A stick is cut into three pieces,each having a different length.Each piece(except the shortest)is twice as long as another piece.What fraction of the whole stick is each piece?
Let 1/x be the length of the shortest stick.
Thus, the expressions for the other two sticks are 2/x and 4/x. From this, you get this equation:
1/x + 2/x + 4/x = 1
Multiply both sides by x
1 + 2 + 4 = x
x = 7
Therefore...
short stick = 1/7 of whole stick
middle stick = 2/7 of whole stick
long stick = 4/7 of whole stick
So good! Is there another way of doing this?
Let's assume the length of the shortest piece is x.
According to the problem, the other two pieces are twice as long as the shortest piece. So, the length of the second piece is 2x, and the length of the third piece is also 2x.
The total length of all three pieces will be the length of the original stick, which we'll call L.
Therefore, we have the equation x + 2x + 2x = L.
Combining like terms, we get 5x = L.
To find the fraction that each piece represents of the whole stick, we need to divide the length of each piece by the total length of the stick.
For the shortest piece:
Fraction of the shortest piece = x / L
For the second piece (2x):
Fraction of the second piece = 2x / L
For the third piece (also 2x):
Fraction of the third piece = 2x / L
Considering the values of x and L may vary, we cannot simplify these fractions any further without specific values.
Therefore, the fractions of each piece of the stick are:
- The shortest piece: x / L
- The second piece: 2x / L
- The third piece: 2x / L
To find the fraction of the whole stick that each piece represents, we need to compare the lengths of the three pieces.
Let's assign variables to the lengths of the pieces. Let the shortest piece be x units long. Since each piece (except the shortest) is twice as long as another piece, we can assign the lengths of the other two pieces accordingly.
The second longest piece will be 2x units long, and the longest piece will be 4x units long.
Now, let's find the total length of the stick. The total length is the sum of the lengths of all three pieces:
Total length = x + 2x + 4x = 7x
To find the fraction of the whole stick that each piece represents, we divide the length of each piece by the total length.
The fraction for the shortest piece is x/7x, which simplifies to 1/7.
The fraction for the second longest piece is 2x/7x, which simplifies to 2/7.
The fraction for the longest piece is 4x/7x, which simplifies to 4/7.
So, the fractions for each piece are:
- Shortest piece: 1/7
- Second longest piece: 2/7
- Longest piece: 4/7