A 180-m wire is cut into three pieces. The third piece is 20 m longer than the first. The second is two thirds as long as the first. How long is each piece?
x = first piece
x + x + 20 + (2/3)x = 180
2 2/3x = 160
x = 160/(8/3)
x = 160 * (3/8)
x = 60
To solve this problem, we can break it down into smaller steps:
Step 1: Assign variables
Let's assign variables to represent the lengths of the three pieces. We'll call the first piece "x", the second piece "y", and the third piece "z".
Step 2: Set up equations
From the given information, we can establish the following equations:
First equation: z = x + 20 (The third piece is 20 meters longer than the first)
Second equation: y = (2/3)x (The second piece is two-thirds as long as the first)
Third equation: x + y + z = 180 (The total length of the wire is 180 meters)
Step 3: Solve the system of equations
Substitute the second and third equations into the first equation to eliminate z:
x + y + (x + 20) = 180
2x + 20 + (2/3)x = 180
Multiplying through by 3 to get rid of the fraction:
6x + 60 + 2x = 540
8x + 60 = 540
8x = 480
x = 60
Substitute the value of x back into the second equation to find y:
y = (2/3) * 60
y = 40
Now, substitute the values of x and y into the first equation to find z:
z = 60 + 20
z = 80
Step 4: Answer the question
The first piece is 60 meters long, the second piece is 40 meters long, and the third piece is 80 meters long.