A poultry keeper had a number of quail, hen, and goose eggs to sell. At the end of the day,
she had sold two dozen eggs for a total of 103 quadrons.
If she sold an equal number of quail and goose eggs, but the goose eggs cost twice as much, and the hen eggs cost 4 quadrons apiece,
what was the price of the various eggs, and how many of each did she sell?
Solution to the Problem:
There are two answers:Answer #1
1 quail egg at 5 quadrons each for a total of 5 quadrons
22 hen eggs at 4 quadrons each for a total of 88 quadrons
1 goose egg at 10 quadrons each for a total of 10 quadrons
Answer #2
7 quail eggs at 3 quadrons each for a total of 21 quadrons
10 hen eggs at 4 quadrons each for a total of 40 quadrons
7 goose eggs at 6 quadrons each for a total of 42 quadrons
I used three variables to solve the problem, but I noticed that there were only two equations, so that alerted me that there may be more than one answer.
Let x = # of quail eggs
Let h = # of hen eggs
Then x also equals the # of goose eggs
Let y = cost of quail eggs
We are told that 4 = cost of hen eggs
Then 2y = cost of the goose eggs.
My two equations are:
x + h + x = 24 and
xy + 4h + 2xy = 103
From equation 1, we get h = 24 - 2x
This tells me that x must be between 0 and 12 in order for h to be a non-negative number.
From the second equation, I get the following:
3xy + 4h = 103
Substituting equation 1 in equation 2, we get
3xy + 4(24 - 2x) = 103
3xy + 96 - 8x = 103
Solving for y:
y = (8x + 7) / 3x
Now, I set up a table for these three variables for the values in the domain (using the equations above):
| x | h | y | works? |
|---|---|---|---|
| 0 | 24 | 7 | no |
| 1 | 22 | 5 | yes |
| 2 | 20 | 3.83 | no |
| 3 | 18 | 3.44 | no |
| 4 | 16 | 3.25 | no |
| 5 | 14 | 3.13 | no |
| 6 | 12 | 3.05 | no |
| 7 | 10 | 3 | yes |
| 8 | 8 | 2.95 | no |
| 9 | 6 | 2.92 | no |
| 10 | 4 | 2.9 | no |
| 11 | 2 | 2.87 | no |
| 12 | 0 | 2.86 | no |
So, the answers are:
#1
1 quail egg at 5 quadrons each for a total of 5 quadrons
22 hen eggs at 4 quadrons each for a total of 88 quadrons
1 goose egg at 10 quadrons each for a total of 10 quadrons
24 eggs for 103 quadrons
#2
7 quail eggs at 3 quadrons each for a total of 21 quadrons
10 hen eggs at 4 quadrons each for a total of 40 quadrons
7 goose eggs at 6 quadrons each for a total of 42 quadrons
24 eggs for 103 quadrons
Correctly solved by:
| 1. Veena Mg | Bangalore, Karnataka, India |
| 2. Colin (Yowie) Bowey | Beechworth, Victoria, Australia |
| 3. Jordan Sollinger |
Woodridge High School, Peninsula, Ohio |
| 4. Hari Kishan | Meerut, Uttar Pradesh, India |
| 5. Ivy Joseph | Pune, Maharashtra, India |
| 6. Kelly Stubblefield | Mobile, Alabama |