Each person paid a DIFFERENT Integral amount towards the bill. Let’s label the five friends A through E where the amount each paid:
A < B < C < D < E
Since the average is $64, the SUM paid by all 5 friends is = $320
A < B < C < D < E = $320
The person who paid the 2nd highest (D) paid = $124
And the average of the 2 friends who paid the least is $23 ———> meaning that A and B together paid $46
To maximize the amount paid by E, the assumed friend who paid the most, we want to keep A and B as close together as we possible can ———> because C has to pay at the very least $1 dollar more than B (because they all pay different integers amounts)
For instance, if we had A pay 1 dollar ——-> B would have to pay 45 dollars ——-> and C would have to pay at least 46 dollars
If instead, we apportioned 10 to A ——-> B would have to pay 36 ———> and C would have to pay at least 37.
As the amounts A and B pay come closer to being equal, we MINIMIZE the amount that we can apportion to C.
In other words, If we put A and B’s payments as close as possible, this will result in the smallest value we can apportion to C.
We can apportion 22 to A and 24 to B —-> this is the closest we can make the values, have them be Different Integral Values, and add up to $46.
And since C must pay a different integral amount, we can minimize C’s contribution by making it just $1 more than B’s ——>$25
(A = 22) < (B = 24) < (C= 25) < (D = 124) < (E = ?) ——-> amounts must SUM to $320
By apportioning the amounts above, we have fulfilled the conditions of the question and Maximized the highest possible payment that could have been made by friend E
MAX E = (320) - (22 + 24 + 25 + 124)
MAX E = (320) - (195)
MAX -E = $125
(A)
The highest amount a friend can pay is $125
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