Question 129726: a farmer buys 100 animals for $100
chicks=10 cents each
pigs=$2 each
sheep=$5 each
how many of each does he buy
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You can't solve this equation directly. You have to use a little intuition. Why? Because you
have three unknowns (the number of Sheep, Pigs, and Chicks) and you only can generate two
independent equations from the information given. If you let S be the number of Sheep, P be
the number of Pigs, and C be the number of Chicks, then you know that the total number of
Sheep, Pigs and Chicks is 100. In equation form this is:
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S + P + C = 100
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You also know the total amount of money spent must be $100. Since you spend $5 for each Sheep
the amount of money you spend on Sheep is $5 times the number of Sheep or 5S. Similarly the amount
spent on Pigs will be $2 per Pig times the number of Pigs or 2P. And finally, the amount
of money spent on Chicks will be $0.10 per Chick times the number of Chicks or 0.1C. Adding
up these expenditures in equation form and setting them equal to $100 results in the equation:
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5S + 2P + 0.1C = 100
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These are the only two independent equations you can write and with three unknowns you
need three independent equations to solve for all three unknowns.
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Let's analyze the problem a little bit by concentrating on the Chicks. You know that you
can't just buy 100 Chicks. Why? Because you have then bought all 100 animals and at 10 cents
per Chick you have spent only $10 buying the 100 animals. So you must have less than 100 Chicks.
Something else important to note. The number of Chicks that you buy must be evenly divisible
by 10. To see this you might picture buying 95 Chicks. The cost of that purchase would be $9.50.
But the odd cents (fifty cents) can't be because the Sheep and Pigs are in even dollars. So when
you add the expenditures for animals you end up having the fifty cents in the total so you
can never spend exactly $100. So in buying Chicks, knowing that you need to have less than 100 of
them, you can guess that you buy 90, or 80, or 70, or 60 or ....whatever.
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If you buy 90 chicks, you spend $9 on that purchase. That means that you can spend $91 on
Sheep and Pigs, and the combined number of Sheep and Pigs must total to 10 so that with the
90 Chicks you end up with 100 animals. So we can write 2 equations with 2 unknowns. The
first equation sums the numbers of Sheep and Pigs which must be a total of 10 and the second
adds the amounts spent on Sheep and Pigs which must be a total of $91. The equations are:
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S + P = 10
5S + 2P = 91
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Solve by variable elimination. Multiply the top equation (both sides and all terms) by 5
and the equation set then becomes:
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5S + 5P = 50
5S + 2P = 91
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If you subtract the two equations in vertical columns, the result of this subtraction is:
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3P = -41
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You can now see that if you divide both sides of this equation by 3 you end up with P
being a negative number of Pigs in addition to not being a whole number. That doesn't
make sense. So it was a mistake to guess that we had bought 90 Chicks.
.
Using the same process, let's guess that we bought 80 Chicks. This would mean that the
number of Sheep added to the number of Pigs must be the 20 animals we need to add to the
80 Chicks. Also the 80 Chicks will cost $8 so the amount spent on Sheep and Pigs must be the
$92 left to spend. Writing these in equation form we get the two equations:
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S + P = 20
5S + 2P = 92
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Multiply the top equation by 5 (both sides and all terms) and the equation set then becomes:
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5S + 5P = 100
5S + 2P = 92
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Subtract the two equations in vertical columns and you end up with:
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3P = 8
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If you divide both sides by 3 the result is P = 8/3. Not good because it's not a whole number
of Pigs.
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Guess that the number of Chicks we bought was 70. That means we still have 30 animals to purchase.
The 70 Chicks cost $7 so we still have $93 to spend. Write the equations:
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S + P = 30
5S + 2P = 93
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As usual, multiply the top equation by 5 to make the set:
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5S + 5P = 150
5S + 2P = 93
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Subtract in vertical columns and you have 3P = 57. Solve for P by dividing both sides by 3
and you get P = 57/3 = 19. This looks promising ... a whole number of Pigs. Since we were
looking for the number of animals to be 30 and we now know the number of Pigs is 19, then the
number of Sheep must be 30 minus 19 or 11 Sheep.
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Let's check this answer out. 11 Sheep plus 19 Pigs + 70 Chicks does total to 100 animals.
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At $5 per Sheep the 11 Sheep cost $55. At $2 per Pig, the 19 Pigs cost $38. And at $0.10 per Chick,
the 70 Chicks cost $7. The total amount spent is $55 + $38 + $7 = $100.
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Everything checks. Therefore, the answer is 11 Sheep, 19 Pigs, and 70 Chicks were bought.
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Hope this helps you to see your way through the problem.
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