NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Ratio and rate modelling

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A walking route in the Dandenong Ranges near Melbourne is designed with two segments: the first segment is 90 minutes long at a constant walking speed of 4.5 km/h, and the second segment is 97 minutes long at a constant speed of 3.2 km/h.

What is the exact total distance walked, in kilometres, expressed as a simplified fraction?

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Two red counters for every three blue counters. Red:blue = 2:3. There are 5 counters in all, so the red fraction of the whole is 2/5.RedRedBlueBlueBlue2 red : 3 blue · 5 counters in all
Red:blue = 2:3. There are 5 counters in all, so the red fraction of the whole is 2/5.
Translate a situation into a model. With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.$4 + $2/kmDefine tariff→C = 4 + 2dBuild rule→d=3 → C=$10Interpret
With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.

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Worked solution and answer (click to reveal)

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Worked solution

  1. The first segment is 90 minutes long at 4.5 km/h. Convert time from minutes to hours: 90 ÷ 60 = 1.5 hours.
  2. Use distance = speed × time for the first segment: 4.5 km/h × 1.5 h = 6.75 km.
  3. The second segment is 97 minutes long at 3.2 km/h. Convert time: 97 ÷ 60 = 97/60 hours.
  4. Calculate distance for second segment: 3.2 km/h × (97/60) h = (3.2 × 97) / 60 km.
  5. Compute 3.2 × 97: 3.2 × 100 = 320, minus 3.2 × 3 = 9.6, so 320 - 9.6 = 310.4.
  6. Now divide: 310.4 / 60 = 3104 / 600 (multiply numerator and denominator by 10 to eliminate decimal).
  7. Simplify 3104 / 600: divide numerator and denominator by 8: 3104 ÷ 8 = 388, 600 ÷ 8 = 75. So 388/75 km.
  8. Now add both distances: 6.75 km + 388/75 km. Convert 6.75 to fraction: 6.75 = 27/4.
  9. Find common denominator for 27/4 and 388/75. LCM of 4 and 75 is 300.
  10. Convert: 27/4 = (27 × 75) / 300 = 2025 / 300; 388/75 = (388 × 4) / 300 = 1552 / 300.
  11. Add: 2025 + 1552 = 3577. So total distance = 3577 / 300 km.
  12. Check if 3577/300 can be simplified. GCD of 3577 and 300 is 1, so it is already simplified. Total distance is 3577/300 km.

Reference answer: 3577/300 km

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