What is the length of the original rectangle?
The perimeter of a rectangle is equal to 40. If the length is halved and the width is divided by 3, the new perimeter is decreased by 24.
Select one:
a. 12
b. 8
c. 4
d. 16
P = Original perimeter = 40
P = 2 w + 2 l = 2 ( w + l )
40 = 2 ( w + l )
Divide both sides by 2
20 = w + l
w + l = 20
Subtract w to both sides
w + l - w = 20 - w
l = 20 - w
If the length is halved and the width is divided by 3 mean:
New lenth l1 = l / 2 = ( 20 - w ) / 2 = 10 - w / 2
New width w1 = w / 3
The new perimeter is decreased by 24 mean:
P1 = New perimeter = 40 - 24 = 16
P1 = 2 w1 + 2 l1 = 2 ( w1 + l1 )
16 = 2 ( w1 + l1 )
Divide both sides by 2
8 = w1 + l1
w1 + l1 = 8
w / 3 + 10 - w / 2 = 8
Subtract 10 to both sides
w / 3 + 10 - w / 2 - 10 = 8 - 10
w / 3 - w / 2 = - 2
2 w / 6 - 3 w / 6 = - 2
- w / 6 = - 2
Multiply both sides by - 6
( - 6 ) ∙ ( - w / 6 ) = ( - 2 ) ∙ ( - 6 )
w = 12
l = 20 - w = 20 - 12 = 8
Proof:
Original perimeter:
P = 2 w + 2 l = 2 ( w + l ) = 2 ∙ ( 12 + 8 ) = 2 ∙ 20 = 40
New lenth:
l1 = l / 2 = 8 / 2 = 4
New width:
w1 = w / 3 = 12 / 3 = 4
New rectangle will be the square. ( the square is a special case of the rectangle )
New perimeter:
P1 = 2 w1 + 2 l1 = 2 ( 4 + 4 ) = 2 ∙ 8 = 16
The length of the original rectangle:
l = 8
Answer b
To find the length of the original rectangle, we can set up a system of equations based on the given information.
Let's say the length of the original rectangle is represented by "L" and the width is represented by "W".
According to the given information:
1. The perimeter of the original rectangle is 40, so we have the equation: 2(L + W) = 40
2. If the length is halved, the new length becomes L/2. If the width is divided by 3, the new width becomes W/3. The new perimeter is decreased by 24, so we have the equation: 2(L/2 + W/3) = 40 - 24
Simplifying both equations:
1. L + W = 20 (dividing both sides by 2)
2. (L/2 + W/3) = 8 (dividing both sides by 2 and subtracting 24 from 40)
By solving the system of equations, we can find the values of L and W.
Multiplying equation 1 by 3, we have: 3(L + W) = 60
Expanding equation 2, we have: (3L + 2W) = 24
Now we have a new system of equations:
3L + 3W = 60
3L + 2W = 24
Subtracting equation 2 from equation 1, we eliminate L:
3L + 3W - (3L + 2W) = 60 - 24
3W - 2W = 36
W = 36
Now, substitute the value of W back into equation 1:
L + 36 = 20
L = 20 - 36
L = -16
Since a negative length doesn't make sense, we made an error in our calculations. It's possible that there is a mistake in the given information or in our equations. Please double-check the information provided.