The first and most important step is to introduce a variable:
$\ \ \ \ \ \ $Let $x$ be the common price of each of the remaining $10$ suits.
The second step is to find an equation in $x$. Of course, we may obtain one by knowing that the average sale price of the twenty suits is $55$ dollars. Once we have our equation, we'll solve it for $x$.
The average sale price of the suits is obtained by computing the total revenue from the $60$ suits and dividing by $60$.
Let's find the total revenue first. Thirty suits were sold at $60$ dollars and twenty suits were sold at $50$ dollars. The revenue from these is $30\cdot60+20\cdot 50$ dollars. The remaining ten suits are sold at $x$ dollars and the revenue from these is $x\cdot10$ dollars. So, adding these together, the total revenue from the $60$ suits is $30\cdot 60+ 20 \cdot 50 +x\cdot 10$ dollars.
Thus, the average sale price of the twenty suits is $\tag{1}{\text{total revenue}\over 60}={30\cdot 60+ 20 \cdot 50 +x\cdot 10\over 60}.$
Now we find our equation. You want $(1)$ to turn out to be $55$. So set $\tag{2} {30\cdot 60+ 20 \cdot 50 +x\cdot 10\over 60}=55. $
Now the third and final step: we solve equation $(2)$ for $x$. Towards that end, first multiply both sides of $(2)$ by $60$: $ {30\cdot 60+ 20 \cdot 50 +x\cdot 10 }=55\cdot 60. $ Simplify the arithmetic: $\tag{3} 2800+10x=3300, $ or, subtracting $2800$ from both sides of $(3)$, $\tag{4} 10x=500. $ Finally, divide both sides of $(4)$ by $10$ to obtain $x=50$ dollars.