[ Arithmetic operations. Numerical identities ]

In a test, each of the 25 students received one of the grades "3", "4", or "5". By how many more were there fives than threes, if the sum of all grades is 106? 

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Let $a$ students received a grade of 3, $b$ students received a grade of 4, and $c$ students received a grade of 5. From the problem, we have $a+b+c=25$ and $3a+4b+5c=106$.

Multiply both sides of the first equation by 4: $4a+4b+4c=100$. Now subtract the obtained equation from the second equation, then $c-a=6$.

## Answer

6.