The $75 cost for a party was to be shared equally by all of those attending. Since five more people attended than was expected, the price per person dropped by 50 cents. How many people attended the party?
original cost per person = 75/x
new cost per person = 75/(x+5)
75/x - 75(x+5) = 1/2
multiply each term by x(x+5)
75(x+5) - 75x = (1/2)x(x+5)
times 2
150x + 750 - 150x = x^2 + 5x
x^2 + 5x - 750 = 0
(x-25)(x+30) = 0
x = 25 or x = -30 , but clearly x > 0
x=25
originally 25 had planned to attend, but 30 actually did.
check:
cost with 25 = 75/25 = $3.00
cost with 30 = 75/30 = $2.50
it cost 50 cents less.
Answer is correct.
To determine the number of people who attended the party, we need to set up an equation based on the given information.
Let's assume the expected number of people who were initially supposed to attend the party was x. Therefore, the cost per person would be 75/x dollars.
However, since five more people attended than expected, the new number of people who attended the party would be x + 5. This means that the new cost per person is now (75/x) - 0.50 dollars.
According to the given information, the new cost per person decreased by 50 cents, so we can set up the equation:
(75/x) - 0.50 = 75/x
To solve this equation, we'll multiply both sides by x to eliminate the denominators:
75 - 0.50x = 75
Next, we'll isolate the variable on one side of the equation:
0.50x = 0
By subtracting 75 from both sides, we eliminated the constant terms. Now, we're left with:
0.50x = -75
To solve for x, we'll divide both sides by 0.50:
x = -75 / 0.50
Simplifying this equation gives:
x = -150
However, it does not make sense to have a negative number of people attending the party. Hence, we realize that there is no solution for this equation.
Therefore, there is something incorrect or missing in the given information. Please double-check the problem statement or provide additional details.