NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Equivalent fractions and decimals

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A student in Brisbane walks a route that consists of two segments. The first segment is 4200 metres long, and the second segment is 5700 metres long.

What is the exact fraction, in simplest form, of the distance walked in the first segment compared to the total distance walked? Express your answer as a simplified fraction.

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Half in three forms. 1/2 = 0.5 = 50%. The whole is divided into two equal parts.1 of 2 equal parts
1/2 = 0.5 = 50%. The whole is divided into two equal parts.
Common denominators as equal strips. The first strip selects 4/12 = 1/3; the second selects 3/12 = 1/4. The result selects 7/12. Every strip represents the same whole.1/31/47/124/12 + 3/12 = 7/12
The first strip selects 4/12 = 1/3; the second selects 3/12 = 1/4. The result selects 7/12. Every strip represents the same whole.

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Explanation

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Worked solution

  1. The first segment is 4200 metres and the second segment is 5700 metres. Total distance walked is the sum of both segments.
  2. Calculate total distance: 4200 m + 5700 m = 9900 m.
  3. The fraction of the first segment compared to the total distance is 4200 / 9900.
  4. Simplify the fraction 4200 / 9900 by finding the greatest common divisor (GCD) of 4200 and 9900.
  5. Factor 4200: 4200 = 2^3 × 3 × 5^2 × 7.
  6. Factor 9900: 9900 = 2^2 × 3^2 × 5^2 × 11.
  7. Identify common prime factors: 2^2, 3, and 5^2. Multiply them: 2^2 × 3 × 5^2 = 4 × 3 × 25 = 300.
  8. Divide both numerator and denominator by 300: 4200 ÷ 300 = 14, 9900 ÷ 300 = 33.
  9. The simplified fraction is 14 / 33.
  10. Check that 14 and 33 have no common factors other than 1. 14 = 2 × 7, 33 = 3 × 11. No common factors.
  11. The fraction 14/33 is in simplest form and represents the exact ratio of the first segment to the total distance.
  12. The answer is 14/33, as it is the exact simplified fraction of the first segment distance to the total distance walked.

Reference answer: 14/33

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