Math - Linear algebra

posted by .

Cats cost $1, dogs cost $15, and mice cost only $0.25. I want a herd of 100 animals, and I have $100, so being a cat lover, it's tempting to get only cats and mice. But I also want a dog, and my cat(s) want mice to "play" with, so the numbers of dogs, cats and mice are each positive. And no cash left over. What's the solution?

  • Math - Linear algebra -

    C = cats
    D = dogs
    M = mice

    1(C) + 15D + .25M = 100 or
    4C + 60D + M = 400 (1)

    Also, C + D + M = 100 (2)

    Subtracting (1) from (2) yields 3C + 59D = 300

    Dividing through by the lowest coefficient yields
    C + 19D + 2D/3 = 100

    Let 2D/3 = an integer "k" making D = 3k/2

    Substituting, 3C + 88.5k = 300 yielding C = 100 - 29.5k

    "k" must be even and can only be 2 making C = 100 - 59 = 41 from which D = 3 and m = 56.

    41 + 45 + 14 = 100 animals
    41(1) + 45(15) + 56/4 = $100

  • Math - Linear algebra -

    Another way:

    Will use the same notations, C,D and M for cats, dogs and mice, and with the same constraints where C,D and M must be integers.

    From the cost,
    C+15D+M/4=100....(1)
    from the number of animals,
    C+D+M = 100......(2)

    (1)-(2)
    14D=(3/4)M, or
    56D=3M
    Since 56 and 3 are coprime, we can only have 3 dogs, 56 mice, or any multiple thereof.
    However, 6 dogs and 112 mice would exceed the budget, so we go with
    3 dogs, 56 mice, and are left with 41 cats.
    Check:
    3+56+41=100
    3*15+56/4+41=100
    So the solution is correct.

  • Math - Linear algebra -

    Thank you, that was the answer I got too, but in a slightly different method, but I want to learn for how you figure this out, with this problem. I feel like you just you the same method but I am having a little problem with this one:

    "Suppose you want to make $5 using exactly 100 common US coins.Easy, you say: just use nickels (5 cents each). But I say, no - we have NO nickels, only pennies (1 cent), dimes (10 cents) and quarters (25 cents) - is it possible to make $5 now with exactly 100 of these coins? Why or why not?"

  • Math - Linear algebra -

    These problems belong to a class called diophantine equations. If these are what you are doing in class, they will be solved in a different way.

    For linear algebra, I will solve it similar to the previous problem, as follows.

    Let integers
    p=number of pennies
    d=number of dimes
    q=number of quarters
    then again,
    p+10d+25q=500...(1) (number of cents)
    p+d+q=100 ....(2) (number of coins)
    Subtract (2) from (1),
    9d+24q=400 or
    3d+8q = 400/3 ...(3)
    It is evident that (3) has no solution in integers because we cannot have the sum of two integers to be a mixed fraction.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Why dogs are better than Cats!

    why are dogs better than cats? many opins needed! I dunno. They're more active. More playful. Not afraid to get dirty. ANything specific that you need?
  2. algebra math

    The following pie chart represents the results of a survey about whether people in a certain town have cats or dogs as pets. Neither 6% Both 22% Cats only 28% Dogs only 44% If there is 300 people in the town, how many people have neither …
  3. Essay Writing

    I need help with making my outline parallel. II. Body A. Needs 1. Feeding a. Cats b. Dogs 2. Training a. Cats b. Dogs 3. Grooming a. Cats b. Dogs 4. Medicating a. Cats b. Dogs B. Personalities 1. Cats a. Playing b. Vocalizing c. Independence
  4. Essay Writing

    I need help with parallel structure. My essay is contrasting cats and dogs, with cats being the superior (sorry dog lovers). One paragraph is to be block method the other is to be point by point. II. Body A. Needs 1. Feeding a. Cats …
  5. Math

    Two cats are taking a walk. 1 cat runs off. three cats come back. 1 cat sits down. How many cats in all are there if 6 more cats came back with one of the cats being the one that ran off?
  6. Statistics

    The American Veterinary Association claims that the annual cost of medical care for dogs averages $100, with a standard deviation of $30, and for cats averages $120, with a standard deviation of $35. A)Whats the expected difference …
  7. math

    If there were only cats and dogs at the pound and the ratio of cats to dogs was 3:4 if there were 105 cats, how many animals altogether?
  8. Algebra 1- Math

    The owner of a pet store spent $236 on 10 dogs and cats. The cats cost &18 each and the dogs cost &26 each. How many of each did the owner buy?
  9. Math

    In its monthly report, the local animal shelter states that it currently has 24 dogs and 18 cats available for adoption. 8 of the dogs and 6 of the cats are male. If two animals are selected at random, what is the probability that …
  10. Mathematics question

    In my town some of the animals are really strange. Ten percent of the dogs think that they are cats and ten percent of the cats think they are dogs. All the other cats and dogs are perfectly normal. One day I tested all the cats and …

More Similar Questions