SOLUTION: The senior class at a high school had a banquet that cost $150. If there had been one 5 persons less, the cost would have been $1.00 more. how many persons were at the banquet?

Question 1116185: The senior class at a high school had a banquet that cost $150. If there had been one 5 persons less, the cost would have been $1.00 more. how many persons were at the banquet?
Found 3 solutions by greenestamps, ikleyn, MathTherapy:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

If a solution without formal algebra is acceptable, then by far the easiest way to find the answer to the problem is by trial and error.

The number of people is certainly going to be a whole number; and the cost per person is very likely to be a whole number. So look for ways to write 150 as the product of two whole numbers and find two pairs that satisfy the conditions of the problem.

Pairs of whole numbers with a product of 150:

1 150
2 75
3 50
5 30
6 25
10 15

The pairs 5*30 and 6*25 satisfy the conditions of the problem: 30 people cost $5 each; 25 people cost $6 each -- 5 fewer people increases the cost for each person by $1.

Here is an algebraic solution....

Let x be the number of people and y the cost per person. Then
(1)
(2)

Solve (1) for y:

Substitute into (2):

or

Of course reject the negative answer; so the number of people is x = 30.

That was a whole lot of work; and in the end you still had to use some trial and error to find the 30 and 25 to factor the quadratic equation....
Answer by ikleyn(52780)   (Show Source): You can put this solution on YOUR website!
.

            It is a STANDARD ALGEBRA problem, and it should be solved by Algebra method.

The standard and literal translation of the condition to Math is this equation - = 1 dollar. where x is the number of people at the banquet, and each fraction in the left side is the cost (= price) per person. Simplify and solve it 150x - 150(x-5) = x*(x-5) 750 = x^2 - 5x x^2 - 5x - 750 = 0 (x-30)*(x+25) = 0 The only positive root x= 30 works as the solution to the problem. Answer. 30 persons. Check. - = 6 - 5 = 1. ! Correct !

Solved.

----------------

Again:   It is a standard Algebra problem, and a well trained student must know how to solve it algebraically.
            The translation from the condition to Math is the key. The rest is just technique.

============

To see MANY OTHER SIMILAR SOLVED problems, look into the lesson
    - Had they sold . . .
in this site.

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
The senior class at a high school had a banquet that cost $150. If there had been one 5 persons less, the cost would have been $1.00 more. how many persons were at the banquet?

This is a SIMPLE math problem, and should be just that!!
Let the number of people at the banquet be N
Then cost to each would be:
If there were 5 less people, then the count would be N - 5
We then get the following per person cost-equation:
150(N - 5) = 150N + N(N - 5) ------- Multiplying by LCD, N(N - 5)

(N - 30)(N + 25) = 0
N, or number of people at the banquet = OR N = - 25 (ignore)
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