At a party, 55 handshakes took place as everyone shook hands with everyone else. How many people were at the party?
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Well, it's a party, so I'm guessing there were some party animals in attendance! Let's see if we can solve this puzzle.
Now, if we think about it, each person shakes hands with every other person exactly once. Let's assume there were "x" people at the party.
The first person will shake hands with "x-1" people, the second person will shake hands with "x-2" people (since they already shook hands with the first person), and so on, until the last person shakes hands with 1 person.
So, we need to find the sum of all the numbers from "x-1" to 1. And we know that the sum of numbers from 1 to "n" is given by the formula: (n * (n+1)) / 2.
Therefore, applying this formula to our case, we have (x-1 * x) / 2 = 55.
Solving the equation, we get x^2 - x - 110 = 0.
Using the quadratic formula, we find that x ≈ 11.94. Since we can't have a fraction of a person, we can round it up to the nearest whole number.
So, there were approximately 12 people at the party. We can start the conga line now! 🎉
To find the number of people at the party, we can use a simple formula based on the number of handshakes.
Let's assume there were "n" people at the party. Each person shakes hands with "n-1" other people as they don't shake hands with themselves. So, if we multiply the number of people by the number of handshakes per person, we should get the total number of handshakes.
In this case, we have 55 handshakes, so we can set up the equation:
n * (n-1) = 55
Now, we need to find two numbers that multiply to 55. The pairs of numbers that multiply to 55 are (1, 55), (5, 11), and (11,5).
So, we can try to substitute these pairs into the equation:
1 * 54 = 54
5 * 10 = 50
11 * 4 = 44
None of these pairs equals 55. Let's try the next pair:
2 * 54 = 108
This also doesn't match our equation. Let's try the next pair:
3 * 52 = 156
Again, this doesn't match our equation. Let's try the next pair:
4 * 51 = 204
This also doesn't match our equation. Let's try the next pair:
6 * 49 = 294
This doesn't match our equation either. Let's try the next pair:
7 * 48 = 336
This is also not equal to 55. Let's try the next pair:
8 * 47 = 376
This is still not equal to 55. Let's try the next pair:
9 * 46 = 414
This is also not equal to 55. Let's try the next pair:
10 * 45 = 450
This is not equal to 55. Let's try the next pair:
12 * 43 = 516
This is not equal to 55. Let's try the next pair:
13 * 42 = 546
This is equal to 55.
So, if we consider that there were 13 people at the party, the equation holds true:
13 * (13-1) = 13 * 12 = 156
Therefore, there were 13 people at the party.
To solve this problem, we can use a combination of logic and mathematics. Let's break it down step by step:
1. Assume there are "n" people at the party.
2. Each person at the party needs to shake hands with every other person. So, the first person shakes hands with (n - 1) people, the second person shakes hands with (n - 1) people (excluding themselves and the person they already shook hands with), and so on.
3. We need to count the total number of handshakes that take place. This can be calculated by summing up the number of handshakes each person is involved in.
4. The first person shakes hands with (n - 1) people. The second person shakes hands with (n - 2) people (excluding themselves and the first person). The third person shakes hands with (n - 3) people (excluding themselves, the first person, and the second person), and so on.
5. The total number of handshakes can be calculated by summing up these individual handshakes for each person. So the total number of handshakes is (n - 1) + (n - 2) + (n - 3) + ... + 3 + 2 + 1.
To simplify the calculation, we can use a formula to find the sum of the first "n" numbers, known as the arithmetic series formula:
Sum = (n/2) * (first term + last term)
In this case, the first term is 1, and the last term is (n - 1). So the formula becomes:
Sum = (n/2) * [1 + (n - 1)]
Now, we know that the total number of handshakes is 55, so we can set up an equation:
55 = (n/2) * [1 + (n - 1)]
Simplifying further:
55 = (n/2) * n
Multiplying both sides by 2:
110 = n^2
Taking the square root:
n = sqrt(110)
Calculating the square root, we find:
n ≈ 10.49
Since we cannot have a fraction of a person, we need to round this value up to the nearest whole number. Therefore, there were approximately 11 people at the party.