# Bumblebees and Honey bees

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).

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.