Prove that the following rational numbers are equal:
- (i) 23 and 46
- (ii) 54 and 108
- (iii) −35 and −610
- (iv) 93 and 3
Answer: Use the rule ab = cd if ad = bc. (i) 2 × 6 = 12 = 3 × 4 (ii) 5 × 8 = 40 = 4 × 10 (iii) (−3) × 10 = −30 = 5 × (−6) (iv) 9 × 1 = 9 = 3 × 3. Each pair is equal.
Step-by-step solution
Idea: Two rational numbers ab and cd are equal when the cross products are equal: ad = bc. An integer such as 3 is written as 31 first.
(i) 23 and 46
- a = 2, b = 3, c = 4, d = 6. ad = 2 × 6 = 12 and bc = 3 × 4 = 12. Since ad = bc, 23 = 46.½ mark
2 × 6 = 3 × 4 = 12, so 23 = 46
(ii) 54 and 108
- ad = 5 × 8 = 40 and bc = 4 × 10 = 40. Equal, so 54 = 108.½ mark
5 × 8 = 4 × 10 = 40, so 54 = 108
(iii) −35 and −610
- Write them as −35 and −610. ad = (−3) × 10 = −30 and bc = 5 × (−6) = −30. Equal, so −35 = −610.½ mark
(−3) × 10 = 5 × (−6) = −30, so −35 = −610
(iv) 93 and 3
- Write 3 = 31. ad = 9 × 1 = 9 and bc = 3 × 3 = 9. Equal, so 93 = 3.½ mark
9 × 1 = 3 × 3 = 9, so 93 = 3
In every pair the cross products are equal (12 = 12, 40 = 40, −30 = −30, 9 = 9), so each pair of rational numbers is equal.
Check: Simplify instead: 46 = 23 (÷2), 108 = 54 (÷2), 610 = 35 (÷2), 93 = 3 (÷3) ✓.
Answer to write in the exam
(i)
2 × 6 = 12, 3 × 4 = 12
∴ 23 = 46
(ii)
5 × 8 = 40, 4 × 10 = 40
∴ 54 = 108
(iii)
(−3) × 10 = −30, 5 × (−6) = −30
∴ −35 = −610
(iv)
3 = 31
9 × 1 = 9, 3 × 3 = 9
∴ 93 = 3
Common mistakes that cost marks
- Multiplying the wrong pairs (numerator × numerator, 2 × 4 and 3 × 6). Cross-multiply: a × d and b × c.
- In (iii), dropping the minus signs and comparing 3 × 10 with 5 × 6. That works here, but keep the signs: a positive and a negative number are never equal.
- In (iv), not writing 3 as 31 first, so there is no ‘d‘ to multiply by.
How this can come in the exam
MCQ (1 mark)
Which rational number is equal to −49?
- 8−18
- 818
- −94
- 49
Show answer
(A) 8−18
(−4) × (−18) = 72 and 9 × 8 = 72, so −49 = 8−18.
Short answer (2 marks)
Find k if k12 = 1520.
Show answer
Cross products are equal: 20k = 12 × 15 = 180 (1 mark), so k = 9 (1 mark).Try one yourself
Show that −721 and 2−6 are equal.
Show answer
(−7) × (−6) = 42 and 21 × 2 = 42; equal cross products, so the numbers are equal (both are −13).