Solution:
Here are the weights of the four women before and after the weight loss program:
| Name |
Before |
After |
| June |
260 |
180 |
| Sophie |
220 |
160 |
| Penny |
170 |
130 |
| Ariana |
130 |
90 |
Let J = June's weight before the program
Let S = Sophie's weight before the program
Let P = Penny's weight before the program
Let A = Ariana's weight before the program
Similarly,
Let x = June's weight after the program
Let y = Sophie's weight after the program
Let w = Penny's weight after the program
Let z = Ariana's weight after the program
Then you can write eight equations with these eight variables corresponding to clues 2 - 9 above:
(2) (J - x) + (S - y) + (P - w) + (A - z) = S
(3) J = 2w
(4) S + P + A = 4w
(5) J + S = 3y
(6) P - w = A - z
(7) 2(S - y) = 3(A - z)
(8) (S - y) = (P - w) + 20
(9) (J - x) + (S - y) = 140
Sub (6) in (8):
(S - y) = (A - z) + 20 (10)
Multiply -2 x (10):
-2(S - y) = -2(A - z) - 40 (11)
Add (7) + (11):
0 = (A - z) - 40
So, A - z = 40 pounds. So, Ariana lost 40 pounds.
From (6):
P - w = 40. So, Penny lost 40 pounds.
From (8):
S - y = 60. So, Sophie lost 60 pounds.
From (9):
J - x = 80. So, June lost 80 pounds.
Subsituting these four values in (2):
S = 220 pounds. So, Sophie weighed 220 pounds before the program.
From (8):
S - y = 60
220 - y = 60
y = 160. So, Sophie weighed 160 pounds after the program.
From (5):
J + S = 3y
J + 220 = 3(160)
J = 260. So, Sophie weighed 260 pounds before the program.
From (3):
J = 2w
w = 130. So, Penny weighed 130 pounds after the program.
From (6):
P - w = 40
P = 170. So, Penny weighed 170 pounds before the program.
From (9):
(260 - x) + (220 - 160) = 140
x = 180. So, June weighed 180 pounds after the program.
From (4):
S + P + A = 4w
220 + 170 + A = 4(130)
A = 130. So, Ariana weighed 130 pounds before the program.
From (6):
P - w = A - z
170 - 130 = 130 - z
z = 90. So, Ariana weighed 90 pounds after the program.