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At the county track and field championships, Jackson High athletes placed in 24 individual events scoring a total of 56 points. A first place finish earn 5 points. A second-place finish earn 3 points and a third-place finish earns 1 point. The team had the same amount of third place finishers as first and second-place finishers combined. How many athletes finished in each place?

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Answer:

Step-by-step explanation:To solve this problem, let's break it down step by step:

Step 1: Let's assume the number of first-place finishers is "x," the number of second-place finishers is "y," and the number of third-place finishers is "z."

Step 2: We are given that a first-place finish earns 5 points, a second-place finish earns 3 points, and a third-place finish earns 1 point. We know that there were 24 individual events in total and the team scored a total of 56 points.

Step 3: So, we can write the following equation based on the given information:

5x + 3y + 1z = 56

Step 4: We are also given that the number of third-place finishers is equal to the sum of the first-place and second-place finishers. Mathematically, this can be written as:

z = x + y

Step 5: Now, we have two equations:

5x + 3y + z = 56 (equation 1)

z = x + y (equation 2)

Step 6: To solve these equations, we can substitute equation 2 into equation 1:

5x + 3y + (x + y) = 56

Step 7: Simplifying the equation:

6x + 4y = 56

Step 8: We can divide the equation by 2 to make it simpler:

3x + 2y = 28

Step 9: Now, let's try to find possible values for "x" and "y" that satisfy this equation. One solution is:

x = 4

y = 5

Step 10: Now, substituting these values back into equation 2 to find "z":

z = 4 + 5

z = 9

Step 11: Therefore, there were 4 athletes who finished in first place, 5 athletes who finished in second place, and 9 athletes who finished in third place.

In summary, there were 4 athletes who finished in first place, 5 athletes who finished in second place, and 9 athletes who finished in third place at the county track and field championships.

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