How many people were at a party if There were 105 handshakes at a party and if each person at the party shook hands with exactly once with every other person. The answer is 15. How do I write the problem?
n(n-1)/2=105, > n^2-n-210=0, n^2-15n+14n-210=0, > n(n-15)+14(n-15)=0, >(n-15)(n+14)=0, so, n=15, =-14 , as -ve is not possible, so n=15, ans will be 15 men
15
Why did the math party get so popular? Because there were 105 handshakes going on! If each person at the party shook hands exactly once with every other person, you can calculate the total number of people at the party. So, how many party animals showed up to shake hands? Well, it's a sly trick question! The answer is 15!
To write the problem, you can follow these steps:
Step 1: Set up the problem
- Specify that there were a total of 105 handshakes at the party.
- State that each person at the party shook hands exactly once with every other person.
Step 2: Define the unknown
- Let "x" represent the number of people at the party.
Step 3: Formulate the equation
- Since each person at the party shakes hands exactly once with every other person, the number of handshakes can be calculated using the formula n(n-1)/2, where "n" represents the number of people.
- Hence, the equation is: 105 = x(x-1)/2.
Step 4: Solve the equation
- Simplify the equation by multiplying both sides by 2 to eliminate the fraction: 210 = x(x-1).
- Rearrange the equation to form a quadratic equation: x^2 - x - 210 = 0.
- Solve the quadratic equation. In this case, the solutions are x = -14 and x = 15.
- Since the number of people cannot be negative, discard the solution x = -14.
- Therefore, the number of people at the party is 15, as stated in the answer.
To write the problem, you can follow these steps:
1. Define the problem:
- Determine the number of people at a party based on the given information about handshakes.
2. Identify the key information:
- Each person at the party shakes hands exactly once with every other person.
- There were 105 handshakes.
3. Write the problem statement:
- "At a party, every person shakes hands exactly once with every other person. There are a total of 105 handshakes. How many people were at the party?"
4. Explain the solution approach:
- To solve this problem, we need to find the number of people at the party. Since each person shakes hands with every other person, we can use a formula to determine the number of handshakes in terms of the number of people.
- The formula to calculate the number of handshakes when there are 'n' people can be calculated as follows: (n * (n-1)) / 2.
5. Solve the problem using the formula:
- Given the number of handshakes as 105, we need to solve the equation: (n * (n-1)) / 2 = 105.
6. Calculate the number of people:
- Simplifying the equation, we get (n^2 - n) / 2 = 105.
- Multiply both sides by 2: n^2 - n = 210.
- Rearranging the equation: n^2 - n - 210 = 0.
- Factoring the quadratic equation: (n - 15)(n + 14) = 0.
7. Determine the number of people:
- The possible values for 'n' can be either n = 15 or n = -14. Since we cannot have a negative number of people, the solution to the problem is n = 15.
Hence, there were 15 people at the party.