A grassland can support 19 sheep and 16 cows for a maximum of 9 days; OR; 21 sheep and 12 buffalos for a maximum of 7 days; OR; 18 cows and 6 buffalos for a maximum of 6 days; OR; 8 sheep, 5 cows and 7 buffalos indefinitely.

Three friends Abner, Bert and Claude jointly arranged for use of the grassland for $11,466 on the understanding that the share of rent paid by any given friend would be determined in accordance with the total amount of grass consumed by his pets.

The grass grows at a constant rate per unit time in the grassland. For example, if the rate of growth of grass is 6 units per day and the initial amount of grass in the grassland is 5000 units, then the total amount of grass at the end of 5 days in the grassland would be 5030 units.

Abner grazes 13 sheep and 4 buffalos for 48 days.
Bert puts in 10 cows and 2 buffalos for 38 days and:
Claude puts in 7 sheep and 6 cows for a period of 56 days.

Determine the respective share of rent payable by Abner, Bert and Claude on the grassland in terms of information inclusive of the foregoing statements.

Let
b = one buffalo's daily feed
c = one cow's daily feed
s = one sheep's daily feed

x = initial supply of grass
r = rate of grass growth

Given
9(19s + 16c) = 9r + x
7(21s + 12b) = 7r + x
6(18c + 6b) = 6r + x
8s + 5c + 7b = r

Eliminating r, by subtracting multiples of the last equation:
99s + 99c - 63b = x
91s - 35c + 35b = x
-48s + 78c - 6b = x

Eliminating x:
147s + 21c - 57b = 0
139s - 113c + 41b = 0

or

21c + 147s = 57b
113c - 139s = 41b

Then

-5580c + 13950s = 0,
so that s = 2c/5

and

19530c = 13950b
so that b = 7c/5

Thus

the ratio s:c:b is 2:5:7

Using the numbers 2, 5 and 7, the relative allocations come out:

A: 2592
B: 2432
C: 2464

These add up to 7488. To bring the total up to 11,466, each relative allocation must be multiplied by 1.53125, making the dollar amounts owed by each:

A: 3969
B: 3724
C: 3773

Posted by Charlie on 2006-04-02 20:47:50