(This problem
was presented on the radio show Car Talk on
You have $100 and you want to spend exactly all of it on three animals that are for sale, dogs, cats, and mice. You must buy exactly 100 animals, and at least one of each animal. Dogs cost $15 each, cats cost $1 each, and mice cost 25 cents each. How many of each animal do you buy?
Variables.
x number of dogs;
y number of cats;
z number of mice.
Equations.
15*x + y + .25*z = 100 the total cost is $100,
x + y + z = 100 the number of animals is 100.
Solution.
By subtracting the equations, we have
14*x - .75*z = 0,
so z = 56/3*x.
But x is a whole number between 1 and 6, because x ≥ 1, and 15*x < 100,
and x is divisible by 3, because z (the number of mice) must be a whole number.
Therefore x = 3 or x = 6.
Case 1, x = 3.
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z = 56/3*x = 56
y = 100 - x - z =
41
There are 3 dogs, 41 cats, and 56 mice.
Case 2, x = 6
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z = 112 (too many mice)
y = 100 - x - z = -15
Also impossible, because y must be positive.
Remark.
In order to prove that they found a solution, students should check that, in case 1, BOTH equations are satisfied.