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The current market price of 175 honey bees is more than the price of 125 bumblebees, but less than the price of 126 bumblebees. Is it possible to buy 3 honey bees and 1 bumblebee for altogether at most 100 cents?

(Note: all prices are integers, and all prices are measured in cents).

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Assume that a bumble bee costs $x$ cents and that a honey bee costs $y$ cents, where $x$ and $y$ are integers. Then the problem statement gives

  • $125x<175y$, which yields $5x<7y$ and hence $5x+1\le7y$
  • $175y<126x$, which yields $25y<18x$ and hence $25y+1\le18x$

We multiply the first inequality $5x+1\le7y$ by $18$ and the second inequality $25y+1\le18x$ by $5$, and combine them into
$$ 5(25y+1)\le90x\le18(7y-1).$$ This implies $125y+5\le126y-18$, and hence $y\ge23$.
Furthermore, $18x\ge25y+1\ge576$ implies $x\ge32$.
Hence the price of 3 honey bees and 1 bumblebee is $3y+x\ge3\cdot23+32=101$, and lies strictly above the price limit of $100$ cents.

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Short answer:

No

Longer:

Lowest price for the relation to be true is:
honey bee 23 cents
bumblebee 32 cents
3*23 + 32 = 101 cents

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  • $\begingroup$ Seems like I got downvoted. O.o I'm new, but I still think I answered first and put the "spoiler" tags. Anyone knows why? Not really worried, just surprised... $\endgroup$ – Moris Zucca Feb 9 '15 at 13:34

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