How to find the solution

Question

Ali wants to surprise his wife Sara by presenting her some flowers, when he returns back from a work tour.

He plans to spend exactly $24 on a bunch of exactly two dozen flowers.

Sara loves lilies, roses and daisies. At the flower market they are selling lilies for $3 each, roses for $2 each, and daisies $0.50 each. How many flowers of each type can Ali buy?

Answer 1

Let

1. x be the number of lilies.
2. y be the number of roses, and
3. z be the number of daisies.

We assume that x, y, and z are positive integers. (Note that z must be even since each daisy is $0.50 while each of the other two flowers is an integer dollar.)

We have a system of two linear Diophantine equations in three variables.

1. x + y + z = 24 (number of flowers)
2. 3x + 2y + 0.5z = 24 (dollars)

We can multiply the second equation by 2 so that the coefficients are integers:

1. 6x + 4y + z = 48

Subtracting the first equation from the second, we get

1. 5x + 3y = 24

Note that gcd(5, 3) = 1, so we can use the Euclidean Algorithm:

1. 5 = 1 x 3 + 2
2. 3 = 1 x 2 + 1

So 1 = (1 x 3) - (1 x 2) = (1 x 3) - 1 x (5 - 1 x 3) = (1 x 3) - (1 x 5) + (1 x 3) = 2 x 3 + (-1) x 5.

Multiplying both sides by 24, we get 24 = (48) 3 + (-24) 5.

Then

1. x = 48 + (5/1) t = 48 + 5t
2. y = -24 - (3/1) t = -24 - 3t
3. z = 24 - x - y = 24 - (48 + 5t) - (-24 - 3t) = -2t
4. where t is any integer

We want x > 0, y > 0, and z > 0, so

1. 48 + 5t > 0 or t > -48/5 = -9 3/5
2. -24 - 3t > 0 or t < -8
3. -2t > 0 or t < 0

The only integer value of t that satisfies all three inequalities is t = -9. Therefore,

1. x = 48 + 5(-9) = 3
2. y = -24 - 3(-9) = 3
3. z = -2(-9) = 18

Therefore, Ali can buy 3 lilies, 3 roses, and 18 daisies.