Nathan spends $11.15 on 7 cups of coffee ($1.75 per cup) and hot chocolate ($1.20 per cup) for his co-workers. How many of each did he buy?

Answer 1

Hi Lilyanne,

This is a great example of a systems of linear equations word problem. Whenever they give you information with multiple categories and totals (usually total cost and total # of items), go ahead and create 2 different equations with the information they give you. In this case, we have 2 different types of information: the number of cups of coffee and hot chocolate, and the cost. 

First, always assign variables, and make them easy to remember:

c= # cups coffee

h= # cups hot chocolate

Let’s look at the cost:

If coffee costs $1.75 each, then multiply $1.75 by however many c cups of coffee we have, and we will get the total cost of all coffees.

If hot chocolate costs $1.20 each, then multiply $1.20 by however many h cups of hot chocolate we have, and we will get the total cost of all hot chocolates.

Add these costs together to get the total amount paid for everything, $11.15.

$1.75c + $1.20h = $11.15

Then, how many cups do we have?

We have c cups of coffee and h cups of hot chocolate, and they’ve told us there are 7 cups total.

c + h = 7

So, we have 2 equations: 

$1.75c + $1.20h = $11.15

c + h = 7

So, make life less complicated and pick one variable to work with. In this instance, since the question asks to solve for both coffee and hot chocolate, it doesn’t really matter which one we choose. 

So, let’s just isolate a variable using the simpler equation. 

c + h = 7

c + h - h = 7 - h

c= 7 - h

And now, let’s substitute in for c into the longer equation:

$1.75c + $1.20h = $11.15

$1.75(7-h) + $1.20h = $11.15

Now that we only have h to work with, we can solve for h using order of operations:

$1.75*7-$1.75*h + $1.20h = $11.15

$12.25 - $1.75h + $1.20h = $11.15

$12.25 - $0.55h = $11.15

$12.25 - $12.25 - $0.55h = $11.15- $12.25

-$0.55h = - $1.10

-$0.55   -$0.55

h= 2

So, Nathan bought 2 cups of hot chocolate. Plug in 2 for h into the simpler equation to solve for c:

c= 7 - h

c=7 - 2

c=5

So, Nathan bought 5 cups of coffee. 

Feel free to ask me for how to tell which variable to solve for—in this case, it didn’t matter, since they asked for both, and both were the same difficulty to solve for. But, sometimes it can save you significant time and energy by solving for one variable rather than the other one. There are also more tricks to use with systems of equations—I’m happy to answer any questions that come up!

Best,

Ginnie