1993 AIME Problems/Problem 3
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[hide]Problem
The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught fish for various values of .
In the newspaper story covering the event, it was reported that
- (a) the winner caught fish;
- (b) those who caught or more fish averaged fish each;
- (c) those who caught or fewer fish averaged fish each.
What was the total number of fish caught during the festival?
Solution 1
Suppose that the number of fish is and the number of contestants is . The fishers that caught or more fish caught a total of fish. Since they averaged fish,
Similarily, those who caught or fewer fish averaged fish per person, so
Solving the two equation system, we find that and , the answer.
Solution 2
Let be the total number of fish caught by the contestants who didn't catch , or fish and let be the number of contestants who didn't catch , or fish. From , we know that . From we have . Using these two equations gets us . Plug this back into the equation to get . Thus, the total number of fish caught is - Heavytoothpaste
See also
1993 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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