A promotional video that lasts w minutes has 3 segments which last x, y, and z minutes, respectively. A celebrity...

1. Translate the problem requirements: We need to find what fraction of the total video (w minutes) consists of celebrity narration. The celebrity narrates portions of three segments with given ratios, and the segments have a specific length relationship.
2. Establish segment length relationships: Use the constraint \(\mathrm{3x = 3y = z}\) to express all segments in terms of one variable and find the total video length.
3. Calculate narrated time for each segment: Apply the given fractions (\(\frac{1}{3}\), \(\frac{1}{2}\), \(\frac{1}{2}\)) to find how much the celebrity narrates in each segment.
4. Express final fraction: Sum the narrated portions and express as a fraction of the total video length.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in plain English. We have a promotional video that's w minutes long total. This video has three separate segments - think of them like chapters in a book. The first segment is x minutes, the second is y minutes, and the third is z minutes.

A celebrity host doesn't narrate the entire video - only portions of each segment. Specifically:

• In the first segment (x minutes), the celebrity narrates \(\frac{1}{3}\) of it
• In the second segment (y minutes), the celebrity narrates \(\frac{1}{2}\) of it
• In the third segment (z minutes), the celebrity narrates \(\frac{1}{2}\) of it

Our goal is to find what fraction of the entire w-minute video consists of celebrity narration.

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

2. Establish segment length relationships

Now let's work with the constraint \(\mathrm{3x = 3y = z}\). This tells us something important about how the segments relate to each other.

From \(\mathrm{3x = 3y}\), we can see that \(\mathrm{x = y}\) (the first two segments are the same length).
From \(\mathrm{3x = z}\), we know that \(\mathrm{z = 3x}\) (the third segment is 3 times as long as the first segment).

So if we call the length of the first segment 'x', then:

• First segment: x minutes
• Second segment: x minutes (same as first)
• Third segment: 3x minutes

The total video length is: \(\mathrm{w = x + x + 3x = 5x}\) minutes

This means \(\mathrm{x = \frac{w}{5}}\), so each segment length in terms of w is:

• First segment: \(\frac{\mathrm{w}}{5}\) minutes
• Second segment: \(\frac{\mathrm{w}}{5}\) minutes
• Third segment: \(\frac{3\mathrm{w}}{5}\) minutes

3. Calculate narrated time for each segment

Now let's figure out how much time the celebrity actually narrates in each segment:

First segment narration: The celebrity narrates \(\frac{1}{3}\) of the first segment
\(= \frac{1}{3} \times \frac{\mathrm{w}}{5} = \frac{\mathrm{w}}{15}\) minutes

Second segment narration: The celebrity narrates \(\frac{1}{2}\) of the second segment
\(= \frac{1}{2} \times \frac{\mathrm{w}}{5} = \frac{\mathrm{w}}{10}\) minutes

Third segment narration: The celebrity narrates \(\frac{1}{2}\) of the third segment
\(= \frac{1}{2} \times \frac{3\mathrm{w}}{5} = \frac{3\mathrm{w}}{10}\) minutes

4. Express final fraction

To find the total narrated time, we add up all three portions:
Total narrated time = \(\frac{\mathrm{w}}{15} + \frac{\mathrm{w}}{10} + \frac{3\mathrm{w}}{10}\)

To add these fractions, we need a common denominator. The LCD of 15, 10, and 10 is 30:

• \(\frac{\mathrm{w}}{15} = \frac{2\mathrm{w}}{30}\)
• \(\frac{\mathrm{w}}{10} = \frac{3\mathrm{w}}{30}\)
• \(\frac{3\mathrm{w}}{10} = \frac{9\mathrm{w}}{30}\)

Total narrated time = \(\frac{2\mathrm{w}}{30} + \frac{3\mathrm{w}}{30} + \frac{9\mathrm{w}}{30} = \frac{14\mathrm{w}}{30} = \frac{7\mathrm{w}}{15}\)

The fraction of the video that consists of celebrity narration is:
(Total narrated time) ÷ (Total video time) = \(\frac{7\mathrm{w}}{15} \div \mathrm{w} = \frac{7}{15}\)

Final Answer

The celebrity host narrates \(\frac{7}{15}\) of the w-minute video.

This matches answer choice E: \(\frac{7}{15}\).

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting the constraint equation \(\mathrm{3x = 3y = z}\)
Students often struggle with this type of multi-part equality. They might incorrectly conclude that all three segments are equal in length (\(\mathrm{x = y = z}\)) instead of recognizing that \(\mathrm{x = y}\) and \(\mathrm{z = 3x}\). This fundamental misunderstanding would lead to incorrect segment lengths throughout the entire solution.

Faltering Point 2: Confusing what needs to be calculated
The question asks for the fraction of the entire video that the celebrity narrates, but students might mistakenly think they need to find the fraction of each individual segment that's narrated. This conceptual error would lead them down the wrong path from the start, focusing on segment-by-segment analysis rather than the total narration time.

Errors while executing the approach

Faltering Point 1: Arithmetic errors when finding common denominators
When adding fractions \(\frac{\mathrm{w}}{15} + \frac{\mathrm{w}}{10} + \frac{3\mathrm{w}}{10}\), students frequently make mistakes with finding the least common denominator (30) or converting fractions incorrectly. For example, they might write \(\frac{\mathrm{w}}{15} = \frac{3\mathrm{w}}{30}\) instead of \(\frac{2\mathrm{w}}{30}\), or struggle with converting \(\frac{3\mathrm{w}}{10}\) to \(\frac{9\mathrm{w}}{30}\).

Faltering Point 2: Incorrect calculation of individual segment narration times
Students may correctly identify segment lengths but then make errors when calculating the narrated portions. For instance, they might calculate \(\frac{1}{2}\) of the third segment as \(\frac{1}{2} \times \frac{\mathrm{w}}{5}\) instead of \(\frac{1}{2} \times \frac{3\mathrm{w}}{5}\), forgetting that the third segment is \(\frac{3\mathrm{w}}{5}\) minutes long, not \(\frac{\mathrm{w}}{5}\) minutes.

Errors while selecting the answer

Faltering Point 1: Providing the total narrated time instead of the fraction
After correctly calculating that the total narrated time is \(\frac{7\mathrm{w}}{15}\), students might select this as their final answer without completing the last step of dividing by the total video length w. They need to remember that the question asks for a fraction of the entire video, which requires the final division: \(\frac{7\mathrm{w}}{15} \div \mathrm{w} = \frac{7}{15}\).

Alternate Solutions

Smart Numbers Approach

This problem is well-suited for the smart numbers method because we have the relationship \(\mathrm{3x = 3y = z}\), which allows us to choose convenient concrete values that satisfy this constraint.

Step 1: Choose Strategic Smart Numbers

Since \(\mathrm{3x = 3y = z}\), let's set this common value to 6. We choose 6 because:

• It's divisible by 3 (needed for the \(\frac{1}{3}\) fraction in the first segment)
• It's divisible by 2 (needed for the \(\frac{1}{2}\) fractions in segments 2 and 3)
• This will give us clean, whole number calculations

From \(\mathrm{3x = 3y = z = 6}\):

• \(\mathrm{3x = 6 \rightarrow x = 2}\)
• \(\mathrm{3y = 6 \rightarrow y = 2}\)
• \(\mathrm{z = 6}\)

Step 2: Calculate Total Video Length

\(\mathrm{w = x + y + z = 2 + 2 + 6 = 10}\) minutes

Step 3: Calculate Celebrity Narration Time for Each Segment

• First segment: Celebrity narrates \(\frac{1}{3}\) of x = \(\frac{1}{3} \times 2 = \frac{2}{3}\) minutes
• Second segment: Celebrity narrates \(\frac{1}{2}\) of y = \(\frac{1}{2} \times 2 = 1\) minute
• Third segment: Celebrity narrates \(\frac{1}{2}\) of z = \(\frac{1}{2} \times 6 = 3\) minutes

Step 4: Find Total Narrated Time

Total narrated time = \(\frac{2}{3} + 1 + 3 = \frac{2}{3} + 4 = \frac{2}{3} + \frac{12}{3} = \frac{14}{3}\) minutes

Step 5: Calculate Fraction of Total Video

Fraction of video narrated = (Total narrated time)/(Total video time)

\(= \frac{14}{3} \div 10 = \frac{14}{3} \times \frac{1}{10} = \frac{14}{30} = \frac{7}{15}\)

Answer: \(\frac{7}{15}\)

Why Smart Numbers Work Here

The smart numbers approach is particularly effective for this problem because:

• The constraint \(\mathrm{3x = 3y = z}\) creates a natural relationship that allows us to choose one parameter and derive all others
• We can select a number (6) that makes all fractional calculations clean
• Working with concrete numbers eliminates the complexity of algebraic manipulation while preserving the mathematical relationships