Two food carts calculate their weekly profits for 3 weeks. The medians and the largest

Question

profit values are the same for both carts. Cart A's mean is $27 more than Cart B's mean.
What is the difference in dollars between the lowest profits of Cart A and Cart B?

Answer 1

Let the weekly profits for Cart A be $a_1 \leq a_2 \leq a_3$ and for Cart B be $b_1 \leq b_2 \leq b_3$. We are told that the medians and largest profit values are the same for both carts, so $b_2 = b_3 = a_2 = a_3$. We are also told that the mean for Cart A is $27 more than Cart B, so $\frac{a_1+a_2+a_3}{3} = \frac{b_1+b_2+b_3}{3} + 27$. Thus, $a_1+a_2+a_3 = b_1+b_2+b_3 + 81$. Since $b_1 \leq b_2 \leq b_3$, we know that $3b_1 \leq b_1+b_2+b_3 \leq 3b_3$. Therefore, $3b_1 \leq a_1+a_2+a_3 \leq 3b_3$, and combining this with the equation $a_1+a_2+a_3 = b_1+b_2+b_3 + 81$, we conclude that $3b_1 = a_1+a_2+a_3 = b_1+b_2+b_3 + 81 = 3b_3 + 81$, so $b_1 = b_3 + 27$. Since $b_2 = b_3$, it follows that $b_1 = b_2 + 27$, so the lowest profits differ by $\boxed{27}$ dollars.