Mathematics 10 — Exercise Review · Q7

Full question

Four knives and six forks cost Rs 136. Six knives and five forks cost Rs 164. Find each unit cost.

Step-by-step solution

1. Let x be the price of one knife and y of one fork. Multiply unit price by quantity for each bill.

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

   $$(\left[\begin{array}{cc}4& 6\\ 6& 5\end{array}\right],\left[\begin{array}{c}x\\ y\end{array}\right],\left[\begin{array}{c}136\\ 164\end{array}\right])$$

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

   $$D=4\times 5-6\times 6$$

4. Calculate the first diagonal product.

   $$4\times 5=20$$

5. Calculate the second diagonal product.

   $$6\times 6=36$$

6. Subtract the second product, including its sign.

   $$D=20-36$$

7. Simplify the subtraction.

   $$D=-16$$

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

   $$D=4\times 5-6\times 6$$

9. Calculate the first diagonal product.

   $$4\times 5=20$$

10. Calculate the second diagonal product.

    $$6\times 6=36$$

11. Subtract the second product, including its sign.

    $$D=20-36$$

12. Simplify the subtraction.

    $$D=-16$$

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

    $$-16\ne 0$$

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

    $$\left[\begin{array}{cc}5& 6\\ 6& 4\end{array}\right]$$

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

    $$\left[\begin{array}{cc}5& -6\\ -6& 4\end{array}\right]$$

16. Calculate this operation before combining the remaining terms.

    $$\frac{1}{-16}=-\frac{1}{16}$$

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

    $$\frac{5\times \left(-1\right)}{1\times 16}$$

18. Calculate these two products separately.

    $$-\frac{5}{16}$$

19. Calculate this operation before combining the remaining terms.

    $$5\times \left(-\frac{1}{16}\right)=-\frac{5}{16}$$

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

    $$\frac{5}{-16}=-\frac{5}{16}$$

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

    $$\frac{\left(-6\right)\times \left(-1\right)}{1\times 16}$$

22. Calculate these two products separately.

    $$\frac{6}{16}$$

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

    $$\mathrm{GCD}=2$$

24. Show the two divisions explicitly.

    $$\frac{\frac{6}{2}}{\frac{16}{2}}$$

25. Evaluate the numerator division.

    $$\frac{6}{2}=3$$

26. Evaluate the denominator division.

    $$\frac{16}{2}=8$$

27. Write the reduced fraction.

    $$\frac{3}{8}$$

28. Calculate this operation before combining the remaining terms.

    $$\left(-6\right)\times \left(-\frac{1}{16}\right)=\frac{3}{8}$$

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

    $$-\frac{6}{-16}=\frac{3}{8}$$

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

    $$-\frac{6}{-16}=\frac{3}{8}$$

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

    $$\frac{4\times \left(-1\right)}{1\times 16}$$

32. Calculate these two products separately.

    $$-\frac{4}{16}$$

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

    $$\mathrm{GCD}=4$$

34. Show the two divisions explicitly.

    $$\frac{-\frac{4}{4}}{\frac{16}{4}}$$

35. Evaluate the numerator division.

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

36. Evaluate the denominator division.

    $$\frac{16}{4}=4$$

37. Write the reduced fraction.

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

38. Calculate this operation before combining the remaining terms.

    $$4\times \left(-\frac{1}{16}\right)=-\frac{1}{4}$$

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

    $$\frac{4}{-16}=-\frac{1}{4}$$

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

    $$\left[\begin{array}{cc}-\frac{5}{16}& \frac{3}{8}\\ \frac{3}{8}& -\frac{1}{4}\end{array}\right]$$

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

42. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 1 columns.

    $$(\left[\begin{array}{cc}-\frac{5}{16}& \frac{3}{8}\\ \frac{3}{8}& -\frac{1}{4}\end{array}\right],\left[\begin{array}{c}136\\ 164\end{array}\right])$$

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

    $$\frac{\left(-85\right)\times 1}{1\times 2}$$

44. Calculate these two products separately.

    $$-\frac{85}{2}$$

45. Calculate this operation before combining the remaining terms.

    $$\left(-85\right)\times \frac{1}{2}=-\frac{85}{2}$$

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

    $$\frac{3\times 1}{1\times 8}$$

47. Calculate this operation before combining the remaining terms.

    $$3\times \frac{1}{8}=\frac{3}{8}$$

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

    $$\frac{3\times 164}{8\times 1}$$

49. Calculate these two products separately.

    $$\frac{492}{8}$$

50. Show the two divisions explicitly.

    $$\frac{\frac{492}{4}}{\frac{8}{4}}$$

51. Evaluate the numerator division.

    $$\frac{492}{4}=123$$

52. Evaluate the denominator division.

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

53. Write the reduced fraction.

    $$\frac{123}{2}$$

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

    $$\mathrm{LCD}=2$$

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

    $$-\frac{85}{2}=\frac{\left(-85\right)\times 1}{2\times 1}$$

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

    $$\frac{123}{2}=\frac{123\times 1}{2\times 1}$$

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

    $$\frac{-85+123}{2}=\frac{38}{2}$$

58. Show the two divisions explicitly.

    $$\frac{\frac{38}{2}}{\frac{2}{2}}$$

59. Evaluate the numerator division.

    $$\frac{38}{2}=19$$

60. Evaluate the denominator division.

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

61. Write the reduced fraction.

    $$19\times \frac{1}{1}$$

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

    $$-\frac{85}{2}+\frac{3}{8}\times 164$$

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

    $$\frac{\left(-5\right)\times 1}{1\times 16}$$

64. Calculate this operation before combining the remaining terms.

    $$\left(-5\right)\times \frac{1}{16}=-\frac{5}{16}$$

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

    $$\frac{\left(-5\right)\times 136}{16\times 1}$$

66. Calculate these two products separately.

    $$-\frac{680}{16}$$

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

    $$\mathrm{GCD}=8$$

68. Show the two divisions explicitly.

    $$\frac{-\frac{680}{8}}{\frac{16}{8}}$$

69. Evaluate the numerator division.

    $$-\frac{680}{8}=-85$$

70. Evaluate the denominator division.

    $$\frac{16}{8}=2$$

71. Calculate this individual product, including its signs.

    $$\left(-\frac{5}{16}\right)\times 136=-\frac{85}{2}$$

72. Calculate this individual product, including its signs.

    $$\frac{3}{8}\times 164=\frac{123}{2}$$

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

    $$\frac{123\times 1}{1\times 2}$$

74. Calculate this operation before combining the remaining terms.

    $$123\times \frac{1}{2}=\frac{123}{2}$$

75. Add the calculated products to obtain this entry.

    $$-\frac{85}{2}+\frac{123}{2}=19$$

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

    $$\frac{3\times 136}{8\times 1}$$

77. Calculate these two products separately.

    $$\frac{408}{8}$$

78. Show the two divisions explicitly.

    $$\frac{\frac{408}{8}}{\frac{8}{8}}$$

79. Evaluate the numerator division.

    $$\frac{408}{8}=51$$

80. Evaluate the denominator division.

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

81. Write the reduced fraction.

    $$51\times \frac{1}{1}$$

82. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 41=-41$$

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

    $$\frac{3}{8}\times 136-41$$

84. Calculate this individual product, including its signs.

    $$\frac{3}{8}\times 136=51$$

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

    $$\frac{\left(-1\right)\times 1}{1\times 4}$$

86. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{4}=-\frac{1}{4}$$

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

    $$\frac{\left(-1\right)\times 164}{4\times 1}$$

88. Calculate these two products separately.

    $$-\frac{164}{4}$$

89. Show the two divisions explicitly.

    $$\frac{-\frac{164}{4}}{\frac{4}{4}}$$

90. Evaluate the numerator division.

    $$-\frac{164}{4}=-41$$

91. Evaluate the denominator division.

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

92. Write the reduced fraction.

    $$-41\times \frac{1}{1}$$

93. Calculate this individual product, including its signs.

    $$\left(-\frac{1}{4}\right)\times 164=-41$$

94. Add the calculated products to obtain this entry.

    $$51-41=10$$

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

    $$\left[\begin{array}{c}19\\ 10\end{array}\right]$$

96. Calculate this operation before combining the remaining terms.

    $$4\times 19=76$$

97. Calculate this operation before combining the remaining terms.

    $$6\times 10=60$$

98. Calculate this operation before combining the remaining terms.

    $$76+60=136$$

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

    $$4\times 19+6\times 10=136$$

100. Calculate this operation before combining the remaining terms.

     $$6\times 19=114$$

101. Calculate this operation before combining the remaining terms.

     $$5\times 10=50$$

102. Calculate this operation before combining the remaining terms.

     $$114+50=164$$

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

     $$6\times 19+5\times 10=164$$

Answer

The final answer is shown below.