 140 + 140 = 620

Simplifying the left side:
\(2x + 420 = 620\)

Subtracting 420 from both sides:
\(2x = 620 - 420\)
\(2x = 200\)

Dividing by 2:
\(x = 100\)

Let's verify this makes sense: Our pieces would be 100, 100, 140, 140, 140 cm.

• Average: \((100 + 100 + 140 + 140 + 140) \div 5 = 620 \div 5 = 124\) ✓
• Median: The middle value is 140 ✓
• All pieces are in non-decreasing order ✓

Final Answer

The maximum possible length of the shortest piece of wood is 100 centimeters.

Looking at our answer choices, this corresponds to choice B. 100.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what 'maximum shortest piece' means

Students often get confused by the phrase "maximum possible length of the shortest piece." They might think this is asking for the longest piece overall, or they might not realize that to maximize the shortest piece, they need to minimize all other pieces while respecting the constraints. This leads them to set up the problem incorrectly from the start.

2. Incorrectly applying the median constraint

Many students struggle with the median concept when there are 5 pieces. They might think the median is the average of two middle values (like with an even number of items), or they might place the median value incorrectly in their ordering. Some students also don't realize that pieces 4 and 5 must be greater than or equal to 140, not necessarily greater than 140.

3. Not recognizing the optimization strategy

Students often fail to see that this is an optimization problem where they need to make strategic choices about the lengths. They might try to assign arbitrary values or don't understand that to maximize one variable, they should minimize others within the given constraints.

Errors while executing the approach

1. Arithmetic errors in calculating the total sum

When calculating \(5 \times 124 = 620\), or when subtracting 140 from 620 to get 480, students commonly make basic multiplication or subtraction errors. These arithmetic mistakes cascade through the entire solution.

2. Incorrect equation setup

Even when students understand the approach, they might set up the equation incorrectly. For example, they might write x + x + 140 + 140 + 140 = 124 (using the average instead of the total), or they might forget to subtract the median value of 140 when calculating the remaining sum.

3. Algebraic manipulation errors

When solving \(2x + 420 = 620\), students might make errors like: incorrectly subtracting (\(2x = 620 + 420\) instead of \(620 - 420\)), or dividing incorrectly (\(x = 200 \div 2 = 200\) instead of 100).

Errors while selecting the answer

1. Forgetting to verify the solution

Students might arrive at \(x = 100\) but fail to check if their answer actually satisfies both the average and median conditions. This verification step would catch any errors made during execution, but students often skip it and select their calculated answer without confirmation.

2. Selecting a related but incorrect value

Students might correctly calculate some intermediate values (like 620 for total length, or 480 for the sum of four pieces, or 200 for 2x) but then accidentally select an answer choice that matches one of these intermediate values instead of the final answer of 100.