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A promotional video that lasts \(\mathrm{w}\) minutes has 3 segments which last \(\mathrm{x}\), \(\mathrm{y}\), and \(\mathrm{z}\) minutes, respectively. A celebrity host narrates \(\frac{1}{3}\) of the first segment, \(\frac{1}{2}\) of the second segment, and \(\frac{1}{2}\) of the third segment. If \(3\mathrm{x} = 3\mathrm{y} = \mathrm{z}\), what fraction of the \(\mathrm{w}\)-minute video does the celebrity host narrate?
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:
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
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:
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:
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
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:
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) = \(\frac{7\mathrm{w}}{15} \div \mathrm{w} = \frac{7}{15}\)
The celebrity host narrates \(\frac{7}{15}\) of the w-minute video.
This matches answer choice E: \(\frac{7}{15}\).
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.
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.
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: \(\frac{7\mathrm{w}}{15} \div \mathrm{w} = \frac{7}{15}\).
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.
Since \(\mathrm{3x = 3y = z}\), let's set this common value to 6. We choose 6 because:
From \(\mathrm{3x = 3y = z = 6}\):
\(\mathrm{w = x + y + z = 2 + 2 + 6 = 10}\) minutes
Total narrated time = \(\frac{2}{3} + 1 + 3 = \frac{2}{3} + 4 = \frac{2}{3} + \frac{12}{3} = \frac{14}{3}\) minutes
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: \(\frac{7}{15}\)
The smart numbers approach is particularly effective for this problem because: