Help with this word problem below

Question

An office supply store recently sold a black printer ink cartridge for

​$19.99

and a color printer ink cartridge for

​$27.99

At the start of a recent fall​ semester, a total of

444

of these cartridges was sold for a total of

​$991.56.

How many of each type were​ purchased?

Answer 1

I wonder if you have a typo somewhere in your problem statement. Perhaps there are 44 printer cartridges sold instead of 444? Even if all 444 cartridges sold were the cheapest type, they would still cost significantly more than 991.56. I will show you how to do the problem assuming 44 cartridges, rather than 444, were sold. However, if there is a different issue, the method will still work. Simply substitute in different values but follow the same basic steps.

Let b = the number of black printer cartridges sold, and

Let c = the number of color printer cartridges sold.

We'll start by translating the information given from words into equations. Consider the first statement: "At the start of the fall semester, a total of 44 of these cartridges was sold." This means that:

b + c = 44 (equation 1)

Now consider the second statement: The total cost of all the cartridges was $991.56. In order to use this information, we need a way of converting the number of cartridges of each type to a cost. We can do this using the cost of each cartridge given at the start of the problem. Black cartridges cost $19.99, and color cartridges cost $27.99. Consequently:

19.99*b + 27.99*c = 991.56 (equation 2)

Alright. Now we have two equations with two unknowns (the unknowns are b and c). This means that we can solve the system of equations because we have as many pieces of information (ie equations) as we have unknowns. There are a number of ways that we can solve this system of equations. I will write out the solution using substitution. First, we solve equation 1 for b, and then we plug that into equation 2. Finally, we go back and solve for b.

solving equation 1 for b:

b+c = 44

b = 44-c

plugging this new equation into equation 2:

19.99b +27.99c = 991.56

19.99(44-c) + 27.99c = 991.56

879.56-19.99c+27.99c = 991.56

8c = 112

c = 14

going back to solve for b:

b = 44-c

b = 44-14

b = 30

14 color cartridges and 30 black cartridges were sold. Let me know if you have any questions.