NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Ratios

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Two friends, Sam and Jess, walk different routes to the same park in Melbourne. Sam walks 65 minutes along a path that takes him 1.3 km per hour.

Jess walks 68 minutes along a path that takes her 1.25 km per hour.

What is the ratio of the distance Sam walked to the distance Jess walked, expressed as a simplified fraction?

Concept illustrations

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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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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. Sam walks for 65 minutes at 1.3 km per hour. Convert time from minutes to hours: 65 ÷ 60 = 13/12 hours.
  2. Use distance = speed × time. Sam's distance = 1.3 km/h × 13/12 h. Convert 1.3 to fraction: 1.3 = 13/10.
  3. Substitute: (13/10) × (13/12) = 169/120 km.
  4. Jess walks for 68 minutes at 1.25 km per hour. Convert time: 68 ÷ 60 = 17/15 hours.
  5. Use distance = speed × time. Jess's distance = 1.25 km/h × 17/15 h. Convert 1.25 to fraction: 1.25 = 5/4.
  6. Substitute: (5/4) × (17/15) = 85/60 km. Simplify: divide numerator and denominator by 5 → 17/12 km.
  7. Now find the ratio of Sam's distance to Jess's distance: (169/120) ÷ (17/12).
  8. Dividing fractions: (169/120) × (12/17) = (169 × 12) / (120 × 17).
  9. Simplify before multiplying: 12 and 120 share a factor of 12 → 12/120 = 1/10. So: (169 × 1) / (10 × 17) = 169/170.
  10. Check if 169/170 can be simplified. 169 = 13², 170 = 2×5×17. No common factors. Ratio is 169:170.
  11. The ratio of Sam's distance to Jess's distance is 169/170. This is already in simplest form.
  12. Final answer: 169/170. This matches the simplified fraction of the ratio.

Reference answer: 169/170

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