Problem 20 · 2015 Math Kangaroo
Stretch
Algebra & Patterns
work-backward
Sarah bought three books. For the first book she paid half of her money plus 1 Euro more. For the second book she paid again half of her left-over money plus 2 Euros more. For the third book she paid again half of her left-over money plus 3 Euros more. After which she had spent all of her money. How much money did she have to begin with?
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Answer: C — 34 €
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Hint 1 of 2
Work backwards from the end, when she has spent everything.
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Hint 2 of 2
Just before the last book she had some amount L; she paid half of it plus 3 and was left with nothing, so half of L equals 3.
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Approach: rewind the spending step by step, starting from when she has nothing left
- Work backwards from the end. At the third book she paid half her money plus 3 € and was left with nothing, so that half must have been the 3 €, meaning she had 6 € before book three.
- Before book two she paid half plus 2 € and was left with that 6 €; so after taking the extra 2 € back the 6 € is the other half, making 8 € the half, so she had 16 € before book two.
- Before book one she paid half plus 1 € and was left with that 16 €; adding the 1 € back, 17 € is one half, so she started with 34 €.
Forward check
Start 34: pay 17+1=18, left 16; pay 8+2=10, left 6; pay 3+3=6, left 0. ✓
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