A rancher is given $100 to buy 100 animals. She must spend all $100 and have no change left over. Cows cost $10, pigs cost $5, and chickens cost $0.15 apiece. H

Question

A rancher is given $100 to buy 100 animals. She must spend all $100 and have no change left over. Cows cost $10, pigs cost $5, and chickens cost $0.15 apiece. How many of each animal does she have to buy to spend all $100 and have exactly 100 animals. Cows cost $10, pigs cost $5, and chickens cost $0.15 apiece. How many of each animal does she have to buy to spend all $100 and have exactly 100 animals.

Answer 1

This is actually a pretty interesting problem. Took me a while to see what they wanted.

 

You can make a system of equations representing cost and number of animals:

 

C + P + H = 100

10C + 5P + .15H = 100

 

but you won't be able to use that to solve for values of C, P, and H, because you must have 3 equations to solve for 3 variables.

 

However... you DO know that the farmer's total dollar amount must be an integer, exactly $100 to be precise. And we know that we can't just go counting chickens willy-nilly, because they cost 15c each. In order to end up with even dollar amounts, you must purchase chickens in quantities of 20. No exceptions. Either 0, 20, 40, 60, 80, or 100 chickens, because anything else will leave you with pocket change in your totals.

 

So let's see what happens for each of those values.

 

• H=0: We have 0 chickens, and we have spent $0. We need to buy 100 more animals for $100. At prices of $5 and $10, both cows and pigs are too expensive. This won't work.
• H=20: We have 20 chickens, and we have spent 20*.15 = $3. We need to buy 80 more animals for $97. We're getting closer, but the average price/animal still needs to be 97/80 = $1.21 per animal, so cows and pigs are still too expensive.
• H=40: We have 40 chickens, and we have spent 40*.15 = $6. We need to buy 60 more animals for $94. Still the average price/animal is $1.57, impossible to meet, so keep going.
• H=60: We have 60 chickens, and we have spent 60*.15 = $9. Need to buy 40 more animals for $91; average price/animal is $2.28. Keep going.
• H=80: Have 80 chickens, spent 80*.15 = $12. Need 20 more animals for $88; average price/animal is $4.40. So close! Let's see if the last possible number of chickens will work for us
• H=100: Have 100 chickens -- whoa. stop. Now that we've achieved the correct number of animals, the only other thing that matters is whether we've also achieved the correct price point. Well, 100 chickens runs us $15, but we need to get to $100, so ... no. This one won't work either.

 

So what we did there is we narrowed down the list of possibilities based on the information given, and then inspected the outcome at each possibility. We ended up finding that there is no solution, no way for the rancher to buy exactly 100 animals for exactly $100, but now, at least, you can prove it. :)

 

If you have any questions, let me know!

 

Brianna L.

Answer 2

With the data that we have there is not solution for the problem