Question

John spends $94 to buy packages of cleaning supplies for a camo. He buys t packages of towels; each package is $6 and contains 8 towels. He also buys s packages of soap; each package is $4 and contain 10 bars of soap. John buys 20 packages of cleaning supplies altogether.

a. Write a system of two equations to represent the packages of cleaning supplies John buys.

b. How many packages of towels does John buy?

c. Explain why it would not be possible for John to buy exactly 16 bars of soap.

Answer 1

Answer: a

Explanation:

a. Let t be the number of packages of towels and s be the number of packages of soap. Then the system of equations is:

6t + 4s = 94 (the total cost of cleaning supplies is $94)

8t + 10s = 20 (the total number of packages of cleaning supplies is 20)

b. To solve for t, we can use the second equation and substitute 20 - s for the total number of packages:

8t + 10s = 20

8t + 10(20 - s) = 20

8t + 200 - 10s = 20

8t = 10s - 180

t = (10/8)s - 45/4

We know that t is a positive integer, so we can test different values of s to find the solution. If we try s = 4, then t = 1.25, which is not a whole number. If we try s = 5, then t = 1.875, which is also not a whole number. If we try s = 6, then t = 2.5, which is a whole number. Therefore, John buys 2 packages of towels.

c. It would not be possible for John to buy exactly 16 bars of soap because each package of soap contains 10 bars. Therefore, the total number of bars of soap he buys must be a multiple of 10. If he buys s packages of soap, then the total number of bars he buys is 10s. From the first equation, we know that 4s is a multiple of 2, so the total cost of soap must be a multiple of 2. However, 16 is not a multiple of 2, so it is not possible for John to buy exactly 16 bars of soap.