Calculating Total Cost: Relationship with Units & Rate

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Understanding the relationship between cost, rate, and number of units will allow you to know how much a group of items will cost, determine the cost of a single item, and find the number of things you can buy with a certain amount of money.

The Relationship Between Cost, Rate, and Number of Units

Imagine that you want to purchase 5 books that cost $3 each. What will be the total cost? Before we can figure out how much money you need to buy your new books, there are three important terms that we need to define: cost, rate, and number.

Cost (c) is the total cost of the purchase.

Rate (r) is the cost of EACH individual item.

Number (n) is the number of items you want to purchase.

These are related in the following way:

c = nr

This means that to find the total cost of a group of items, you need to multiply the rate (r) by the number (n).

For example, if you want to buy those 5 books that cost $3 each, the total cost will be $15.

n = 5 and r = $3, so c = (5) ($3) = $15

Find the Number when Cost and Rate are Known

In addition to finding the total cost of a group of items, you may also need to find the number of items when you know the cost and rate. For example, suppose you go to the store to buy some apples and see that apples are $0.88 each. You have $10 and want to know how many apples you can buy.

In this situation, you know the cost ($10) and the rate ($0.88), but you don't know the number you can buy. Even though this seems different from the first problem at first, you can still use the same relationship. It just needs to be rearranged first.

Start with c = nr. Then to find n, you need to divide both sides of the equation by r. This gives you n = c / r. Now, you are ready to find the number of apples you can buy.

n = c / r = $10 / $0.88 = 11.4 apples

Since the store probably won't let you cut up an apple before you buy it, this means you can only buy 11 apples with your $10.

Find the Rate when Number and Cost are Known

Another common problem involves finding the rate when you know the number and cost. For example, if you started a new job and were told you would be paid $100 for each eight-hour day you worked, how much money would you be making per hour?

Once again, start with c = nr. This time, to find r, you need to divide both sides of the equation by n. This gives you r = c / n. In this situation, 8 would be the number, and $100 would be the cost (even though you aren't buying anything, it is still the total amount).

r = c / n = $100/8 = $12.50

So, you will be making $12.50 each hour that you work at your new job.

Practice Problems

Now that you have seen how to find the cost, rate, or number, let's look at a few practice problems to make sure that you really understand all the different ways to use this formula.

Problem #1

Oranges cost $1.50 a pound. If you buy 5 lbs of oranges, how much will it cost?

Solution:

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