Mathematics 10 — Exercise 12.2 · Q10

Full question

At a road junction, 50% of vehicles turn right, 30% turn left and 20% continue straight. Find the probability that a randomly chosen vehicle turns left or right.

Step-by-step solution

1. The two conditions cannot happen together in this trial. Their overlap is zero, so add their probabilities.

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

   $$\frac{1\times 1}{1\times 2}$$

3. Calculate these two products separately.

   $$\frac{1}{2}$$

4. Calculate this operation before combining the remaining terms.

   $$1\times \frac{1}{2}=\frac{1}{2}$$

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

   $$\frac{3\times 1}{1\times 10}$$

6. Calculate this operation before combining the remaining terms.

   $$3\times \frac{1}{10}=\frac{3}{10}$$

7. Substitute the given probabilities or outcome counts.

   $$\frac{1}{2}+\frac{3}{10}$$

8. To add these fractions, first use a common denominator: the least common multiple of their denominators.

   $$\mathrm{LCD}=10$$

9. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

   $$\frac{1}{2}=\frac{1\times 5}{2\times 5}$$

10. Calculate the converted numerator; the denominator is now common.

    $$\frac{5}{10}$$

11. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{3}{10}=\frac{3\times 1}{10\times 1}$$

12. Calculate the converted numerator; the denominator is now common.

    $$\frac{3}{10}$$

13. Calculate this operation before combining the remaining terms.

    $$5+3=8$$

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

    $$\frac{8\times 1}{1\times 10}$$

15. Calculate these two products separately.

    $$\frac{8}{10}$$

16. Calculate this operation before combining the remaining terms.

    $$8\times \frac{1}{10}=\frac{4}{5}$$

17. Keep the common denominator and add the signed numerators.

    $$\frac{5+3}{10}=\frac{8}{10}$$

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

    $$\mathrm{GCD}=2$$

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

    $$\frac{8\times 1}{1\times 2}$$

20. Calculate these two products separately.

    $$\frac{8}{2}$$

21. Show the two divisions explicitly.

    $$\frac{\frac{8}{2}}{\frac{2}{2}}$$

22. Evaluate the denominator division.

    $$\frac{2}{2}=1$$

23. Write the reduced fraction.

    $$4\times \frac{1}{1}$$

24. Calculate this operation before combining the remaining terms.

    $$8\times \frac{1}{2}=4$$

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

    $$\frac{10\times 1}{1\times 2}$$

26. Calculate these two products separately.

    $$\frac{10}{2}$$

27. Show the two divisions explicitly.

    $$\frac{\frac{10}{2}}{\frac{2}{2}}$$

28. Write the reduced fraction.

    $$5\times \frac{1}{1}$$

29. Calculate this operation before combining the remaining terms.

    $$10\times \frac{1}{2}=5$$

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

    $$\frac{4\times 1}{1\times 5}$$

31. Calculate this operation before combining the remaining terms.

    $$4\times \frac{1}{5}=\frac{4}{5}$$

32. Show the two divisions explicitly.

    $$\frac{\frac{8}{2}}{\frac{10}{2}}$$

33. Evaluate the numerator division.

    $$\frac{8}{2}=4$$

34. Evaluate the denominator division.

    $$\frac{10}{2}=5$$

35. Write the reduced fraction.

    $$\frac{4}{5}$$

36. Calculate this operation before combining the remaining terms.

    $$\frac{1}{2}+\frac{3}{10}=\frac{4}{5}$$

37. The simplified result is the required probability.

    $$\frac{4}{5}$$

Answer

The final answer is shown below.