Mary sold 15 widgets last week. The median sale price of all widgets was $130$ and the average sale price of all widgets was $150. What's the least possible price of the most expensive widget?
The total sum of all values is $150*15$ $=$ $2250$. With the median being $130$, we assume all values to the left of the median are $130$ to ensure we maximize the sum of all values left of the median. This implies a sum of $1040$ for all $8$ values ($130*8$). Then we know the sum of the 7 values greater than $130$ is $2250 - 1040$. So the average price is ($2250 - 1040$)/$7$). I know this average is the lowest possible price of the most expensive widget, but I do not find this intuitive. Could someone explain why the lowest possible price of the most expensive widget is the average of the numbers assuming all to left of median are equal to the median?
You sold $15$ widgets for $2250$ and you know you sold $8$ of them for $130$, receiving $1040$. That means the remaining $7$ brought in $2250-1040=1210$. As you say, they could all have sold for $\frac {1210}7$. If any of them sold for less than that, at least one of the others would have to have sold for more to keep the average what it needs to be. If you want to minimize the maximum price when you know the average, you do that by selling them all at the average price.