Mathematics 10 — Exercise 11.1 · Q8 (i–iii)

Full question

Football players took speed and strength tests, each marked out of 200. Plot the paired scores from the table, draw a line of best fit and describe the correlation. Abdullah scored 60 in the second test. Estimate his score in the first test.

Step-by-step solution

1. Each table column forms ONE ordered pair (x,y). Keep each pair together.

2. Label the horizontal and vertical axes, choose uniform scales, then plot every pair.

3. Draw a straight line through the middle of the point cloud, with points on both sides. This is an estimate; different sensible hand-drawn lines give slightly different predictions.

4. Strong negative correlation: y tends to fall as x rises.

5. For reproducible estimates the drawn line uses least squares. Its slope is Σ(x−x̄)(y−ȳ)/Σ(x−x̄)²; intercept is ȳ−slope×x̄.

   $$y=\frac{7740390}{43241}-\frac{43462\times x}{43241}$$

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

   $$\frac{7740390\times 1}{1\times 43241}$$

7. Calculate these two products separately.

   $$\frac{7740390}{43241}$$

8. Calculate this operation before combining the remaining terms.

   $$7740390\times \frac{1}{43241}=\frac{7740390}{43241}$$

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

   $$\frac{\left(-1\right)\times 7740390}{1\times 43241}$$

10. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{7740390}{43241}=-\frac{7740390}{43241}$$

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

    $$\frac{\left(-43462\right)\times 1}{1\times 43241}$$

12. Calculate these two products separately.

    $$-\frac{43462}{43241}$$

13. Calculate this operation before combining the remaining terms.

    $$\left(-43462\right)\times \frac{1}{43241}=-\frac{43462}{43241}$$

14. The given value is on the VERTICAL axis. Rearrange y=mx+c: subtract c, then divide by m to estimate x.

    $$\frac{60-\frac{7740390}{43241}}{-\frac{43462}{43241}}$$

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

    $$\mathrm{LCD}=43241$$

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

    $$60\times \frac{1}{1}=\frac{60\times 43241}{1\times 43241}$$

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

    $$\frac{2594460}{43241}$$

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

    $$\frac{\left(-7740390\right)\times 1}{1\times 43241}$$

19. Calculate this operation before combining the remaining terms.

    $$\left(-7740390\right)\times \frac{1}{43241}=-\frac{7740390}{43241}$$

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

    $$-\frac{7740390}{43241}=\frac{\left(-7740390\right)\times 1}{43241\times 1}$$

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

    $$-\frac{7740390}{43241}$$

22. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 7740390=-7740390$$

23. Calculate this operation before combining the remaining terms.

    $$2594460-7740390=-5145930$$

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

    $$\frac{\left(-5145930\right)\times 1}{1\times 43241}$$

25. Calculate these two products separately.

    $$-\frac{5145930}{43241}$$

26. Calculate this operation before combining the remaining terms.

    $$\left(-5145930\right)\times \frac{1}{43241}=-\frac{5145930}{43241}$$

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

    $$\frac{2594460-7740390}{43241}=-\frac{5145930}{43241}$$

28. Calculate this operation before combining the remaining terms.

    $$60-\frac{7740390}{43241}=-\frac{5145930}{43241}$$

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

    $$\frac{1\times \left(-43241\right)}{1\times 43462}$$

30. Calculate these two products separately.

    $$-\frac{43241}{43462}$$

31. Calculate this operation before combining the remaining terms.

    $$1\times \left(-\frac{43241}{43462}\right)=-\frac{43241}{43462}$$

32. Calculate this operation before combining the remaining terms.

    $$\frac{1}{-\frac{43462}{43241}}=-\frac{43241}{43462}$$

33. Calculate this operation before combining the remaining terms.

    $$\left(-5145930\right)\times \left(-43241\right)=222515159130$$

34. Calculate this operation before combining the remaining terms.

    $$43241\times 43462=1879340342$$

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

    $$\frac{222515159130\times 1}{1\times 1879340342}$$

36. Calculate this operation before combining the remaining terms.

    $$222515159130\times \frac{1}{1879340342}=\frac{2572965}{21731}$$

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

    $$\frac{\left(-5145930\right)\times \left(-43241\right)}{43241\times 43462}$$

38. Calculate these two products separately.

    $$\frac{222515159130}{1879340342}$$

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

    $$\mathrm{GCD}=86482$$

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

    $$\frac{222515159130\times 1}{1\times 86482}$$

41. Calculate these two products separately.

    $$\frac{222515159130}{86482}$$

42. Show the two divisions explicitly.

    $$\frac{\frac{222515159130}{86482}}{\frac{86482}{86482}}$$

43. Evaluate the denominator division.

    $$\frac{86482}{86482}=1$$

44. Write the reduced fraction.

    $$2572965\times \frac{1}{1}$$

45. Calculate this operation before combining the remaining terms.

    $$222515159130\times \frac{1}{86482}=2572965$$

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

    $$\frac{1879340342\times 1}{1\times 86482}$$

47. Calculate these two products separately.

    $$\frac{1879340342}{86482}$$

48. Show the two divisions explicitly.

    $$\frac{\frac{1879340342}{86482}}{\frac{86482}{86482}}$$

49. Write the reduced fraction.

    $$21731\times \frac{1}{1}$$

50. Calculate this operation before combining the remaining terms.

    $$1879340342\times \frac{1}{86482}=21731$$

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

    $$\frac{2572965\times 1}{1\times 21731}$$

52. Calculate this operation before combining the remaining terms.

    $$2572965\times \frac{1}{21731}=\frac{2572965}{21731}$$

53. Show the two divisions explicitly.

    $$\frac{\frac{222515159130}{86482}}{\frac{1879340342}{86482}}$$

54. Evaluate the numerator division.

    $$\frac{222515159130}{86482}=2572965$$

55. Evaluate the denominator division.

    $$\frac{1879340342}{86482}=21731$$

56. Write the reduced fraction.

    $$\frac{2572965}{21731}$$

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

    $$\frac{\left(-43241\right)\times 1}{1\times 43462}$$

58. Calculate this operation before combining the remaining terms.

    $$\left(-43241\right)\times \frac{1}{43462}=-\frac{43241}{43462}$$

59. Calculate this operation before combining the remaining terms.

    $$\left(-\frac{5145930}{43241}\right)\times \left(-\frac{43241}{43462}\right)=\frac{2572965}{21731}$$

Answer

Strong negative correlation. Estimated First test≈118.40.