Mathematics 10 — Exercise Review · Q8

Full question

Five men and three women earn Rs 3500 daily in total. With two men and six women the total is Rs 5000. Find the daily wage of one man and one woman.

Step-by-step solution

1. Let x be one man’s daily wage and y one woman’s. The second group has 3+3=6 women.

2. Write the coefficients in A, the unknowns in X, and constants in B, giving AX = B.

   $$(\left[\begin{array}{cc}5& 3\\ 2& 6\end{array}\right],\left[\begin{array}{c}x\\ y\end{array}\right],\left[\begin{array}{c}3500\\ 5000\end{array}\right])$$

3. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

   $$D=5\times 6-3\times 2$$

4. Calculate the first diagonal product.

   $$5\times 6=30$$

5. Calculate the second diagonal product.

   $$3\times 2=6$$

6. Subtract the second product, including its sign.

   $$D=30-6$$

7. Simplify the subtraction.

   $$D=24$$

8. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

   $$D=5\times 6-3\times 2$$

9. Calculate the first diagonal product.

   $$5\times 6=30$$

10. Calculate the second diagonal product.

    $$3\times 2=6$$

11. Subtract the second product, including its sign.

    $$D=30-6$$

12. Simplify the subtraction.

    $$D=24$$

13. The determinant is nonzero, so the inverse exists.

    $$24\ne 0$$

14. Swap the two diagonal entries to begin the adjoint.

    $$\left[\begin{array}{cc}6& 3\\ 2& 5\end{array}\right]$$

15. Change the signs of the two off-diagonal entries.

    $$\left[\begin{array}{cc}6& -3\\ -2& 5\end{array}\right]$$

16. Divide adjoint entry (1,1) by the determinant.

    $$\frac{6}{24}=\frac{1}{4}$$

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

    $$\frac{\left(-3\right)\times 1}{1\times 24}$$

18. Calculate these two products separately.

    $$-\frac{3}{24}$$

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

    $$\mathrm{GCD}=3$$

20. Show the two divisions explicitly.

    $$\frac{-\frac{3}{3}}{\frac{24}{3}}$$

21. Evaluate the numerator division.

    $$-\frac{3}{3}=-1$$

22. Evaluate the denominator division.

    $$\frac{24}{3}=8$$

23. Write the reduced fraction.

    $$-\frac{1}{8}$$

24. Calculate this operation before combining the remaining terms.

    $$\left(-3\right)\times \frac{1}{24}=-\frac{1}{8}$$

25. Divide adjoint entry (1,2) by the determinant.

    $$-\frac{3}{24}=-\frac{1}{8}$$

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

    $$\frac{\left(-2\right)\times 1}{1\times 24}$$

27. Calculate these two products separately.

    $$-\frac{2}{24}$$

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

    $$\mathrm{GCD}=2$$

29. Show the two divisions explicitly.

    $$\frac{-\frac{2}{2}}{\frac{24}{2}}$$

30. Evaluate the numerator division.

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

31. Evaluate the denominator division.

    $$\frac{24}{2}=12$$

32. Write the reduced fraction.

    $$-\frac{1}{12}$$

33. Calculate this operation before combining the remaining terms.

    $$\left(-2\right)\times \frac{1}{24}=-\frac{1}{12}$$

34. Divide adjoint entry (2,1) by the determinant.

    $$-\frac{2}{24}=-\frac{1}{12}$$

35. Divide adjoint entry (2,2) by the determinant.

    $$\frac{5}{24}=\frac{5}{24}$$

36. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{cc}\frac{1}{4}& -\frac{1}{8}\\ -\frac{1}{12}& \frac{5}{24}\end{array}\right]$$

37. Premultiply AX = B by A⁻¹. Since A⁻¹A = I, X = A⁻¹B.

38. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 1 columns.

    $$(\left[\begin{array}{cc}\frac{1}{4}& -\frac{1}{8}\\ -\frac{1}{12}& \frac{5}{24}\end{array}\right],\left[\begin{array}{c}3500\\ 5000\end{array}\right])$$

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

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

40. Calculate these two products separately.

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

41. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{1\times 3500}{4\times 1}$$

43. Calculate these two products separately.

    $$\frac{3500}{4}$$

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

    $$\mathrm{GCD}=4$$

45. Show the two divisions explicitly.

    $$\frac{\frac{3500}{4}}{\frac{4}{4}}$$

46. Evaluate the numerator division.

    $$\frac{3500}{4}=875$$

47. Evaluate the denominator division.

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

48. Write the reduced fraction.

    $$875\times \frac{1}{1}$$

49. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 625=-625$$

50. Entry (1,1): pair row 1 with column 1 and add their products.

    $$\frac{1}{4}\times 3500-625$$

51. Calculate this individual product, including its signs.

    $$\frac{1}{4}\times 3500=875$$

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

    $$\frac{\left(-1\right)\times 1}{1\times 8}$$

53. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{8}=-\frac{1}{8}$$

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

    $$\frac{\left(-1\right)\times 5000}{8\times 1}$$

55. Calculate these two products separately.

    $$-\frac{5000}{8}$$

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

    $$\mathrm{GCD}=8$$

57. Show the two divisions explicitly.

    $$\frac{-\frac{5000}{8}}{\frac{8}{8}}$$

58. Evaluate the numerator division.

    $$-\frac{5000}{8}=-625$$

59. Evaluate the denominator division.

    $$\frac{8}{8}=1$$

60. Write the reduced fraction.

    $$-625\times \frac{1}{1}$$

61. Calculate this individual product, including its signs.

    $$\left(-\frac{1}{8}\right)\times 5000=-625$$

62. Add the calculated products to obtain this entry.

    $$875-625=250$$

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

    $$\frac{\left(-875\right)\times 1}{1\times 3}$$

64. Calculate these two products separately.

    $$-\frac{875}{3}$$

65. Calculate this operation before combining the remaining terms.

    $$\left(-875\right)\times \frac{1}{3}=-\frac{875}{3}$$

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

    $$\frac{5\times 1}{1\times 24}$$

67. Calculate these two products separately.

    $$\frac{5}{24}$$

68. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{24}=\frac{5}{24}$$

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

    $$\frac{5\times 5000}{24\times 1}$$

70. Calculate these two products separately.

    $$\frac{25000}{24}$$

71. Show the two divisions explicitly.

    $$\frac{\frac{25000}{8}}{\frac{24}{8}}$$

72. Evaluate the numerator division.

    $$\frac{25000}{8}=3125$$

73. Evaluate the denominator division.

    $$\frac{24}{8}=3$$

74. Write the reduced fraction.

    $$\frac{3125}{3}$$

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

    $$\mathrm{LCD}=3$$

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

    $$-\frac{875}{3}=\frac{\left(-875\right)\times 1}{3\times 1}$$

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

    $$\frac{3125}{3}=\frac{3125\times 1}{3\times 1}$$

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

    $$\frac{-875+3125}{3}=\frac{2250}{3}$$

79. Show the two divisions explicitly.

    $$\frac{\frac{2250}{3}}{\frac{3}{3}}$$

80. Evaluate the numerator division.

    $$\frac{2250}{3}=750$$

81. Evaluate the denominator division.

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

82. Write the reduced fraction.

    $$750\times \frac{1}{1}$$

83. Entry (2,1): pair row 2 with column 1 and add their products.

    $$-\frac{875}{3}+\frac{5}{24}\times 5000$$

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

    $$\frac{\left(-1\right)\times 1}{1\times 12}$$

85. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{12}=-\frac{1}{12}$$

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

    $$\frac{\left(-1\right)\times 3500}{12\times 1}$$

87. Calculate these two products separately.

    $$-\frac{3500}{12}$$

88. Show the two divisions explicitly.

    $$\frac{-\frac{3500}{4}}{\frac{12}{4}}$$

89. Evaluate the numerator division.

    $$-\frac{3500}{4}=-875$$

90. Evaluate the denominator division.

    $$\frac{12}{4}=3$$

91. Calculate this individual product, including its signs.

    $$\left(-\frac{1}{12}\right)\times 3500=-\frac{875}{3}$$

92. Calculate this individual product, including its signs.

    $$\frac{5}{24}\times 5000=\frac{3125}{3}$$

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

    $$\frac{3125\times 1}{1\times 3}$$

94. Calculate this operation before combining the remaining terms.

    $$3125\times \frac{1}{3}=\frac{3125}{3}$$

95. Add the calculated products to obtain this entry.

    $$-\frac{875}{3}+\frac{3125}{3}=750$$

96. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{c}250\\ 750\end{array}\right]$$

97. Calculate this operation before combining the remaining terms.

    $$5\times 250=1250$$

98. Calculate this operation before combining the remaining terms.

    $$3\times 750=2250$$

99. Calculate this operation before combining the remaining terms.

    $$1250+2250=3500$$

100. Check in this original equation by multiplying the coefficients by the calculated values.

     $$5\times 250+3\times 750=3500$$

101. Calculate this operation before combining the remaining terms.

     $$2\times 250=500$$

102. Calculate this operation before combining the remaining terms.

     $$6\times 750=4500$$

103. Calculate this operation before combining the remaining terms.

     $$500+4500=5000$$

104. Check in this original equation by multiplying the coefficients by the calculated values.

     $$2\times 250+6\times 750=5000$$

Answer

The final answer is shown below.