paid before February 12th the social cost $2.00. If you paid at the door the social cost $5.00. Altogether, 110 students attended the social collecting a total of $454. How many students paid before February 12th and how many paid at the door?
e = paid early
d = paid at door
e + d = 110
2e + 5d = 454
3d = 234
d = 78
e = 32
check:
2*32 + 5*78 = 64 + 390 = 454
To solve this problem, let's assume that the number of students who paid before February 12th is represented by the variable 'x' and the number of students who paid at the door is represented by the variable 'y'.
According to the given information:
1) The cost for students who paid before February 12th is $2.00, so the total amount collected from those students would be 2x.
2) The cost for students who paid at the door is $5.00, so the total amount collected from those students would be 5y.
3) Altogether, 110 students attended the social, so we have the equation x + y = 110.
4) The total collected from all the students is $454, so we have the equation 2x + 5y = 454.
Now we have a system of two equations with two variables:
x + y = 110
2x + 5y = 454
We can solve this system of equations using various methods, such as substitution or elimination. Let's use the substitution method in this case.
1) Solve the first equation for one variable in terms of the other. We can do this by isolating 'x':
x = 110 - y
2) Substitute the value of 'x' into the second equation:
2(110 - y) + 5y = 454
220 - 2y + 5y = 454
3y = 234
y = 78
We have found that y, the number of students who paid at the door, is 78.
3) Substitute the value of 'y' back into the first equation to find 'x':
x + 78 = 110
x = 110 - 78
x = 32
We have found that x, the number of students who paid before February 12th, is 32.
Therefore, 32 students paid before February 12th, and 78 students paid at the door.