Evaluate:
(i) [tex]\left(\frac{13}{18} \times \frac{-12}{39}\right)-\left(\frac{8}{9} \times \frac{-3}{4}\right)+\left[-\frac{7}{-9} \div \frac{63}{-36}\right][/tex]
(ii) [tex]\frac{2}{5} \times\left(-\frac{3}{7}\right)-\frac{1}{6} \times \frac{3}{2}+\frac{1}{14} \times \frac{2}{5}[/tex]
(iii) [tex]\frac{5}{6} \times\left(-\frac{2}{3}\right)-\frac{16}{5} \times \frac{3}{4}+\frac{3}{2} \times\left(-\frac{3}{9}\right)[/tex]
Answer (2)
Evaluate the first expression: Simplify each multiplication and division, then combine the terms to get 0.
Evaluate the second expression: Simplify each multiplication, find a common denominator, and combine the terms to get 28 − 11 .
Evaluate the third expression: Simplify each multiplication, find a common denominator, and combine the terms to get 90 − 311 .
The final answers are 0, 28 − 11 , and 90 − 311 .
Explanation
Problem Overview We are asked to evaluate three expressions involving fractions. We will simplify each expression step by step, following the order of operations (PEMDAS/BODMAS).
Evaluating Expression (i) Let's evaluate the first expression: ( 18 13 × 39 − 12 ) − ( 9 8 × 4 − 3 ) + [ − − 9 7 ÷ − 36 63 ] First, simplify each multiplication and division: 18 13 × 39 − 12 = 18 × 39 13 × − 12 = 6 × 3 × 13 × 3 13 × − 2 × 6 = 9 − 2 9 8 × 4 − 3 = 9 × 4 8 × − 3 = 3 × 3 × 4 2 × 4 × − 3 = 3 − 2 − − 9 7 ÷ − 36 63 = 9 7 ÷ 36 − 63 = 9 7 × 63 − 36 = 9 × 63 7 × − 36 = 9 × 7 × 9 7 × − 4 × 9 = 9 − 4 Now, substitute these back into the original expression: 9 − 2 − ( 3 − 2 ) + [ 9 − 4 ] = 9 − 2 + 3 2 − 9 4 To add these fractions, we need a common denominator, which is 9: 9 − 2 + 3 × 3 2 × 3 − 9 4 = 9 − 2 + 9 6 − 9 4 = 9 − 2 + 6 − 4 = 9 0 = 0
Evaluating Expression (ii) Now, let's evaluate the second expression: 5 2 × ( − 7 3 ) − 6 1 × 2 3 + 14 1 × 5 2 First, simplify each multiplication: 5 2 × ( − 7 3 ) = 5 × 7 2 × − 3 = 35 − 6 6 1 × 2 3 = 6 × 2 1 × 3 = 12 3 = 4 1 14 1 × 5 2 = 14 × 5 1 × 2 = 70 2 = 35 1 Now, substitute these back into the original expression: 35 − 6 − 4 1 + 35 1 To add these fractions, we need a common denominator. The least common multiple of 35 and 4 is 140: 35 × 4 − 6 × 4 − 4 × 35 1 × 35 + 35 × 4 1 × 4 = 140 − 24 − 140 35 + 140 4 = 140 − 24 − 35 + 4 = 140 − 55 = 28 − 11
Evaluating Expression (iii) Finally, let's evaluate the third expression: 6 5 × ( − 3 2 ) − 5 16 × 4 3 + 2 3 × ( − 9 3 ) First, simplify each multiplication: 6 5 × ( − 3 2 ) = 6 × 3 5 × − 2 = 18 − 10 = 9 − 5 5 16 × 4 3 = 5 × 4 16 × 3 = 5 × 4 4 × 4 × 3 = 5 12 2 3 × ( − 9 3 ) = 2 × 9 3 × − 3 = 18 − 9 = 2 − 1 Now, substitute these back into the original expression: 9 − 5 − 5 12 + ( − 2 1 ) = 9 − 5 − 5 12 − 2 1 To add these fractions, we need a common denominator. The least common multiple of 9, 5, and 2 is 90: 9 × 10 − 5 × 10 − 5 × 18 12 × 18 − 2 × 45 1 × 45 = 90 − 50 − 90 216 − 90 45 = 90 − 50 − 216 − 45 = 90 − 311
Final Answer Therefore, the results are: (i) 0 (ii) -11/28 (iii) -311/90
Examples
Fractions are used in everyday life, such as when cooking, measuring ingredients, or splitting a bill with friends. Understanding how to perform operations with fractions is essential for accurate calculations in these situations. For example, if you are halving a recipe that calls for 3 2 cup of flour, you need to calculate 2 1 × 3 2 to determine the new amount of flour needed. Similarly, when splitting a bill, you might need to calculate fractions of the total amount to determine each person's share.
The evaluations of the expressions are as follows: (i) 0, (ii) -\frac{11}{28}, and (iii) -\frac{311}{90}. Each expression was simplified step-by-step. The process involved proper multiplication, division, and finding common denominators for addition and subtraction of fractions.
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