In Professor Tuffguy's mathematics class, 36 students took the final exam. If the average passing grade was 78, the average failing grade was 60, and the class average was 71, how many of these 36 student passed the final?
If x passed, then the rest (36-x) failed. So, adding up all the grades, we have
78x + 60(36-x) = 71*36
This is just a mixture problem in other guise.
To solve this problem, we need to use the concept of averages and some basic arithmetic. Here's how you can solve it step by step:
1. Let's assume the number of students who passed the final exam is 'x' and the number of students who failed is 'y'. Since there are 36 students in total, we can express this as:
x + y = 36 -- Equation 1 (the total number of students)
2. We are given that the average passing grade is 78, so the sum of passing grades will be equal to the number of passing students multiplied by the average passing grade:
78x -- Sum of passing grades
3. Similarly, given that the average failing grade is 60:
60y -- Sum of failing grades
4. The class average is given as 71, so the total sum of all grades will be equal to the class average multiplied by the total number of students:
71 * 36 -- Total sum of all grades
5. Since the sum of all grades is equal to the sum of passing grades plus the sum of failing grades, we can write:
78x + 60y = 71 * 36 -- Equation 2 (the sum of all grades)
6. Now we have two equations (Equation 1 and Equation 2) and two variables (x and y) representing the number of passing and failing students. We can solve this system of equations to find the value of 'x', which represents the number of students who passed the final.
Using these steps, we can solve the equations.