Difference between revisions of "1993 AIME Problems/Problem 3"

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:(b) those who caught <math>3</math> or more fish averaged <math>6</math> fish each;  
 
:(b) those who caught <math>3</math> or more fish averaged <math>6</math> fish each;  
 
:(c) those who caught <math>12</math> or fewer fish averaged <math>5</math> fish each.  
 
:(c) those who caught <math>12</math> or fewer fish averaged <math>5</math> fish each.  
What was the total number of fish caught during the festival?  
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What was the total number of fish caught during the festival?
  
 
== Solution ==
 
== Solution ==

Revision as of 12:56, 4 November 2018

Problem

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 $n\,$.

$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ \hline \text{number of contestants who caught} \ n \ \text{fish} & 9 & 5 & 7 & 23 & \dots & 5 & 2 & 1 \\ \hline \end{array}$

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?

Solution

Suppose that the number of fish is $x$ and the number of contestants is $y$. The $y-21$ fishers that caught 3 or more fish caught a total of $x - \left(0(9) + 1(5) + 2(7)\right) = x - 19$ fish. Since they averaged 6 fish,

$6 = \frac{x - 19}{y - 21} \Longrightarrow x - 19 = 6y - 126.$

Similarily, those whom caught 12 or fewer fish averaged 5 fish per person, so

$5 = \frac{x - (13(5) + 14(2) + 15(1))}{y - 8} = \frac{x - 108}{y - 8} \Longrightarrow x - 108 = 5y - 40.$

Solving the two equation system, we find that $y = 175$ and $x = \boxed{943}$, the answer.

See also

1993 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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