1. June, Sophie, Penny, and Ariana all went through a weight loss diet program.

2. Their combined weight loss after the program was equal to Sophie's weight before the program.

3. June's weight before the program was twice Penny's weight after the program.

4. The combined weight of Sophie, Penny and Ariana before the program was four times Penny's weight after the program.

5. The combined weight of June and Sophie before the program was three times Sophie's weight after the program.

6. Penny and Ariana both lost the same amount of pounds.

7. Twice Sophie's weight loss was equal to three times Ariana's weight loss.

8. Sophie lost 20 pounds more than Penny.

9. The combined weight loss of June and Sophie is equal to 140 pounds.

Determine the weights of June, Sophie, Penny and Ariana before and after the weight loss program.
Solution to the Problem:


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.


Correctly solved by:

1. Davit Banana Istanbul, Turkey
2. Mrs. Painter's Honors Precalculus Class Broome High School,
Spartanburg, South Carolina, USA
3. Dr. Hari Kishan D.N. College,
Meerut, Uttar Pradesh, India
4. Colin (Yowie) Bowey Beechworth, Victoria, Australia
5. Seth Cohen Concord, New Hampshire, USA