NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Accuracy and measurement errors

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Required skills (click to reveal)

A hiker in the Blue Mountains near Sydney measures two segments of a walking route: the first segment is 42 metres long with a maximum error of ±0.25 metres, and the second segment is 28 metres long with a maximum error of ±0.15 metres. The hiker calculates the total route length using the maximum measured values for both segments.

What is the exact total length of the route, in metres, when both segments are at their maximum measured values? Express your answer as a simplified fraction.

Concept illustrations

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Measurement compared with a reference. Measured length 49 cm differs from the reference 50 cm by 1 cm. Absolute error is 1 cm and percentage error is 2%. The axis is a close-up from 48 to 51 cm.48 cm49 cm50 cm51 cmMeasured: 49Reference: 50|49 − 50| = 1 cm
Measured length 49 cm differs from the reference 50 cm by 1 cm. Absolute error is 1 cm and percentage error is 2%. The axis is a close-up from 48 to 51 cm.
Nearest hundredth on a number line. 6.347 lies between 6.34 and 6.35. It is beyond the midpoint 6.345, so rounding to the nearest hundredth gives 6.35.6.346.3456.356.3476.347 ≈ 6.35
6.347 lies between 6.34 and 6.35. It is beyond the midpoint 6.345, so rounding to the nearest hundredth gives 6.35.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The first segment is measured as 42 metres with a maximum error of ±0.25 metres. The maximum measured value is 42 + 0.25 = 42.25 metres.
  2. The second segment is measured as 28 metres with a maximum error of ±0.15 metres. The maximum measured value is 28 + 0.15 = 28.15 metres.
  3. To find the total route length at maximum measured values, add the two maximum values: 42.25 + 28.15.
  4. Convert both decimals to fractions: 42.25 = 42 + 1/4 = 169/4, and 28.15 = 28 + 15/100 = 28 + 3/20 = (28×20 + 3)/20 = 563/20.
  5. Find a common denominator for 169/4 and 563/20. The least common denominator of 4 and 20 is 20.
  6. Convert 169/4 to a fraction with denominator 20: (169×5)/(4×5) = 845/20.
  7. Add the two fractions: 845/20 + 563/20 = (845 + 563)/20 = 1408/20.
  8. Simplify 1408/20 by dividing numerator and denominator by their greatest common divisor. GCD of 1408 and 20 is 4.
  9. 1408 ÷ 4 = 352, and 20 ÷ 4 = 5. So, 1408/20 = 352/5.
  10. Check that 352/5 is in simplest form: 352 and 5 share no common factors other than 1, so it is simplified.
  11. The total length when both segments are at their maximum measured values is 352/5 metres.
  12. Verify: 352 ÷ 5 = 70.4, and 42.25 + 28.15 = 70.4, confirming correctness.

Reference answer: 352/5

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