NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Survey means and medians

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A survey was conducted in Sydney to compare the average walking distances of two groups of students on their way to school. Group A, consisting of 43 students, reported a total walking distance of 1247 metres.

Group B, consisting of 66 students, reported a total walking distance of 2178 metres.

What is the exact mean walking distance, in metres, for all students surveyed across both groups?

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A sample is part of a population. The large outline represents a population of 24 members. Six filled members illustrate a sample. This picture does not establish that a selection process was random.6 selected from 24
The large outline represents a population of 24 members. Six filled members illustrate a sample. This picture does not establish that a selection process was random.
Median, mode and range in sorted data. The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.23389↑ medianmode 3 range 7
The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The total number of students surveyed is the sum of students in Group A and Group B: 43 + 66 = 109 students.
  2. The total walking distance for all students is the sum of the distances reported by both groups: 1247 m + 2178 m = 3425 m.
  3. To find the mean walking distance, divide the total walking distance by the total number of students: 3425 ÷ 109.
  4. Perform the division: 3425 ÷ 109. First, estimate: 109 × 30 = 3270, which is less than 3425. 109 × 31 = 3379. 109 × 32 = 3488, which is too high.
  5. Subtract: 3425 - 3379 = 46. So, 3425 = 109 × 31 + 46.
  6. Express the result as a mixed number: 31 + 46/109. Since 46 and 109 share no common factors (gcd = 1), the fraction is in simplest form.
  7. Convert the mixed number to a decimal for verification: 46 ÷ 109 ≈ 0.4220. So, 31 + 0.4220 = 31.4220 m.
  8. Check the calculation: 109 × 31.4220 ≈ 109 × 31.422 = 3424.998, which rounds to 3425 m. The result is consistent.
  9. The mean walking distance for all students is exactly 3425/109 metres.
  10. Simplify 3425/109: since 109 is a prime number and does not divide 3425 evenly, the fraction is already in simplest form.
  11. The exact mean walking distance is 3425/109 metres, which is approximately 31.42 metres.
  12. The final answer is the exact value 3425/109 metres, as required by the question.

Reference answer: 3425/109 m

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