Box W and Box V each contain several blue sticks and red sticks, and all of the red sticks have...

1. Translate the problem requirements: We need to find the difference between the average length of sticks in Box W minus the average length of sticks in Box V. We're told that red sticks are 18 inches shorter than Box W's average and 6 inches longer than Box V's average.
2. Set up variable relationships: Use the red stick length as our reference point since it relates to both box averages, then express both averages in terms of this common reference.
3. Apply the constraint relationships: Write equations showing how the red stick length relates to each box's average, then solve for the difference between the two averages.
4. Calculate the final difference: Substitute the relationships to find the numerical difference between the two box averages.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in plain English. We have two boxes of sticks - Box W and Box V. Each box contains blue and red sticks of various lengths. We need to find how much bigger the average stick length in Box W is compared to Box V.

The key information is about the red sticks: all red sticks are the same length, and this length gives us a way to compare the two boxes. Think of the red stick length as a measuring stick that tells us about both boxes.

Specifically, we're told:

• Red stick length = Box W average minus 18 inches
• Red stick length = Box V average plus 6 inches

Our goal: Find (Box W average) - (Box V average)

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

2. Set up variable relationships

Since the red stick length connects both boxes, let's use it as our reference point. Let's call the length of each red stick "R" inches.

From the problem, we can write two simple relationships:

• \(\mathrm{R = (Box\ W\ average) - 18}\)
• \(\mathrm{R = (Box\ V\ average) + 6}\)

This means:

• \(\mathrm{Box\ W\ average = R + 18}\)
• \(\mathrm{Box\ V\ average = R - 6}\)

Notice how both averages are now expressed in terms of the same red stick length R. This makes our calculation much simpler.

3. Apply the constraint relationships

Now we can find the difference between the two averages using our relationships from step 2.

The difference we want is:

\(\mathrm{(Box\ W\ average) - (Box\ V\ average)}\)

Substituting our expressions:

\(\mathrm{= (R + 18) - (R - 6)}\)
\(\mathrm{= R + 18 - R + 6}\)
\(\mathrm{= 18 + 6}\)
\(\mathrm{= 24}\)

Notice something interesting: the actual value of R (the red stick length) cancels out completely! This makes sense because we only care about the difference between the two averages, not their absolute values.

Process Skill: MANIPULATE - Algebraic substitution to eliminate variables and simplify

4. Calculate the final difference

Our calculation shows that the average length of sticks in Box W is exactly 24 inches longer than the average length of sticks in Box V.

Let's verify this makes sense: if red sticks are 18 inches shorter than Box W's average but 6 inches longer than Box V's average, then Box W's average must be 18 + 6 = 24 inches higher than Box V's average.

Final Answer

The average length of sticks in Box W minus the average length of sticks in Box V is 24 inches.

The answer is E. 24

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the relationship descriptions

Students often struggle with the phrase "18 inches less than the average length of sticks in Box W." They may incorrectly set up the relationship as: Box W average = Red stick length - 18, when it should be: Red stick length = Box W average - 18. The key is recognizing that "18 inches less than X" means "X - 18," not "18 - X."

2. Attempting to find individual averages instead of the difference

Many students think they need to calculate the actual average length in each box before finding the difference. They may try to assign specific values to the red stick length or search for additional information about the number of sticks. However, the problem is designed so that only the difference between averages can be determined, not the individual averages themselves.

3. Confusion about what the red sticks represent

Students may get distracted thinking they need to know how many red sticks are in each box or their proportion relative to blue sticks. They might assume the red sticks are the only sticks or that they somehow determine the averages directly, rather than understanding that red sticks simply provide a common reference point to compare the two boxes.

Errors while executing the approach

1. Sign errors in algebraic manipulation

When expanding \(\mathrm{(R + 18) - (R - 6)}\), students frequently make mistakes with the negative signs. A common error is: \(\mathrm{(R + 18) - (R - 6) = R + 18 - R - 6 = 12}\), forgetting that subtracting a negative number (−6) becomes adding a positive number (+6). The correct expansion should yield \(\mathrm{R + 18 - R + 6 = 24}\).

2. Incorrect variable substitution

Students may mix up which box corresponds to which relationship. They might write \(\mathrm{Box\ W\ average = R - 18}\) instead of \(\mathrm{R + 18}\), or \(\mathrm{Box\ V\ average = R + 6}\) instead of \(\mathrm{R - 6}\). This stems from not carefully tracking which average is greater or smaller than the red stick length.

Errors while selecting the answer

No likely faltering points - once students correctly execute the algebra, the final answer of 24 clearly matches one of the given choices, making selection straightforward.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient length for red sticks

Let's assign a concrete value to the red stick length. Since we're dealing with differences of 18 and 6 inches, let's choose \(\mathrm{red\ stick\ length = 30\ inches}\) (this gives us clean, positive numbers for both box averages).

Step 2: Calculate Box W average using the red stick constraint

We know: \(\mathrm{Red\ stick\ length = Box\ W\ average - 18\ inches}\)
So: \(\mathrm{30 = Box\ W\ average - 18}\)
Therefore: \(\mathrm{Box\ W\ average = 30 + 18 = 48\ inches}\)

Step 3: Calculate Box V average using the red stick constraint

We know: \(\mathrm{Red\ stick\ length = Box\ V\ average + 6\ inches}\)
So: \(\mathrm{30 = Box\ V\ average + 6}\)
Therefore: \(\mathrm{Box\ V\ average = 30 - 6 = 24\ inches}\)

Step 4: Find the difference between the two box averages

\(\mathrm{Difference = Box\ W\ average - Box\ V\ average}\)
\(\mathrm{Difference = 48 - 24 = 24\ inches}\)

Verification with different red stick length:

Let's verify by choosing \(\mathrm{red\ stick\ length = 42\ inches}\):

• \(\mathrm{Box\ W\ average = 42 + 18 = 60\ inches}\)
• \(\mathrm{Box\ V\ average = 42 - 6 = 36\ inches}\)
• \(\mathrm{Difference = 60 - 36 = 24\ inches}\) ✓

The difference remains constant at 24 inches regardless of the specific red stick length we choose, confirming our answer.