42 students took a test in school where the passing score was 70. the mean score of the 42 students taking the exam was 75.5 while the mean score of the students who passed the test was 81, and the mean score of the students who failed was 60. How many students passed the test?
To find out how many students passed the test, we need to determine the number of students who scored higher than or equal to the passing score of 70.
First, we should calculate the number of students who failed the test. We know that the mean score for these students is 60. However, we don't have the exact number of students who failed. To find this, we need to find the number of students who passed the test.
Since the mean score of the entire group is given as 75.5, and the mean score of those who failed is 60, we can calculate the number of students who passed by subtracting the number of students who failed from the total number of students.
Let's represent the number of students who passed the test as "P" and the number of students who failed as "F."
We can set up the following equation:
P + F = 42 (since the total number of students is 42)
Now, let's calculate the number of students who failed:
Mean score of failed students = 60
Sum of scores from failed students = Mean score of failed students × Number of failed students
= 60 × F
The sum of scores from passed students can be calculated similarly:
Mean score of passed students = 81
Sum of scores from passed students = Mean score of passed students × Number of passed students
= 81 × P
Finally, we can set up another equation using the given mean score of the entire group:
[(60 × F) + (81 × P)] / 42 = 75.5
Now, we can solve these two equations to find the values of P and F.
P + F = 42 (equation 1)
[(60 × F) + (81 × P)] / 42 = 75.5 (equation 2)
Solving these two equations, we will find the value of P, which represents the number of students who passed the test.