The sum of three numbers $a$, $b$, and $c$ is 99. If we increase $a$ by 6, decrease $b$ by 6 and multiply $c$ by 5, the three resulting numbers are equal. What is the value of $b$?
I need to find a way to express the relationship between the three numbers and use it to find $b$. I know that $a + b + c = 99$, so I can use that to eliminate one of the variables. I also know that $a + 6 = b - 6 = 5c$, so I can use that to express $a$ and $b$ in terms of $c$. Solving for $a$, I get $a = 5c - 6$. Solving for $b$, I get $b = 5c + 6$. Substituting these expressions into the first equation, I get $(5c - 6) + (5c + 6) + c = 99$. Simplifying, I get $11c = 99$, so $c = 9$. Now that I have $c$, I can plug it into the expression for $b$ and get $b = 5 \times 9 + 6$. Simplifying, I get $b = 51$.

