Ahmad received \$a$ in his Chinese New Year ang pow. He donated $\dfrac{1}{4}$ of the money to charity. Of what was LEFT, he spent $\dfrac{3}{5}$ on storybooks. He had $\$36$ remaining. Find the value of $a$.
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Worked Solution
Step 1: Set up. Ahmad starts with \$a$. The two operations happen IN ORDER: donate 1/4 first, THEN spend 3/5 of the REMAINDER on books.\n
Step 2: After donating 1/4: amount remaining = $a - \dfrac{1}{4}a = \dfrac{3}{4}a$.\n
Step 3: He spends 3/5 of this remainder on books, so the books cost $\dfrac{3}{5} \times \dfrac{3}{4}a = \dfrac{9a}{20}$.\n
Step 4: The amount LEFT after both steps = $\dfrac{3}{4}a - \dfrac{9a}{20} = \dfrac{15a}{20} - \dfrac{9a}{20} = \dfrac{6a}{20} = \dfrac{3a}{10}$. (Equivalently, after donating he has 3/4 a; he keeps the OTHER 2/5 of that, so 2/5 × 3/4 = 6/20 = 3/10.)\n
Step 5: Set this equal to \$36: $\dfrac{3a}{10} = 36$, so $a = 36 \times \dfrac{10}{3} = 120$.\nCheck: $a = 120$ → charity = $30$, remainder = $90$; books = $\dfrac{3}{5} \times 90 = 54$; left = $90 - 54 = 36$. ✓
Answer: $120
Correct answer: $120
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