Chickens and Rabbits

Chickens and Rabbits

​Chickens and Rabbits

​There are two types of animals on Old Macdonald's farm: chickens and rabbits.

Each chicken has one head and two legs. Each rabbit has one head and four legs.

Below are some types of chickens and rabbits problems:

(1) Given the total number of chicken and rabbit heads and the total number of chicken and rabbit legs, find the number of animals of each type.

(2) Given the difference between the number of chicken and rabbit heads, and the total number of chicken and rabbit legs, find the number of animals of each type.

​(3) Given the difference between the number of chicken legs and rabbit legs, and the total number of chicken and rabbit legs, find the number of animals of each type.

(4) Given the difference between the number of chicken and rabbit heads, and the difference between the number of chicken legs and rabbit legs, find the number of animals of each type.

(5) There are problems similar to chicken and rabbit problems, but with three or four animals at Old Macdonald's farm.

(6) There are problems similar to chicken and rabbit problems but instead of chickens and rabbits, there can be three-leg stools and four-leg tables, bicycles and tricycles, octopus and sea otters, cows and ducks, spiders and dragonflies, etc.

​Basic Formulas

(3) Generalized formula

(1) Given the total number of chicken and rabbit heads and the total number

of chicken and rabbit legs, find the number of animals of each type.

(1) Given the total number of chicken and rabbit heads and the total number

of chicken and rabbit legs, find the number of animals of each type.

Open Ended

​Example 1. Joe has a cage for his chickens and rabbits. He counts 40 heads from the top of the cage and 130 legs from the bottom of the cage. How many chickens and how many rabbits does he have in the cage?

​Example 1. Joe has a cage for his chickens and rabbits. He counts 40 heads from the top of the cage and 130 legs from the bottom of the cage. How many chickens and how many rabbits does he have in the cage?

Open Ended

Example 2. Alex has a cage for his chickens and rabbits. Alex counts 46 heads

from the top of the cage and 128 legs from the bottom of the cage. How many

chickens and how many rabbits does he have?

Example 2. Alex has a cage for his chickens and rabbits. Alex counts 46 heads from the top of the cage and 128 legs from the bottom of the cage. How many chickens and how many rabbits does he have?

(2) Given the difference between the number of chicken and rabbit heads, and the total number of chicken and rabbit legs, find the number of animals of each type.

Open Ended

Example 3. Alex has a cage for his chickens and rabbits. Alex counts 274 legs

from the bottom of the cage. He counts 26 more chickens than rabbits. How

many chickens and how many rabbits does he have?

Example 3. Alex has a cage for his chickens and rabbits. Alex counts 274 legs from the bottom of the cage. He counts 26 more chickens than rabbits. How many chickens and how many rabbits does he have?

Open Ended

Example 4. Alex has a cage for his chickens and rabbits. Alex counts 32 legs

from the bottom of the cage. He counts 4 more chickens than rabbits. How many chickens and how many rabbits does he have?

Example 4. Alex has a cage for his chickens and rabbits. Alex counts 32 legs

from the bottom of the cage. He counts 4 more chickens than rabbits. How many chickens and how many rabbits does he have?

(3) Given the difference between the number of chicken legs and rabbit legs, and the total number of chicken and rabbit legs, find the number of animals of each type.

Open Ended

Example 5. Alex has a cage for his chickens and rabbits. Alex counts 100 heads

from the top of the cage. He counts 80 more chicken legs than rabbit legs. How

many chickens and how many rabbits are there?

Example 5. Alex has a cage for his chickens and rabbits. Alex counts 100 heads from the top of the cage. He counts 80 more chicken legs than rabbit legs. How many chickens and how many rabbits are there?

Open Ended

Example 6. Alex has a cage for his chickens and rabbits. Alex counts 100 heads

from the top of the cage. He counts 64 more rabbit legs than chicken legs. How

many chickens and how many rabbits does he have?

Example 6. Alex has a cage for his chickens and rabbits. Alex counts 100 heads from the top of the cage. He counts 64 more rabbit legs than chicken legs. How many chickens and how many rabbits does he have?

Open Ended

Example 7. Alex has a cage for his chickens and rabbits. Alex counts 40 heads

from the top of the cage. He counts 70 fewer chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

Example 7. Alex has a cage for his chickens and rabbits. Alex counts 40 heads from the top of the cage. He counts 70 fewer chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

(4) Given the difference between the number of chicken and rabbit heads, and the difference between the number of chicken legs and rabbit legs, find the number of animals of each type.

Open Ended

Example 8. Alex has a cage for his chickens and rabbits. Alex counts 14 more

chicken heads than rabbit heads from the top of the cage. He counts 12 more chicken legs than rabbit legs from the bottom of the cage. How many chickens

and how many rabbits does he have?

Example 8. Alex has a cage for his chickens and rabbits. Alex counts 14 more

chicken heads than rabbit heads from the top of the cage. He counts 12 more chicken legs than rabbit legs from the bottom of the cage. How many chickens and how many rabbits does he have?

​We see that when the number of rabbits is 8 and the number of chickens is 16, the number of chicken and rabbit legs is equal.

We also need to add the 6 chickens that left the cage in the beginning, making the total number of chickens 16 + 6 = 22. The total number of rabbits is 8.

Open Ended

Example 9. Alex has a cage for his chickens and rabbits. Alex counts 14 more

chicken heads than rabbit heads from the top of the cage. He counts 16 more

chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

Example 9. Alex has a cage for his chickens and rabbits. Alex counts 14 more

chicken heads than rabbit heads from the top of the cage. He counts 16 more chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

(5) Problems similar to chicken and rabbit problems, but with three or four animals at Old Macdonald's farm

Open Ended

Example 10. There are 24 three-leg stools and four-leg tables. Together there are 86 legs. How many three-leg stools and how many four-leg tables are there?

Example 10. There are 24 three-leg stools and four-leg tables. Together there are 86 legs. How many three-leg stools and how many four-leg tables are there?

Solution: 10 three-leg stools and 14 four-leg tables.

First, we remove one leg from each table. Each table now has 3 legs. The number of legs of the 24 three-leg stools and three-leg tables is 24 × 3 = 72 legs. Given that there were 86 legs originally, 86 – 72 = 14 legs were removed. Since one leg was removed from one table, there are 14 tables. It follows that the number of stools is 24 – 14 = 10.

Open Ended

Example 11. Bob spent $10 to buy 35 stamps worth either 20-cents or 50-cents.

How many 20-cent stamps and 50-cent stamps did he buy?

Example 11. Bob spent $10 to buy 35 stamps worth either 20-cents or 50-cents. How many 20-cent stamps and 50-cent stamps did he buy?

Open Ended

Example 12. Joe has a cage for his chickens and rabbits. Joe counts 110 legs in

the cage. If the number of chickens and rabbits are switched, there are 100 legs. How many chickens and how many rabbits does he have?

Example 12. Joe has a cage for his chickens and rabbits. Joe counts 110 legs in the cage. If the number of chickens and rabbits are switched, there are 100 legs. How many chickens and how many rabbits does he have?

Open Ended

Example 13. Joe has a cage for his chickens and rabbits. Joe counts 144 legs in

the cage. If the number of chickens and rabbits are switched, there are 156 legs.

How many chickens and how many rabbits does he have?

Example 13. Joe has a cage for his chickens and rabbits. Joe counts 144 legs in the cage. If the number of chickens and rabbits are switched, there are 156 legs. How many chickens and how many rabbits does he have?

(6) Problems similar to chicken and rabbit problems but instead of chickens and rabbits, there can be three-leg stools and four-leg tables, bicycles and tricycles, octopus and sea otters, cows and ducks, spiders and dragonflies, etc.

Open Ended

Example 14. A spider has 8 legs and no wings. A dragonfly has 6 legs and 2 pairs

of wings. A cicada has 6 legs and one pair of wings. We have all 3 kinds of

insects and a total of 18 in a cage. Given that there are 118 legs and 20 pairs of

wings in the cage, how many insects do we have of each kind?

Example 14. A spider has 8 legs and no wings. A dragonfly has 6 legs and 2 pairs of wings. A cicada has 6 legs and one pair of wings. We have all 3 kinds of insects and a total of 18 in a cage. Given that there are 118 legs and 20 pairs of wings in the cage, how many insects do we have of each kind?

Solution: 5 spiders, 7 dragonflies, and 6 cicadas.

Method 1:

Since both the dragonfly and cicada have 6 legs, we first consider these two

insects to be of one type.

The problem becomes the following:

A spider has 8 legs. A dragonfly-cicada has 6 legs. We have these two kinds and a total of 18 insects in a cage with 118 legs. How many spiders are there?

Open Ended

Example 15. A spider has 8 legs and no wings. A dragonfly has 6 legs and 2 pairs

of wings. A cicada has 6 legs and one pair of wings. We have all 3 kinds and a

total of 21 insects in a cage. This gives us a total of 140 legs and 23 pairs of wings.

How many insects do we have of each kind?

Example 15. A spider has 8 legs and no wings. A dragonfly has 6 legs and 2 pairs of wings. A cicada has 6 legs and one pair of wings. We have all 3 kinds and a total of 21 insects in a cage. This gives us a total of 140 legs and 23 pairs of wings. How many insects do we have of each kind?

Solution: 7 spiders, 9 dragonflies, and 5 cicadas.

Method 1:

Since both the dragonfly and cicada have 6 legs, we first consider these two

insects to be of one type.

The problem becomes the following:

A spider has 8 legs. A dragonfly-cicada has 6 legs. We have these two kinds and a total of 21 insects in a cage with 140 legs. How many spiders are there?

PROBLEMS

Open Ended

Problem 1. Joe has a cage for his chickens and rabbits. Joe counts 25 heads from

the top of the cage and 80 legs from the bottom of the cage. How many chickens

and how many rabbits does he have?

Problem 1. Joe has a cage for his chickens and rabbits. Joe counts 25 heads from the top of the cage and 80 legs from the bottom of the cage. How many chickens and how many rabbits does he have?

Open Ended

Problem 2. Alex has a cage for his chickens and rabbits. Alex counts 30 heads

from the top of the cage and 88 legs from the bottom of the cage. How many

chickens and how many rabbits does he have?

Problem 2. Alex has a cage for his chickens and rabbits. Alex counts 30 heads from the top of the cage and 88 legs from the bottom of the cage. How many chickens and how many rabbits does he have?

Open Ended

Problem 3. Alex has a cage for his chickens and rabbits. Alex counts 100 legs

from the bottom of the cage. He counts 1 more rabbit than chickens. How many

chickens and how many rabbits does he have?

Problem 3. Alex has a cage for his chickens and rabbits. Alex counts 100 legs from the bottom of the cage. He counts 1 more rabbit than chickens. How many chickens and how many rabbits does he have?

Open Ended

Problem 4. Alex has a cage for his chickens and rabbits. Alex counts 100 legs

from the bottom of the cage. He counts 10 more rabbits than chickens. How

many chickens and how many rabbits does he have?

Problem 4. Alex has a cage for his chickens and rabbits. Alex counts 100 legs from the bottom of the cage. He counts 10 more rabbits than chickens. How many chickens and how many rabbits does he have?

Open Ended

Problem 5. Alex has a cage for his chickens and rabbits. Alex counts 107 heads

from the top of the cage. He counts 56 more rabbit legs than chicken legs. How

many rabbits are there?

Problem 5. Alex has a cage for his chickens and rabbits. Alex counts 107 heads from the top of the cage. He counts 56 more rabbit legs than chicken legs. How many rabbits are there?

Open Ended

Problem 6. Alex has a cage for his chickens and rabbits. Alex counts 176 heads

from the top of the cage. He counts 214 more chicken legs than rabbit legs. How

many chickens and how many rabbits are there?

Problem 6. Alex has a cage for his chickens and rabbits. Alex counts 176 heads from the top of the cage. He counts 214 more chicken legs than rabbit legs. How many chickens and how many rabbits are there?

Open Ended

Problem 7. Alex has a cage for his chickens and rabbits. Alex counts 60 heads

from the top of the cage. He counts 30 fewer chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

Problem 7. Alex has a cage for his chickens and rabbits. Alex counts 60 heads from the top of the cage. He counts 30 fewer chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

Open Ended

Problem 8. Alex has a cage for his chickens and rabbits. Alex counts 24 more

chicken heads than rabbit heads from the top of the cage. He counts 20 more

chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

Problem 8. Alex has a cage for his chickens and rabbits. Alex counts 24 more chicken heads than rabbit heads from the top of the cage. He counts 20 more chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

Open Ended

Problem 9. Alex has a cage for his chickens and rabbits. Alex counts 16 more

chicken heads than rabbit heads from the top of the cage. He counts 14 more chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

Problem 9. Alex has a cage for his chickens and rabbits. Alex counts 16 more chicken heads than rabbit heads from the top of the cage. He counts 14 more chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

Open Ended

Problem 10. Alex has a cage for his chickens and rabbits. Alex counts 11 more

chicken heads than rabbit heads from the top of the cage. He counts 12 more

chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

Problem 10. Alex has a cage for his chickens and rabbits. Alex counts 11 more chicken heads than rabbit heads from the top of the cage. He counts 12 more chicken legs than rabbit legs. How many chickens and how many rabbits does he have?

Open Ended

Problem 11. Alex counts 32 cars and tricycles in a parking lot. He also counts

108 wheels from these cars and tricycles. How many cars and how many tricycles are there in a parking lot?

Problem 11. Alex counts 32 cars and tricycles in a parking lot. He also counts 108 wheels from these cars and tricycles. How many cars and how many tricycles are there in a parking lot?

Open Ended

Problem 12. Bob spent $2.64 to buy 78 stamps each worth either 2-cents or 5-

cents. How many 2-cent stamps and how many 5-cent stamps did he buy?

Problem 12. Bob spent $2.64 to buy 78 stamps each worth either 2-cents or 5-cents. How many 2-cent stamps and how many 5-cent stamps did he buy?

Open Ended

Problem 13. Joe has a cage for his chickens and rabbits. Joe counts 100 legs. If

the number of chickens and rabbits were switched, there would be 86 legs. How many chickens and how many rabbits does he have?

Problem 13. Joe has a cage for his chickens and rabbits. Joe counts 100 legs. If the number of chickens and rabbits were switched, there would be 86 legs. How many chickens and how many rabbits does he have?

Open Ended

Problem 14. Joe has a cage for his chickens and rabbits. Joe counts 118 legs. If

the number of chickens and rabbits were switched, there would be 128 legs. How

many chickens and how many rabbits does he have?

Problem 14. Joe has a cage for his chickens and rabbits. Joe counts 118 legs. If the number of chickens and rabbits were switched, there would be 128 legs. How many chickens and how many rabbits does he have?

Open Ended

Problem 15. A spider has 8 legs and no wings. A dragonfly has 6 legs and 2 pairs

of wings. A cicada has 6 legs and one pair of wings. We have all 3 kinds and a

total of 21 insects in a cage. Given that there are a total of 140 legs and 24 pairs of

wings, how many insects do we have of each kind?

Problem 15. A spider has 8 legs and no wings. A dragonfly has 6 legs and 2 pairs of wings. A cicada has 6 legs and one pair of wings. We have all 3 kinds and a total of 21 insects in a cage. Given that there are a total of 140 legs and 24 pairs of wings, how many insects do we have of each kind?

Open Ended

Problem 16. A spider has 8 legs and no wings. A dragonfly has 6 legs and 2 pairs

of wings. A cicada has 6 legs and one pair of wings. We have all 3 kinds and a

total of 16 insects in a cage. Given that there are a total of 110 legs and 14 pairs of

wings, how many insects do we have of each kind?

Problem 16. A spider has 8 legs and no wings. A dragonfly has 6 legs and 2 pairs of wings. A cicada has 6 legs and one pair of wings. We have all 3 kinds and a total of 16 insects in a cage. Given that there are a total of 110 legs and 14 pairs of wings, how many insects do we have of each kind?