farmer brown just bought 100 animals. He spent $600 but can't remember how many of each animal he bought. Geese are $21, guinea pigs are $8 and chooks are $3. He remembers he bought an even number of geese. How many of each did he buy?

Question

farmer brown just bought 100 animals. He spent $600 but can't remember how many of each animal he bought. Geese are $21, guinea pigs are $8 and chooks are $3. He remembers he bought an even number of geese. How many of each did he buy?

Answer 1

Facts as presented:

g+p+c = 100
21g+8p+3c=600

since c = 100-(g+p),
21g+8p+3(100-g-p) = 600
21g+8p+300-3g-3p = 600
18g+5p = 300

Since 5p = 300-18g, g must be a multiple of 5. But, since g is even, it must be 10,20,30,...

If g=10, p=24
If g=20, 18g=360, so any number g>10 will not work.

So, we have 10 geese, 24 pigs, and 66 chooks

Answer 2

So, we have 10 geese, 24 pigs, and 66 chooks

Answer 3

Let's assume Farmer Brown bought x geese, y guinea pigs, and z chooks.

According to the given information:
x + y + z = 100 [Equation 1]
21x + 8y + 3z = 600 [Equation 2]
x is an even number.

We can start by solving Equation 1 for z:
z = 100 - x - y [Equation 3]

Next, substitute Equation 3 into Equation 2:
21x + 8y + 3(100 - x - y) = 600

Simplifying the equation:
21x + 8y + 300 - 3x - 3y = 600
18x + 5y = 300 [Equation 4]

Since we know x is an even number, we can substitute x = 2n into Equation 4:
18(2n) + 5y = 300
36n + 5y = 300 [Equation 5]

Now, we need to find the values of n and y that satisfy Equation 5.

Let's solve for y in terms of n:
5y = 300 - 36n
y = (300 - 36n) / 5 [Equation 6]

Since y must be a whole number, we need to find values of n that make (300 - 36n) divisible by 5.

Let's check the values of n from 1 to 8:
For n = 1, y = (300 - 36) / 5 = 52.8 (not a whole number)
For n = 2, y = (300 - 72) / 5 = 45.6 (not a whole number)
For n = 3, y = (300 - 108) / 5 = 38.4 (not a whole number)
For n = 4, y = (300 - 144) / 5 = 31.2 (not a whole number)
For n = 5, y = (300 - 180) / 5 = 24 (whole number)
For n = 6, y = (300 - 216) / 5 = 16.8 (not a whole number)
For n = 7, y = (300 - 252) / 5 = 9.6 (not a whole number)
For n = 8, y = (300 - 288) / 5 = 2.4 (not a whole number)

Therefore, n = 5 is the only valid value. Substituting n = 5 into Equation 6, we get:
y = (300 - 36*5) / 5 = 24

Now, substitute n = 5 and y = 24 into Equation 1:
x + 24 + z = 100
x + z = 76 [Equation 7]

Since we know x and z must be even numbers, let's try different combinations:

For x = 2 and z = 74, Equation 7 is satisfied, but this doesn't satisfy Equation 2 (21*2 + 8*24 + 3*74 = 645)

For x = 4 and z = 72, Equation 7 is satisfied, but this doesn't satisfy Equation 2 (21*4 + 8*24 + 3*72 = 630)

For x = 6 and z = 70, Equation 7 is satisfied, but this doesn't satisfy Equation 2 (21*6 + 8*24 + 3*70 = 615)

For x = 8 and z = 68, Equation 7 is satisfied, and this satisfies Equation 2 (21*8 + 8*24 + 3*68 = 600)

Therefore, Farmer Brown bought 8 geese, 24 guinea pigs, and 68 chooks.

Answer 4

To find out how many of each animal Farmer Brown bought, we can use algebraic equations.

Let's assume that Farmer Brown bought x geese, y guinea pigs, and z chooks.

From the information given in the problem:

1. He bought a total of 100 animals, so we can write the equation:
x + y + z = 100

2. He spent a total of $600, so we can also write the equation:
21x + 8y + 3z = 600

3. He bought an even number of geese, which means x must be an even number.

Now, we need to solve these equations simultaneously to find the values of x, y, and z.

Before we move forward with the solution, there is something important to note. We have three variables (x, y, and z) but only two equations. This means we cannot find the unique values for x, y, and z because there are multiple solutions possible. However, we can find a set of values that satisfy both equations.

One way to solve these equations is by using a method called substitution. Let's solve them step by step:

Step 1: Solve the first equation for x:
x = 100 - y - z

Step 2: Substitute the value of x in the second equation:
21(100 - y - z) + 8y + 3z = 600

Step 3: Simplify the equation:
2100 - 21y - 21z + 8y + 3z = 600
-13y - 18z = -1500

Step 4: Rearrange the equation:
13y + 18z = 1500 (multiply both sides by -1 to get rid of the negative)

Now we have two equations:
-13y - 18z = -1500
13y + 18z = 1500

Step 5: Add both equations:
-13y - 18z + 13y + 18z = -1500 + 1500
0 = 0

As we can see, the two equations result in 0 = 0. This means that the system of equations is dependent, and there is not a unique solution. Instead, an infinite number of solutions exist.

Therefore, we cannot determine the exact number of each animal Farmer Brown bought with the given information.