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
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.