Mathematics 10 — Exercise 12.1 · Q3 (ii)

Full question

The ratio P(A) : P(Aᶜ) is 17 : 15 and the sample space contains 640 equally likely outcomes. Find n(A), the number of outcomes favourable to A.

Step-by-step solution

1. The event occupies 17 of32 parts, so n(A)=640×17/32.

2. Substitute the given probabilities or outcome counts.

   $$640\times \frac{17}{17+15}$$

3. Calculate this operation before combining the remaining terms.

   $$17+15=32$$

4. Calculate this operation before combining the remaining terms.

   $$17\times 1=17$$

5. Calculate this operation before combining the remaining terms.

   $$1\times 32=32$$

6. For multiplication of fractions, multiply the numerators together and the denominators together.

   $$\frac{17\times 1}{1\times 32}$$

7. Calculate these two products separately.

   $$\frac{17}{32}$$

8. For multiplication of fractions, multiply the numerators together and the denominators together.

   $$\frac{1\times 1}{1\times 32}$$

9. Calculate these two products separately.

   $$\frac{1}{32}$$

10. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{32}=\frac{1}{32}$$

11. Calculate this operation before combining the remaining terms.

    $$17\times \frac{1}{32}=\frac{17}{32}$$

12. Calculate this operation before combining the remaining terms.

    $$640\times 17=10880$$

13. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{10880\times 1}{1\times 32}$$

14. Calculate this operation before combining the remaining terms.

    $$10880\times \frac{1}{32}=340$$

15. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{640\times 17}{1\times 32}$$

16. Calculate these two products separately.

    $$\frac{10880}{32}$$

17. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=32$$

18. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{32\times 1}{1\times 32}$$

19. Calculate these two products separately.

    $$\frac{32}{32}$$

20. Show the two divisions explicitly.

    $$\frac{\frac{32}{32}}{\frac{32}{32}}$$

21. Write the reduced fraction.

    $$\frac{1}{1}$$

22. Calculate this operation before combining the remaining terms.

    $$32\times \frac{1}{32}=1$$

23. Calculate this operation before combining the remaining terms.

    $$\frac{1}{1}=1$$

24. Calculate this operation before combining the remaining terms.

    $$340\times 1=340$$

25. Show the two divisions explicitly.

    $$\frac{\frac{10880}{32}}{\frac{32}{32}}$$

26. Evaluate the numerator division.

    $$\frac{10880}{32}=340$$

27. Evaluate the denominator division.

    $$\frac{32}{32}=1$$

28. Write the reduced fraction.

    $$340\times \frac{1}{1}$$

29. Calculate this operation before combining the remaining terms.

    $$640\times \frac{17}{32}=340$$

30. The simplified result is the required probability.

    $$340$$

31. This result is an outcome count, not a probability.

Answer

The final answer is shown below.