A carpenter is working on a project that calls for three pieces of wood.
The longest piece must be twice the length of the middle-sized piece.
The shortest piece must be 6 inches shorter than the middle-sized piece.
If the three pieces sum to a total of 80 inches, how long must each piece be?
Let x= length of the middle-sized piece
Write an expression involving x for the length of each piece.
Longest piece =
Middle-sized piece =
Shortest piece =
Write an equation taking the sum of the three pieces and solve.
Longest piece = 2x
Middle-sized piece = x
Shortest piece = x - 6
The equation for the sum of the three pieces is:
2x + x + (x - 6) = 80
Simplifying the equation:
4x - 6 = 80
4x = 86
x = 21.5
Therefore, the length of the middle-sized piece is 21.5 inches.
The longest piece is 2 * 21.5 = 43 inches.
The shortest piece is 21.5 - 6 = 15.5 inches.
Let's generate the expressions for the length of each piece:
Longest piece = 2 * (middle-sized piece) = 2x
Shortest piece = (middle-sized piece) - 6 = x - 6
Now let's write the equation for the sum of the three pieces:
Longest piece + Middle-sized piece + Shortest piece = 80
2x + x + (x - 6) = 80
Simplifying the equation:
4x - 6 = 80
4x = 86
x = 21.5
Therefore, the middle-sized piece is 21.5 inches.
Using this value, we can find the lengths of the other two pieces:
Longest piece = 2 * (21.5) = 43 inches
Shortest piece = 21.5 - 6 = 15.5 inches
So the lengths of the three pieces are:
Longest piece = 43 inches
Middle-sized piece = 21.5 inches
Shortest piece = 15.5 inches
To find the lengths of the three pieces, let's write expressions involving x for each piece.
Longest piece = 2x
The longest piece must be twice the length of the middle-sized piece.
Middle-sized piece = x
This is already given in the question.
Shortest piece = x - 6
The shortest piece must be 6 inches shorter than the middle-sized piece.
Now, let's write an equation taking the sum of the three pieces and solve it.
Longest piece + Middle-sized piece + Shortest piece = 80
Substituting the expressions we found earlier,
2x + x + (x - 6) = 80
Now, combine like terms,
4x - 6 = 80
Next, add 6 to both sides to isolate the variable,
4x = 86
Finally, divide both sides by 4,
x = 21.5
So the length of the middle-sized piece is 21.5 inches.
To find the lengths of the other two pieces, substitute this value back into the expressions we found earlier.
Longest piece = 2 * 21.5 = 43 inches
Shortest piece = 21.5 - 6 = 15.5 inches
Therefore, the lengths of the three pieces are 43 inches, 21.5 inches, and 15.5 inches, respectively.