Question 18 of 58beginner🔧 ApplyShort Answer2 marks

Show that: (12+34)×83=12×83+34×83\left(\frac{1}{2} + \frac{3}{4}\right) \times \frac{8}{3} = \frac{1}{2} \times \frac{8}{3} + \frac{3}{4} \times \frac{8}{3}.

Correct Answer

Evaluating LHS: (12+34)×83\left(\frac{1}{2} + \frac{3}{4}\right) \times \frac{8}{3} First, add the rational numbers inside the brackets: 12+34=24+34=54\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}. Next, multiply the sum by 83\frac{8}{3}: 54×83=4012=103\frac{5}{4} \times \frac{8}{3} = \frac{40}{12} = \frac{10}{3}.

Evaluating RHS: 12×83+34×83\frac{1}{2} \times \frac{8}{3} + \frac{3}{4} \times \frac{8}{3} Multiply the terms first: 12×83=86=43\frac{1}{2} \times \frac{8}{3} = \frac{8}{6} = \frac{4}{3} and 34×83=2412=2\frac{3}{4} \times \frac{8}{3} = \frac{24}{12} = 2. Adding these products: 43+2=43+63=103\frac{4}{3} + 2 = \frac{4}{3} + \frac{6}{3} = \frac{10}{3}.

Since LHS = RHS, the equation is verified. This demonstrates the law of distributivity for rational numbers.

Exercise: Exercise Set 3.3 | Q: 6 | (Page 10)
For More Understanding

Explanation

The question requires verifying the distributive property of multiplication over addition for rational numbers, which is stated in the text as p(q+r) = pq+pr. By calculating both the LHS and RHS separately and showing they equal 10/3, the property is confirmed.

Solution Steps

  1. Step 1: Evaluate LHS by adding fractions inside brackets: 1/2 + 3/4 = 2/4 + 3/4 = 5/4

  2. Step 2: Multiply the sum by 8/3: 5/4×84 \times 8/3 = 40/12 = 10/3

  3. Step 3: Evaluate RHS by multiplying first terms: 1/2×82 \times 8/3 = 8/6 = 4/3

  4. Step 4: Multiply next terms: 3/4×84 \times 8/3 = 24/12 = 2

  5. Step 5: Add the products: 4/3 + 2 = 4/3 + 6/3 = 10/3

  6. Step 6: Conclude LHS = RHS, verifying the law of distributivity

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