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The table gives information about the distances, in miles, that some Year 10 students live from school - Edexcel - GCSE Maths - Question 18 - 2022 - Paper 3
Question 18
The table gives information about the distances, in miles, that some Year 10 students live from school. | Distance (d miles) | Frequency | |-------------------|----... show full transcript
Worked Solution & Example Answer:The table gives information about the distances, in miles, that some Year 10 students live from school - Edexcel - GCSE Maths - Question 18 - 2022 - Paper 3
Step 1
Drawing a histogram for this information
Answer
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Identify the intervals: First, we note the intervals given in the table which are:
- 0 < d ≤ 1.0
- 1.0 < d ≤ 1.5
- 1.5 < d ≤ 2.0
- 2.0 < d ≤ 3.0
- 3.0 < d ≤ 5.0
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Calculate frequency density: Frequency density can be calculated using the formula:
- For 0 < d ≤ 1.0: ( \text{Frequency Density} = \frac{90}{1} = 90 )
- For 1.0 < d ≤ 1.5: ( \text{Frequency Density} = \frac{48}{0.5} = 96 )
- For 1.5 < d ≤ 2.0: ( \text{Frequency Density} = \frac{22}{0.5} = 44 )
- For 2.0 < d ≤ 3.0: ( \text{Frequency Density} = \frac{8}{1} = 8 )
- For 3.0 < d ≤ 5.0: ( \text{Frequency Density} = \frac{12}{2} = 6 )
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Draw the bars: On the histogram, you should accurately represent the frequency densities for each interval. Make sure the heights of the bars reflect the calculated frequency densities for visual clarity.
Step 2
Find an expression, in terms of n, for the number of Year 11 students who live between 3 and 5 miles from school.
Answer
To find the number of Year 11 students who live between 3 and 5 miles from school, we need to analyze the total frequency of the histogram for this range, which includes:
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Analyze proportions: Given that the number of students between 1 and 2 miles is n, the intervals associated with this group will help us express the number of students between 3 and 5 miles as a proportion.
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Calculate the total frequency: Say the total frequency of students represented in the histogram is ( F = n + x ), where x is the unknown number of students that live between 3 and 5 miles.
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Use ratios: Utilizing the bar heights for the ranges:
- Since the ratio of students in this area compared with the interval's areas gives the insight needed:
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Derive expression for x: From the ratios, rearranging gives:
= n \cdot \frac{1}{16} $$ Hence, the expression for the number of Year 11 students who live between 3 and 5 miles from school is \( \frac{n}{16} \).