Using two variables - A carpenter wants to cut a 24-foot board into two pieces so that one piece is 5 times as long as the other. How long should each piece be?
Represent the unknowns:
Let x = one piece
Since according to the problem, the other piece is 5 times as long as the first one,
Let 5x = other piece
Since their total length must be 24 ft,
x + 5x = 24
6x = 24
x = 24/6
x = 4 ft.
5x = 20 ft.
To solve this problem, we can set up a system of equations. Let's use two variables, x and y, to represent the length of the two pieces of the board.
Let x represent the length of the shorter piece,
and let y represent the length of the longer piece.
Based on the problem, we have two pieces of the board, so their lengths should add up to the total length of the board, which is 24 feet:
x + y = 24 -- Equation 1
According to the problem statement, the longer piece should be 5 times as long as the shorter piece:
y = 5x -- Equation 2
Now, we have a system of two equations with two variables. We can solve it using substitution or elimination.
Let's use substitution to solve this system:
1. Solve Equation 2 for x:
y = 5x
Divide both sides by 5:
x = y/5
2. Substitute the value of x from Equation 2 into Equation 1:
x + y = 24
(y/5) + y = 24
Multiply both sides by 5 to eliminate the fraction:
y + 5y = 120
6y = 120
Divide both sides by 6:
y = 20
3. Substitute the value of y back into Equation 2 to find the value of x:
x = y/5 = 20/5 = 4
So, the shorter piece should be 4 feet long, and the longer piece should be 20 feet long.