Question 769376: a famer has $100 doallrs to spend on 100 animals.
he has to end up with 100 animals.
pigs cost $10 each, turkeys cost .50cents and chickens cost 12.5cents
how many of each can he have for his $100?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! 1 pig - $10
1 turkey - 50 cents
1 chicken - 12.5 cents
I am assuming the farmer has to get exactly 100 animals and spend exactly $100 dollars. Otherwise there would be too many possible solutions.
The easiest strategy seems to be casework (checking the possibilities case by case) with a little thinking to reduce the number of cases.
If each pig costs $10, the farmer could get 0, 1, 2, 3, ....7, 8, 9, 10 pigs.
Low numbers of pigs will not work.
If the farmer buys only 5 pigs, he would spend $50 om the pigs and would have $50 left to spend, which would get him at least 100 more animals (if he spends the remaining $50 on just turkeys).
High numbers of pigs will not work either.
Ten pigs would use up the $100 and he would be getting only 10 animals.
Nine pigs would use up $90, and leave $10 left, which at most would get him 80 animals (if he spends it all on chickens).
That leaves 6, 7, or 8 pigs as the only possibilities.
Buying 6 pigs uses $60, leaving $40 to buy another 100-6=94 animals.
If the farmer buys 6 pigs, he needs to also buy 94 birds (between turkeys and chickens).
$40 buys only 80 turkeys, so to get 94 birds, the farmer would need some chickens.
At 50 cents=4x(12.5 cents), each turkey cost the same as 4 chickens.
For $40, instead of 80 turkeys, the farmer could get:
80-1=79 turkeys and 4 chickens, for a total of 79+4=83 birds, or
80-2=78 turkeys and 2x4=8 chickens, for a total of 78+8=86 birds, or
80-3=77 turkeys and 3x4=12 chickens, for a total of 77+12=89 birds, or
80-4=76 turkeys and 4x4=16 chickens, for a total of 76+16=92 birds, or
80-5=75 turkeys and 5x4=20 chickens, for a total of 75+20=95 birds.
The number of birds that can be bought for $40 (or for any fixed amount of many varies by 3's (86, 89, 92, 95).
With 6 pigs, there is no way of buying the 94 birds needed.
Buying 8 pigs does not work either.
Buying 7 pigs uses $70, leaving $30 to buy another 100-7=93 animals.
If the farmer buys 7 pigs, he needs to also buy 93 birds (between turkeys and chickens).
$30 buys only 60 turkeys, so to get 93 birds, the farmer would need quite a few chickens.
Again, trading each turkey for 4 chickens, the number of birds the farmer can buy for $30 varies by 3's.
He can get 60, 63, 66, 69, .....
He can numbers that are multiples of 3.
Each time he reduces the number of turkeys by 1, the total number of birds increases by 3.
He needs  less than 60 turkeys.
He can get 93 birds using 11 less turkeys (60-11=49)and 4x11=44 more chickens.
He could get  ,  , and 
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