Test Average Word Problem
Date: 02/20/2004 at 10:22:51 From: Rich Subject: Difficult Word Problem In a certain class there are more than 20 and fewer than 40 students. On a recent test the average passing mark was 75. The average failing mark was 48 and the class average was 66. The teacher then raised every grade 5 points. As a result the average passing mark became 77.5 and the average failing mark became 45. If 65 is the established minimum for passing, how many students had their grades changed from failing to passing?
Date: 02/20/2004 at 11:22:59 From: Doctor Greenie Subject: Re: Difficult Word Problem Hi, Rich -- Thanks for sending the interesting problem. It was unlike any I had seen before; and in solving it I got to use one of my favorite mathematical "tricks". Here's what we have.... Before changing the grades: average passing grade: 75 class average: 66 average failing grade: 48 After changing the grades: average passing grade: 77.5 class average: 71 average failing grade: 45 All of these numbers were given specifically in the problem except for the class average after the grades were raised; because every grade was raised by 5 points, the class average was raised by 5 points. What we do with these numbers is determine the ratio of passing grades to failing grades before and after. To do this, we use my pet technique for solving "mixture" problems. For the case before the grades were raised, the average passing grade is 9 points above the class average, and the average failing grade is 18 points below the class average. So what we have is a "mixture" problem, where a certain number of students with average grades 18 points below average and another number of students with average grades 9 points above average combine to produce that average. With the average for one group twice as far from the overall average as the average for the other group, the ratio of students in the two groups must be 2:1. And since the overall average is closer to the average of the students who passed, the larger group of students is the group who passed. So now we know that we have "2x" students who passed and "x" students who failed. This means the total number of students in the class is 3x, so the total number of students is divisible by 3. For the case after the grades have been raised, we perform the same analysis. The passing average and failing average are, respectively, 6.5 points above and 26 points below the overall average. These two numbers are in the ratio 1:4; this means that now the ratio of students who passed to students who failed is 4:1; and this means the number of students in the class is divisible by 5. So now we know that the number of students in the class is divisible by both 3 and 5, and has to be between 20 and 40. Can you determine how many students there are from that information? Once you know the total number of students, can you use the pass:fail ratios to determine how many students passed and failed with each of the two scorings? That will let you answer the actual question posed in the problem. I hope this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/
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