NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Two-way tables

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A walking-route survey was conducted in Melbourne over a 130-day period to investigate the relationship between the day of the week (weekday or weekend) and the type of walking route preferred (urban path, park trail, or coastal walk). The results are summarised in the following two-way table: | Route Type | Weekday | Weekend | Total | |----------------|---------|---------|-------| | Urban Path | 31 | 40 | 71 | | Park Trail | 42 | 39 | 81 | | Coastal Walk | 28 | 30 | 58 | | Total | 101 | 109 | 210 | What is the exact proportion of all walking routes taken on weekdays that were park trails?

Express your answer as a simplified fraction.

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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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HH, HT, TH and TT are equally likely. Two outcomes have exactly one head: probability = 2/4 = 1/2.

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Worked solution and answer (click to reveal)

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Worked solution

  1. The question asks for the proportion of walking routes taken on weekdays that were park trails. This means we need to find the fraction of weekday routes that were park trails.
  2. From the two-way table, the number of park trails taken on weekdays is 42.
  3. The total number of walking routes taken on weekdays is 101, as given in the 'Total' column under 'Weekday'.
  4. The proportion is calculated as: (number of park trails on weekdays) / (total weekday routes). This gives 42 / 101.
  5. We now simplify the fraction 42/101. First, check for common factors between 42 and 101.
  6. The prime factors of 42 are 2 × 3 × 7. The number 101 is a prime number, as it is not divisible by any prime less than its square root (approximately 10), and 101 is not divisible by 2, 3, 5, or 7.
  7. Since 101 is prime and does not divide 42, the greatest common divisor (GCD) of 42 and 101 is 1.
  8. Therefore, the fraction 42/101 is already in its simplest form.
  9. The proportion is exactly 42/101.
  10. We verify: 42 ÷ 101 ≈ 0.4158, and 42/101 cannot be simplified further.
  11. The exact proportion of all walking routes taken on weekdays that were park trails is 42/101.
  12. This matches the requirement to express the answer as a simplified fraction.

Reference answer: 42/101

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