## Seventh Student Score

Six students in a small class took an exam. The average of their grades was $75$. The seventh student in the class was ill that day and took the exam another day. When this student’s score was included, the class average rose to $78$. What was the seventh student’s score?
Source: NCTM Mathematics Teacher, February 2006

SOLUTION
Imagine without loss of generality that the six students’ scores are near the average of $75$, that is, some scores are below $75$ and some above $75$ but they don’t deviate too much from it. Let’s further assume that the six scores area as follows with the amount above or below $75$ shown in parentheses
$76(+1)$
$74(-1)$
$77(+2)$
$74(-1)$
$75(+0)$
$74(-1)$
By a property of average, the amounts above and below the average sum to zero
$1-1+2-1+0-1=0$
When the seventh student took the exam, the average shoots up to $78$. The seven students’ scores are listed again below but now compared to the new average
$76(-2)$
$74(-4)$
$77(-1)$
$74(-4)$
$75(-3)$
$74(-4)$
$??(x)$, the seventh student score
The amounts above and below the average sum to zero
$-2-4-1-4-3-4+x=0$
$x = 18$
Score of seventh student equals $78+18=96$.

Answer: $96$