A 51 meter rope is cut into 2 pieces so that one piece is 6 meters longer than four times the shorter piece. Find the lengths of both pieces.
Let x = the shorter piece.
x + 4x - 6 = 51
5x = 45
x = 9
4654
Let's assume the length of the shorter piece of rope is x meters.
According to the given information, the longer piece of rope is 6 meters longer than four times the shorter piece. Thus, the length of the longer piece can be expressed as 4x + 6 meters.
We are also given that the total length of the rope is 51 meters, so we can set up the equation:
x + (4x + 6) = 51
Now, let's solve this equation to find the value of x:
5x + 6 = 51 (combining like terms)
5x = 45 (subtracting 6 from both sides)
x = 9 (dividing both sides by 5)
So, the shorter piece of rope has a length of 9 meters.
Now we can find the length of the longer piece:
4x + 6 = 4(9) + 6 = 36 + 6 = 42
Therefore, the length of the longer piece of rope is 42 meters.
To solve this problem, we can set up a system of equations.
Let's define two variables:
- Let x represent the length of the shorter piece of the rope.
- Let y represent the length of the longer piece of the rope.
According to the problem, we are given that the total length of the rope is 51 meters. So we have the equation:
x + y = 51 (Equation 1)
We are also given that the longer piece is 6 meters longer than four times the shorter piece. Mathematically, this can be written as:
y = 4x + 6 (Equation 2)
Now we can solve this system of equations by substituting Equation 2 into Equation 1.
Substituting y = 4x + 6 into Equation 1, we get:
x + (4x + 6) = 51
Simplifying, we have:
5x + 6 = 51
Subtracting 6 from both sides, we get:
5x = 45
Dividing both sides by 5, we have:
x = 9
Now, we can substitute this value of x back into Equation 2 to find the value of y:
y = 4x + 6
y = 4(9) + 6
y = 36 + 6
y = 42
Therefore, the lengths of the two pieces of the rope are:
- The shorter piece is 9 meters long.
- The longer piece is 42 meters long.