A 20 -foot piece of string is cut into two pieces so that the longer piece is 5 feet longer than twice the shorter piece. Find the lengths of both pieces. What is the length of the shorter piece?

Step 1 ：Let's denote the length of the shorter piece as \(x\). According to the problem, the longer piece is 5 feet longer than twice the shorter piece, so it can be represented as \(2x + 5\). The total length of the string is 20 feet, so we can set up the equation \(x + 2x + 5 = 20\) to find the value of \(x\).

Step 2 ：Solving the equation gives the length of the shorter piece of string as \(x = 5\) feet.

Step 3 ：Now, we will calculate the length of the longer piece using the relationship given in the problem. The longer piece is \(2x + 5\) feet long.

Step 4 ：Substituting \(x = 5\) into the equation gives the length of the longer piece as \(15\) feet.

Step 5 ：Final Answer: The length of the shorter piece is \(\boxed{5}\) feet and the length of the longer piece is \(\boxed{15}\) feet.