Aisha and Bala each have some cards. Aisha gives Bala 1/3 of her cards. Bala then gives Aisha 1/4 of all the cards he has at that point. After both exchanges, Aisha has 18 cards and Bala has 12 cards. Express the original number of cards Aisha had as an improper fraction of the original number of cards Bala had. Give your answer in its simplest form.
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Worked Solution
Step 1: Use the conservation of total cards. Total = 18 + 12 = 30. So A + B = 30, where A = Aisha's original cards and B = Bala's original cards.
Step 2: Build an expression for Aisha's final count. After Aisha gives 1/3 of her cards: Aisha has 2A/3, Bala has B + A/3. After Bala gives back 1/4 of his cards: Aisha's final count = 2A/3 + (1/4)(B + A/3) = 2A/3 + B/4 + A/12 = 8A/12 + A/12 + 3B/12 = (9A + 3B)/12 = (3A + B)/4.
Step 3: Set up the equation: (3A + B)/4 = 18 → 3A + B = 72.
Step 4: Solve simultaneously: subtract A + B = 30 from 3A + B = 72 → 2A = 42 → A = 21. Then B = 30 − 21 = 9.
Step 5: Write as an improper fraction in simplest form: A/B = 21/9. HCF(21, 9) = 3. So 21/9 = 7/3.
Answer: 7/3
Correct answer: 7/3
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