To determine the number of cows, chickens, and spiders in the barn given that there are a total of 320 legs, with the number of chickens being twice the number of cows and the number of spiders being twice the number of chickens, we can follow these steps:
1. Define Variables:
- Let [tex]\( c \)[/tex] represent the number of cows.
- According to the problem, the number of chickens ([tex]\( k \)[/tex]) is twice the number of cows: [tex]\( k = 2c \)[/tex].
- Similarly, the number of spiders ([tex]\( s \)[/tex]) is twice the number of chickens: [tex]\( s = 2k \)[/tex], which converts to [tex]\( s = 2(2c) = 4c \)[/tex].
2. Establish Equations:
- Each cow has 4 legs, thus the total number of legs contributed by cows is [tex]\( 4c \)[/tex].
- Each chicken has 2 legs, making the total number of legs contributed by chickens [tex]\( 2k \)[/tex].
- Each spider has 8 legs, so the total number of legs contributed by spiders is [tex]\( 8s \)[/tex].
- We know the total number of legs is 320, so our equation becomes:
[tex]\[
4c + 2k + 8s = 320
\][/tex]
3. Substitute Variables:
- Substituting [tex]\( k = 2c \)[/tex] into the equation gives:
[tex]\[
4c + 2(2c) + 8s = 320
\][/tex]
- This simplifies to:
[tex]\[
4c + 4c + 8s = 320
\][/tex]
- Substituting [tex]\( s = 4c \)[/tex] into the equation simplifies further to:
[tex]\[
4c + 4c + 8(4c) = 320
\][/tex]
- Combining like terms yields:
[tex]\[
4c + 4c + 32c = 40c = 320
\][/tex]
4. Solve for [tex]\( c \)[/tex]:
- Solving [tex]\( 40c = 320 \)[/tex] for [tex]\( c \)[/tex] gives:
[tex]\[
c = \frac{320}{40} = 8
\][/tex]
5. Calculate [tex]\( k \)[/tex] and [tex]\( s \)[/tex]:
- The number of chickens ([tex]\( k \)[/tex]) is:
[tex]\[
k = 2c = 2(8) = 16
\][/tex]
- The number of spiders ([tex]\( s \)[/tex]) is:
[tex]\[
s = 4c = 4(8) = 32
\][/tex]
Therefore, the numbers of cows, chickens, and spiders in the barn are [tex]\( 8 \)[/tex], [tex]\( 16 \)[/tex], and [tex]\( 32 \)[/tex] respectively.
### Extra Hot:
To find how many spiders there are specifically:
- From the above calculations, we found that the number of spiders is [tex]\( 32 \)[/tex].
### Door Problem Solution:
Let's break down the conditions one by one:
- Three of the numbers are prime numbers: The primes are 2, 11, and 17.
- Two of the numbers are square numbers: The squares are 4 and 16.
- Two of the numbers are multiples of 5: The multiples are 10 and 15.
- Five of the numbers are even: The evens are 2, 4, 10, 16, and 18.
- Three is a factor of two of the numbers: The multiples of 3 are 15 and 18.
- There is a pair of consecutive numbers, and four consecutive numbers:
- Consecutive pairs are: (10, 11) or (17, 18)
- Four consecutive numbers are: 15, 16, 17, 18.
All conditions can be met with the door numbers being:
[tex]\[
2, 4, 10, 11, 15, 16, 17, 18
\][/tex]
Which sums to 93, and the solution satisfies all the constraints given in the problem.
