NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Financial modelling with percentages

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A hiker in Darwin plans a two-segment walking route. The first segment is 56 km long and is completed at a constant speed.

The second segment is 37 km long and is walked at a speed that is 20% faster than the first segment's speed. If the hiker takes exactly 7 hours to complete the first segment, what is the total time, in hours, to complete both segments?

Express your answer as an exact fraction.

Concept illustrations

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A quarter is 25 percent. One of four equal parts is 25/100. For a whole of 80, each part is 20.1 of 4 equal parts
One of four equal parts is 25/100. For a whole of 80, each part is 20.
Compare the increase with the original. A price grows from 80 to 100. The added 20 is one quarter of the original 80: a 25% increase. Bar lengths share the same scale.BeforeAfter80802020 ÷ 80 × 100% = 25%
A price grows from 80 to 100. The added 20 is one quarter of the original 80: a 25% increase. Bar lengths share the same scale.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The hiker completes the first segment of 56 km in 7 hours. Speed is distance divided by time, so calculate the speed for the first segment: 56 km ÷ 7 h = 8 km/h.
  2. The second segment is walked at a speed 20% faster than the first segment. Convert 20% to a decimal: 20 ÷ 100 = 0.2.
  3. Calculate the increased speed: 8 km/h × (1 + 0.2) = 8 × 1.2 = 9.6 km/h.
  4. The second segment is 37 km long. Time = distance ÷ speed, so time for second segment = 37 km ÷ 9.6 km/h.
  5. Perform the division: 37 ÷ 9.6. Multiply numerator and denominator by 10 to eliminate decimal: 370 ÷ 96.
  6. Simplify 370/96 by dividing numerator and denominator by 2: 185/48.
  7. The total time is the sum of the times for both segments: 7 h + 185/48 h.
  8. Convert 7 h to a fraction with denominator 48: 7 = 336/48.
  9. Add the times: 336/48 + 185/48 = 521/48 h.
  10. The total time is 521/48 hours. This is an improper fraction and cannot be simplified further since 521 is a prime number and 48 has no common factors with 521.
  11. Check: 521 ÷ 48 ≈ 10.854 h. First segment: 7 h. Second segment: 37 ÷ 9.6 ≈ 3.854 h. Total ≈ 10.854 h, which matches 521/48 h.
  12. The total time to complete both segments is 521/48 hours.

Reference answer: 521/48

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