Problem 23 · 1996 AJHSME
Hard
Algebra & Patterns
system-equations
The manager of a company planned to give a $50 bonus to each employee from the company fund, but the fund was $5 short of what was needed. Instead the manager gave each employee a $45 bonus and kept the remaining $95 in the fund. How much money was in the company fund before any bonuses were paid?
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Answer: E — 995 dollars.
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Hint 1 of 2
The fund never changed — only the plans for it did. Write the SAME fund two different ways: once for the $50 plan (which fell $5 short) and once for the $45 plan (which left $95 over). Two expressions for one quantity means you can set them equal.
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Hint 2 of 2
With n employees: the $50 plan needs 50n but the fund is $5 less, so fund = 50n − 5. The $45 plan uses 45n and leaves $95, so fund = 45n + 95. Equate them to find n.
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Approach: write the same fund two ways and equate
- Let n be the number of employees. The fund equals 50n − 5 (it was $5 short of giving everyone $50) and also 45n + 95 (after $45 each, $95 stayed in). Same fund, so 50n − 5 = 45n + 95.
- That gives 5n = 100, so n = 20, and the fund is 45·20 + 95 = $995.
- Why this transfers: when one quantity is described two ways, set the two descriptions equal — the unknown pops out. No need to find the fund first to get n.
Another way — follow the $5-per-person savings:
- Cutting each bonus from $50 to $45 frees up $5 per employee. That freed-up money is exactly what turns a $5 shortfall into a $95 surplus — a total swing of 5 + 95 = $100.
- So 5 × (employees) = 100, meaning 20 employees, and the fund = 45·20 + 95 = $995.
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