Problem 18 · AMC 8 Stretch
Core
Fractions, Decimals & Percents
Arithmetic & Operations
specification-without-loss-of-generality
A store takes \(10\%\) off a price, and then takes another \(8\%\) off the new (already reduced) price. What single discount percentage gives the same final price?
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Answer: 17.2 percent
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Hint 1 of 4
The starting price isn't given, so pick an easy one to work with — try \(100\) dollars. (The answer as a percent won't depend on the price.)
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Hint 2 of 4
Take \(10\%\) off \(100\) dollars first. What's the new price?
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Hint 3 of 4
Now take \(8\%\) off that new price — NOT off the original \(100\). Careful!
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Approach: Pick a convenient price (specification without loss of generality)
- Pick a convenient starting price of \(100\) dollars; the final percent off is the same no matter the price.
- After \(10\%\) off: \(100 - 10 = 90\). After \(8\%\) off the \(90\): \(8\%\) of \(90\) is \(7.20\), so \(90 - 7.20 = 82.80\).
- The price dropped from \(100\) to \(82.80\), a drop of \(17.20\) out of \(100\), which is \(17.2\%\).
- Note the two discounts of \(10\%\) and \(8\%\) do NOT add to \(18\%\): they give \(17.2\%\), because the second discount comes off a smaller amount.
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