120 inch length of ribbon is to be cut into three pieces. The longest piece is to be 40 inches longer than the shortest piece, and the third piece is to be half the length of the longest piece. Find the length of each piece of ribbon.
Let x = length of shortest piece. You want to express everything in terms of x, then add them in an equation you can solve for x.
Longest piece = 40 + x.
Medium piece = (40 + x)/2 OR 20 + 1/2x.
Shortest piece = x.
You know all the pieces combined = 120 inches, so add all the expressions:
40 + x + 20 + 1/2x + x = 120. Add the whole numbers first, then add the variables:
60 + 2.5x = 120. (note that I made 1/2x into 2.5x for the purpose of plugging in to a calculator in the next step).
Subtract 60 from the left side and the right side:
2.5x = 60
x = 60/2.5 = 24
So, finally...
shortest = 24 in
longest = 64
medium = 32
Want to check your work? Add shortest, longest, and medium, and you'll see it all adds up to 120 inches! :)
Also just thought I should clarify where I got the 2.5x from - x + x = 2x and 1/2x = .5x, so 2x + .5x = 2.5x.
I didn't want it to seem like I said that ONLY 1/2x simplifies to 2.5x - that's x + x + 1/2x!
shortest = x
longest = x+40
middle = (x+40)/2
so
x +(x+40) + (x+40)/2 = 120
x + x +40 +.5 x +20 = 120
2.5 x = 60
x = 24
(x+40)/2 = 32
(x+40) = 64
To solve this problem, let's assume the length of the shortest piece of ribbon as 'x' inches.
According to the given conditions:
- The longest piece is to be 40 inches longer than the shortest piece, so its length would be 'x + 40' inches.
- The third piece is to be half the length of the longest piece, so its length would be '(x + 40) / 2' inches.
Now, we know that the sum of the lengths of the three pieces should be equal to the total length of the ribbon, which is 120 inches.
Therefore, we can write the equation:
x + (x + 40) + (x + 40)/2 = 120
To solve the equation:
1. Multiply through by 2 to eliminate the fraction:
2x + 2(x + 40) + (x + 40) = 240
2x + 2x + 80 + x + 40 = 240
5x + 120 = 240
2. Move the constant term to the other side of the equation:
5x = 240 - 120
5x = 120
3. Divide both sides of the equation by 5 to solve for x:
x = 120/5
x = 24
Therefore, the length of the shortest piece of ribbon is 24 inches.
Now we can substitute this value back into the expressions we derived earlier to find the length of the longest piece and the third piece:
- Length of the longest piece = x + 40
= 24 + 40
= 64 inches
- Length of the third piece = (x + 40) / 2
= (24 + 40) / 2
= 64 / 2
= 32 inches
Therefore, the length of each piece of ribbon is:
- Shortest piece: 24 inches
- Longest piece: 64 inches
- Third piece: 32 inches