Farmer Ben has only ducks and cows. He can't remember how many each he has, but he really doesn't need to remember. He does know that he has 22 animals, which is his age. He also remembers that those animals have a total of 56 legs, which is his father's age. Assuming that each animal is normal, how many of each does Farmer Ben have?

Question

Farmer Ben has only ducks and cows. He can't remember how many each he has, but he really doesn't need to remember. He does know that he has 22 animals, which is his age. He also remembers that those animals have a total of 56 legs, which is his father's age. Assuming that each animal is normal, how many of each does Farmer Ben have?

Answer 1

Suppose he's got D ducks and C cows. Then we know that D + C = 22.

Now: a duck has two legs, and a cow has four. So 2D + 4C = 56, which is the total number of legs. But D + C = 22, so 2D + 2C = 44. So if you subtract this equation from the one that tells you the number of legs, you'll get 2C = 56-44 = 12, so C = 6. But D + C = 22, so D = 16.
Check it: If C = 6 and D = 16, then D + C = 22, and 2D + 4C = 32 + 24 = 56. Correct.

Answer 2

Well, it seems Farmer Ben's memory is a bit fuzzy, but luckily math is here to save the day! Let's solve this animal mystery.

Let's use a little algebra to find the solution. Let's say Farmer Ben has "d" ducks and "c" cows.

Given that he has 22 animals, we can write the equation: d + c = 22.

Since ducks have 2 legs and cows have 4 legs, we can also write the equation: 2d + 4c = 56.

Now we have a system of equations.

Let's start solving! Multiply the first equation by 2 to eliminate the variable "d":
2d + 2c = 44.

Now we have the equations:
2d + 4c = 56,
2d + 2c = 44.

Subtract the second equation from the first equation to eliminate the variable "d":
(2d + 4c) - (2d + 2c) = 56 - 44,
2c = 12.

Divide both sides by 2:
c = 6.

Now that we know c = 6, we can substitute this value back into the first equation to find d:
d + 6 = 22,
d = 22 - 6,
d = 16.

So, Farmer Ben has 16 ducks and 6 cows. I hope he can count on his animals (and math) a little better next time!

Answer 3

Let's denote the number of ducks as "D" and the number of cows as "C."

From the information given, we can form two equations:

Equation 1: D + C = 22 (since Farmer Ben has 22 animals)
Equation 2: 2D + 4C = 56 (since ducks have 2 legs and cows have 4 legs)

We can now solve these two equations to find the values of D and C.

To do this, we can first solve equation 1 for D:
D = 22 - C

Now substitute this value of D into equation 2:
2(22 - C) + 4C = 56

Distribute the 2:
44 - 2C + 4C = 56

Combine like terms:
2C + 44 = 56

Subtract 44 from both sides:
2C = 56 - 44
2C = 12

Divide both sides by 2:
C = 6

Now substitute the value of C back into equation 1 to find D:
D + 6 = 22
D = 22 - 6
D = 16

Therefore, Farmer Ben has 16 ducks and 6 cows.

Answer 4

To solve this problem, we can set up a system of equations based on the information given.

Let's assume the number of ducks is 'D' and the number of cows is 'C'.

From the given information:
1. The total number of animals is 22: D + C = 22
2. The total number of legs is 56: 2D + 4C = 56

Now we can solve this system of equations to find the values of D and C.

First, let's rearrange the equation D + C = 22 to solve for D:
D = 22 - C

Substituting this value of D in the second equation:
2(22 - C) + 4C = 56

Now, simplify and solve for C:
44 - 2C + 4C = 56
2C = 12
C = 6

Now substitute this value of C back into the equation D + C = 22:
D + 6 = 22
D = 22 - 6
D = 16

Therefore, Farmer Ben has 16 ducks and 6 cows.