Before hosting their annual Chess Tournament and Spelling Bee, a school received 7 boxes of honorary medals: one medal for every participant. After Chess Tournament, two boxes were empty and the rest were still closed. After the Spelling Bee, which had twice as many participants, there were 72 medals left. How many people competed in Chess Tournament?
the Spelling Bee used twice as many boxes as the Chess Tournament
that left one unopened box remaining ... 72 medals per box
two boxes for the Chess Tournament ... (2 * 72) medals and competitors
Let's break down the information given step-by-step:
1. Initially, the school received 7 boxes of honorary medals, with one medal for every participant. This means that there were a total of 7 participants.
2. After the Chess Tournament, two boxes were empty, leaving 7 - 2 = 5 boxes of medals unopened.
3. After the Spelling Bee, which had twice as many participants, there were 72 medals left. This implies that there were a total of 72 + 5 = 77 medals issued.
4. Since each participant received one medal, the number of participants in the Spelling Bee was 77.
5. The Spelling Bee had twice as many participants as the Chess Tournament, so the number of participants in the Chess Tournament was 77 divided by 2, which is 77 / 2 = 38.5.
Since the number of participants must be a whole number, it is not possible for there to be 38.5 participants. Thus, something seems to be incorrect with the given information. Please double-check the details provided.
To determine the number of people who competed in the Chess Tournament, we need to work backwards from the information given.
Let's break down the given information step by step:
1. The school received 7 boxes of honorary medals - one medal for every participant.
2. After the Chess Tournament, two boxes were empty, which means there were 2 * (the number of medals in each box) fewer medals remaining.
3. The rest of the boxes were still closed, meaning the number of remaining medals after the Chess Tournament is equal to the number of medals in all the closed boxes.
4. After the Spelling Bee, which had twice as many participants as the Chess Tournament, there were 72 medals left.
Let's define the variables:
Let x be the number of medals in each box.
Let y be the number of participants in the Chess Tournament.
Based on the given information, we can set up the following equations:
Equation 1: 7x = y (since there was one medal for every participant)
Equation 2: (7 - 2) * x = (x * (y - x)) - 72
Now, let's solve the equations to find the value of y, which represents the number of participants in the Chess Tournament:
Substituting the value of y from Equation 1 into Equation 2:
(5x) = (x * (7x - x)) - 72
5x = 7x^2 - x^2 - 72
0 = 6x^2 - 5x - 72
This is a quadratic equation. To solve it, we can factorize or use the quadratic formula.
Factoring the equation gives us:
(2x - 9)(3x + 8) = 0
Using the zero-product property, we have two possibilities:
1. 2x - 9 = 0, which means x = 4.5 (This cannot be the solution since the number of medals cannot be a fraction.)
2. 3x + 8 = 0, which means x = -8/3 (Again, this is not a valid solution.)
Therefore, there is no valid solution for x, which means we made an error in our calculations or assumptions. Double-checking the information and equations might help identify the error.