A farmer purchased 100 head of livestock for a total cost of

A farmer purchased 100 head of livestock for a total cost of $4000. Prices were as follow: Calves $120 , $50 per lamb, and $25 per piglet. If the farmer obtained at least 80 piglets, how many of each type did he buy?

PLease, help, and show the details.

Solution

Total Stock purchased =100, Total Cost = $4000

Lets say No of Calves bought =C,

No of Lamb bought = L, No of Piglet bought= 80+X(As 80 is the minimum bought by him)

120C+50L+(80+X)25=4000,

120C+50 L+25X=2000

24C+10L+5X=400, As 400 is divisible by 5, this equation will be satisfied only if C is multiple of 5

C can be 5,10 or 15...Above that total will exceed the equation

Case 1: C=5 , 10L+5X =280, 2L+X=56... In this case sum of total items exceed 100, As atleast 28 Lambs ,5 Calves and 80 piglets are bought

Case 2: C=10, 10L+5X =160, 2L+X=32...In this case sum of total items exceed 100, As atleast 16 Lambs ,10Calves and 80 piglets are bought

Case 3:C=15 , 10L+5X =40, 2L+X=8... Given condition will be true incase No of Lambs =3, No of Calves=15, No of piglets = 82.. Total will be 100.

Verification : 15(120)+3(50)+82(25)= 4000, So given condition is satisfied