Initially, Mei Ling has $3$ times as many stamps as her brother Ahmad. Mei Ling then buys $12$ more stamps, while Ahmad sells $8$ of his stamps. The ratio of Mei Ling's stamps to Ahmad's stamps becomes $5 : 1$. How many stamps did Mei Ling have ORIGINALLY?
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Worked Solution
Step 1: Translate '3 times as many' into algebra. Let Ahmad originally have $a$ stamps. Then Mei Ling originally has $3a$ stamps.\n
Step 2: Apply the changes. After buying $12$ more, Mei Ling has $3a + 12$. After selling $8$, Ahmad has $a - 8$.\n
Step 3: Use the after ratio. $(3a + 12) : (a - 8) = 5 : 1$ means $\\dfrac{3a + 12}{a - 8} = 5$, i.e. $3a + 12 = 5(a - 8)$.\n
Step 4: Expand and solve.\n $3a + 12 = 5a - 40$\n $12 + 40 = 5a - 3a$\n $52 = 2a$\n $a = 26$.\n
Step 5: Mei Ling originally $= 3a = 3 \\times 26 = 78$ stamps.\nCheck: After: Mei Ling $= 78 + 12 = 90$, Ahmad $= 26 - 8 = 18$. Ratio $90 : 18 = 5 : 1$. ✓
Answer: 78
Correct answer: 78
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