Part A
S = smallest piece
M = medium length piece
L = largest piece
The ratio of the smallest to largest is 2:3, so we can say S = 2x and L = 3x for some positive real number x. The ratio 2x:3x reduces to 2:3.
The third piece (medium length piece) is 1 & 3/4 feet shorter than the longest piece. This means
M = L - (1 & 3/4)
M = L - (1 + 3/4)
M = L - (4/4 + 3/4)
M = L - (7/4)
M = 3x - (7/4)
Add up the three pieces
T = total length
T = S+M+L
T = ( S ) + ( M ) + ( L )
T = ( 2x ) + ( 3x - (7/4) ) + ( 3x )
T = (2x+3x+3x) - 7/4
T = 8x - 7/4
Set this equal to the expression (8/3)x - 7/4 and solve for x
9x - 7/4 = (8/3)x - 7/4
12* [ 8x-7/4 ] = 12*[ (8/3)x - 7/4 ] ... multiply both sides by 12
96x - 21 = 32x - 21
108x = 32x
108x-32x = 0
76x = 0
x = 0/76
x = 0
This is not a useful answer because for example S = 2x = 2*0 = 0, but we can't have a piece of 0 length.
There isn't enough info for Margaret to find the length of each piece.
Part B
Instead of (8/3)x - 7/4 feet, we are given a total length of 22 & 1/4 feet.
This is equivalent to the improper fraction 89/4 since
22 & 1/4 = 22 + 1/4
22 & 1/4 = 88/4 + 1/4
22 & 1/4 = 89/4
Let's try to find x now. Set the value of T we got earlier equal to 89/4 and solve for x
T = 8x - 7/4
T = 89/4
8x - 7/4 = 89/4
4*( 8x-7/4) = 4*89/4 multiply both sides by 4 to clear out fractions
32x - 7 = 89
32x = 89+7
32x = 96
x = 3
This is a useful x value because...
S = 2x = 2*3 = 6
M = 3x - 7/4 = 3*3 - 7/4 = 9 - 7/4 = 36/4 - 7/4 = 29/4 = 7 & 1/4
L = 3x = 3*3 = 9
None of the lengths are 0.
Then note how,
T = S+M+L
T = 6 + (7 & 1/4) + 9
T = 6 + (7 + 1/4) + 9
T = (6+9+7) + (1/4)
T = 22 + 1/4
T = 22 & 1/4
Which helps confirm the answers.
Answers:
There is enough info for her to find the length of each piece in this case. The lengths are
shortest = 6 feetmedium = 7 & 1/4 feetlongest = 9 feet
