Show that: (12 + 34) × 83 = 12 × 83 + 34 × 83.
Answer: LHS = 54 × 83 = 103 and RHS = 43 + 2 = 103, so LHS = RHS (the distributive law).
Step-by-step solution
Idea: Work out each side separately. This is the distributive law (p + q)r = pr + qr for rational numbers.
- LHS: 12 + 34 = 24 + 34 = 54.½ mark
- 54 × 83 = 4012 = 103.½ mark
- RHS: 12 × 83 = 86 = 43 and 34 × 83 = 2412 = 2. Sum = 43 + 63 = 103.½ mark
- LHS = RHS = 103. Hence shown.½ mark
Both sides equal 10/3, so (1/2 + 3/4) × 8/3 = 1/2 × 8/3 + 3/4 × 8/3.
Check: Decimals: 1.25 × 2.667 ≈ 3.333 and 1.333 + 2 = 3.333 ✓.
Answer to write in the exam
LHS = (12 + 34) × 83 = 54 × 83 = 103
RHS = 12 × 83 + 34 × 83 = 43 + 2 = 103
∴ LHS = RHS. Hence shown.
Common mistakes that cost marks
- Adding 12 + 34 as 46. The correct sum is 54.
- Working on only one side and then writing ‘hence proved’. Both sides must be shown equal.
- Writing 34 × 83 = 247 by adding the denominators.
How this can come in the exam
MCQ (1 mark)
25 × 79 + 35 × 79 =
- 79
- 745
- 1445
- 1
Show answer
(A) 79
Distributive law: (25 + 35) × 79 = 1 × 79 = 79.
Short answer (2 marks)
Verify (23 − 16) × 65 = 23 × 65 − 16 × 65.
Show answer
LHS: 12 × 65 = 35 (1 mark). RHS: 45 − 15 = 35. Equal ✓ (1 mark).Try one yourself
Show that (13 + 16) × 45 = 13 × 45 + 16 × 45.
Show answer
LHS = 12 × 45 = 25. RHS = 415 + 215 = 615 = 25. Equal.
More questions like this
- Simplify the following using the distributive property: 79(67 − 34).
- Find the rational number x such that: 56(x + 35) = 56x + 12.
- Try and represent 85 and −74 on a number line.
- The absolute value of a rational number x, written as |x|, represents its distance from 0 on the number line. |53| = 53, |−53| is also equal to 53 and |0| = 0.
- Try to explain why the average of two rational numbers a and b, which equals (a + b)2, is always a rational number between a and b.