A certain company has $1,000$ to distribute to Bob and nine other employees, with each to receive at least $ 75$. What is the minimum amount of money Bob can receive (in whole dollars) to ensure he receives more than any other employee?
I think bob has to receive $1000-75 \cdot 9$ dollars, but the answer key says $201$. Why?
Somehow we are not giving all the $1000?
You’re not trying to maximize Bob’s earnings. Instead, you’re trying to maximize the other employees’ earnings, given that Bob earns more than all other employees. This is equivalent to asking “what is the maximum earning we can assign to one employee, say Alice, while still ensuring that Bob earns more than Alice, and all other employees earn at most what Alice earns”? Well, if we suppose that Bob earns $x$, then the most Alice could earn is $x-1$. In order to ensure that Alice (and therefore Bob, as wel) earns the most she could possibly earn under the given assumptions, we then require that each other employee only earn $75$. Does this make sense? Can you use this information to formulate an equation in terms of Bob’s earnings, $x$?
You calculated the maximum amount Bob could receive.
Since $1000 \div 10 = 100$, we can minimize the amount Bob receives by giving as many employees as possible $\$100$ each. We cannot give all nine of them $\$100$ each since then Bob would also receive $\$100$, which is not more than his peers. However, if we gave exactly one of those employees $\$99$ and the other eight $\$100$ each, then Bob would receive $$\$1000 - 8 \cdot \$100 - \$99 = \$101$$ which is more than any other employee. Therefore, there appears to be a typographical error in the stated answer.
