### Brain Teasers

# Farmers market

A farmer must spend exactly $100 to buy exactly 100 animals, and must buy at least one of each animal. Cows are $10 each, Goats are $1 each and Chickens are 8 for a $1. What does he buy?

### Answer

7 Cows, 21 Goats, and 72 Chickens.Hide Answer Show Answer

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## Comments

Nov 21, 2001

not my farmer he couldn't be bothered with all the math and bought 100 goats

do you know how many different possible answers there are to this question? i haven't added it up, but there are quite a few...

Actually, there are exactly two answers (the two given). The number of cows must be a multiple of 7, and the number of chickens must be 72/7 times the number of cows. 7 cows gives you the first answer; 0 cows gives you petera's answer.

Do some people have trouble reading, or was "and must buy at least one of each animal" added to the teaser after Jul 31, 2002?

How about a full solution?

I totally missed the must buy one of each animal. So I chose 100 goats. Good one!

Couldn't you buy 9 cows, 9 goats and 8 chickens?

never mind, I skipped the 100 animals part.

i came up with the same answer without a calculator and enjoyed its simplicity, it may have other solutions but i enjoyed this just the way it was

Good one! I wish I had 7 cows.

Here's the logic:

First, forget about the goats for now - each goat contributes 1 to the total # of animals and 1 to the cost, so they're trivial

Let 'a' be the number of cows, and 'b' be the number of chickens. For the total cost and the number of animals to be equal, then:

10a + (1/b = a + b

== 80a + b = 8a + 8b

== 72a = 7b

which means there has to be 72/7 times as many chickens as cows, and 7/72 times as many cows as chickens.

The only way to have this and have the totals to be greater than 0 and less than 100 would be to have 72 chickens and 7 cows. Let the number of goats bring the totals up to 100.

First, forget about the goats for now - each goat contributes 1 to the total # of animals and 1 to the cost, so they're trivial

Let 'a' be the number of cows, and 'b' be the number of chickens. For the total cost and the number of animals to be equal, then:

10a + (1/b = a + b

== 80a + b = 8a + 8b

== 72a = 7b

which means there has to be 72/7 times as many chickens as cows, and 7/72 times as many cows as chickens.

The only way to have this and have the totals to be greater than 0 and less than 100 would be to have 72 chickens and 7 cows. Let the number of goats bring the totals up to 100.

Naturally (1/ should be ( 1/8 )

Easier way to explain the logic:

The digital sum (sum of the two digits) of the number of chickens must equal the cost of the chickens. 7 + 2 = 9; 9 x 8 = 72. Then buy the number of cows represented by the first digit in the number of chickens. This will give you the same number of animals as dollars spent, allowing the remainder to be spent on goats.

The digital sum (sum of the two digits) of the number of chickens must equal the cost of the chickens. 7 + 2 = 9; 9 x 8 = 72. Then buy the number of cows represented by the first digit in the number of chickens. This will give you the same number of animals as dollars spent, allowing the remainder to be spent on goats.

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