Question 1085834: Question
A twenty-foot piece of wire is cut into three pieces. The first piece is twice as long as the second piece, and four times as long as the third piece. How long is the second piece of wire?
I have the following:
First piece is x
Second piece: 2x
I don't understand the four times as long as the third piece because it doesn't specify. How do I define 4x as long as the third piece?
Thanks in advance.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Let x, y and z be positive real numbers. Let's define the following
x = Length of first piece
y = Length of second piece
z = Length of third piece
each length is in feet
We're told that "The first piece is twice as long as the second piece" which means
length of first piece = 2*(length of second piece)
x = 2*(y)
x = 2*y
We're also told that "The first piece is... four times as long as the third piece" meaning
length of first piece = 4*(length of third piece)
x = 4*(z)
x = 4*z
If x = 2*y and x = 4*z, then the two right sides must be equal, leading to
2*y = 4*z
2*y/2 = 4*z/2
y = 2*z
The key thing to take away from all this is that x = 4*z and y = 2*z. Why these two equations? Well the two right sides are in terms of one variable: z. The left sides are the other two variables.
The length of the wire is 20 feet originally. We split that 20 ft wire into 3 pieces (x,y and z ft long). This means
(length of first piece)+(length of second piece)+(length of third piece) = total length
x+y+z = 20
We take the equation x+y+z = 20 and make substitutions. We'll replace 'x' with 4*z (since x = 4*z) and we'll replace 'y' with 2*z (because y = 2*z).
So we go from this
x+y+z = 20
to this
4*z+2*z+z = 20
At this point, we solve for z
4*z+2*z+z = 20
4*z+2*z+1*z = 20
(4+2+1)*z = 20
7*z = 20
7*z/7 = 20/7
z = 20/7
The length of the third piece is exactly 20/7 ft long. To express this as a decimal, you would use a calculator to get 20/7 = 2.85714285714286, which is an approximate value.
Use this to find x
x = 4*z
x = 4*(20/7)
x = 80/7
So the first piece is 80/7 ft exactly and roughly 80/7 = 11.4285714285714 ft long
Use the z value to find y
y = 2*z
y = 2*(20/7)
y = 40/7
The second piece is 40/7 ft exactly and roughly 40/7 = 5.71428571428571 ft long.
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In summary,
Length of first piece = 11.4285714285714 ft
Length of second piece = 5.71428571428571 ft
Length of third piece = 2.85714285714286 ft
Those three values are approximate
Note how those values add back up to 20
11.4285714285714+5.71428571428571+2.85714285714286 = 20
That helps confirm we have the right lengths.
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