# Problem of the Day

A teacher administers a test to two classes. She adds 15 points to each student’s score in the first class and 10 points to each student’s score in the second class, which increases the overall average of the two classes by 13.5 points. If the ratio of the number of students in the first class to the number of students in the second class is where and are positive integers with no common factors, what is the value of ?

A. 8

B. 9

C. 10

D. 11

E. 13

### Solution

[latexpage]

There are at least a couple of ways we can approach this problem. One is to calculate the ratio directly from the definition of an average. To do this, let’s assume that there are exactly $p$ people in the first class and $q$ people in the second class. The increase in the total number of points would therefore be $15p+10q$. The average increase would then be $\displaystyle\frac{15p+10q}{p+q}$, which we are told is equal to 13.5:

$\displaystyle \frac{15p+10q}{p+q}=13.5$

$15p+10q=13.5p+13.5q$

$1.5p=3.5q$

$\displaystyle\frac{p}{q}=\frac{3.5}{1.5}$

$\displaystyle \frac{p}{q}=\frac{7}{3}$

So, $p:q=7:3$ and $p+q=10$

We can also find this ratio more quickly by using the concept of weighted averages. Note that the overall average increase is 3.5 away from the second class’s average increase, and 1.5 away from the first class’s average increase. This implies that the ratio of the number of people in each class is $3.5:1.5$, and because the overall average is closer to the average increase of the first class, the first class must have more people. Thus, $p:q=3.5:1.5=7:3$ and $p+q=7+3=10$

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