NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Two-way tables

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A survey was conducted at a high school in Perth to investigate the relationship between student participation in a science fair (yes or no) and their preferred subject (biology, chemistry, or physics). The results are presented in the following two-way table: | | Biology | Chemistry | Physics | Total | |----------------------|---------|-----------|---------|-------| | Participated | 28 | 32 | 20 | 80 | | Did Not Participate | 42 | 38 | 30 | 110 | | Total | 70 | 70 | 50 | 190 | What is the exact proportion of students who participated in the science fair and preferred chemistry?

Express your answer as a simplified fraction.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The question asks for the proportion of students who participated in the science fair and preferred chemistry. This is a conditional proportion based on the joint occurrence of two categorical variables: participation (yes) and subject preference (chemistry).
  2. From the two-way table, the number of students who participated in the science fair and preferred chemistry is 32.
  3. The total number of students surveyed is 190, as given in the 'Total' row under the 'Total' column.
  4. The proportion is calculated as the number of students who participated and preferred chemistry divided by the total number of students. This is expressed as: 32 / 190.
  5. Simplify the fraction 32/190 by finding the greatest common divisor (GCD) of 32 and 190. The GCD of 32 and 190 is 2.
  6. Divide both numerator and denominator by 2: 32 ÷ 2 = 16, 190 ÷ 2 = 95.
  7. The simplified fraction is 16/95.
  8. Check that 16 and 95 have no common factors other than 1. 16 = 2^4, 95 = 5 × 19. No common prime factors exist.
  9. The fraction 16/95 is in simplest form and represents the exact proportion.
  10. Verify the result by checking that 16/95 ≈ 0.1684 and 32/190 ≈ 0.1684, confirming the simplification is correct.
  11. The proportion of students who participated in the science fair and preferred chemistry is exactly 16/95.
  12. The final answer is a simplified fraction, as required.

Reference answer: 16/95

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