Problem 19 · 2014 Math Kangaroo
Medium
Fractions, Decimals & Percents
substitutionfraction-to-decimal
The triplets Meike, Monika and Zita each want to buy equally expensive hats. However, Meike’s savings were 13, Monika’s 14, and Zita’s 15 smaller than the price of a hat. After the hats were reduced by €9·40, the triplets put their savings together and each bought a hat, with not a single cent left over. How much had a hat cost originally?
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Answer: D — €36
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Hint 1 of 2
Write each girl's savings as a fraction of the original price P.
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Hint 2 of 2
Their combined savings exactly buy three reduced hats; solve for P.
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Approach: sum the three savings and set equal to three reduced prices
- Savings: Meike 2P/3, Monika 3P/4, Zita 4P/5. Together they buy 3 hats at (P − 9.40) each.
- 2P/3 + 3P/4 + 4P/5 = 3(P − 9.40) → (133/60)P = 3P − 28.20.
- Solving gives P = 36 €.
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