Question

The table below displays some of the results of last summer’s Frostbite Falls Fishing Festival, showing how many contestants caught $n$ fish for various values of $n.$

$n$	$0$	$1$	$2$	$3$	$\ldots$
number of contestants who caught $n$ fish	$9$	$5$	$7$	$23$	$\ldots$

$n$	$13$	$14$	$15$
number of contestants who caught $n$ fish	$5$	$2$	$1$

In the newspaper story covering the event, it was reported that

(a) the winner caught $15$ fish;
(b) those who caught $3$ or more fish averaged $6$ fish each;
(c) those who caught $12$ or fewer fish averaged $5$ fish each.

What was the total number of fish caught during the festival?

Answer

Answer: 943

Concepts:data and graph interpretation system of equations weighted mean

Difficulty rating: 2070

Small Hint:

Let $N$ be the total number of contestants and $T$ the total number of fish

Big Hint:

Use the two averages by first subtracting the known groups with fewer than $3$ fish and with more than $12$ fish

Solution:

The $9+5+7=21$ contestants below $3$ fish caught $5+14=19$ fish. Thus condition (b) gives $T-19=6(N-21),$ or $T=6N-107.$ The $5+2+1=8$ contestants above $12$ fish caught $65+28+15=108$ fish, so condition (c) gives $T-108=5(N-8),$ or $T=5N+68.$ Hence $N=175$ and $T=943.$

1993 AIME Problem 3

Answer: 943

Difficulty rating: 2070

Small Hint:

Big Hint:

Solution:

Problem 3 in Other Years