Interpreting the Quotient

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

By reading this lesson, you'll learn a visual way that can help you understand division problems and their resulting answers. Learn how you can use this visual way to help you solve division problems.

What is a Quotient?

When you first learn about the quotient, you learn that it is the answer to a division problem, the result you get when you divide one number by another. For example, look at this problem.


interpret the quotient


The 10 here is the quotient, the answer to your division problem.

While this gives you an intellectual explanation for a quotient, you may not fully understand just what the quotient means. Yes, it's the answer to a division problem, but how does that help you? This is what you'll learn in this lesson, a new way of interpreting the quotient that will help you to really understand what the answer to your division problem means.

A Division Problem

Let's go back to the problem you just saw, the 5 divided by 1/2.


interpret the quotient


It equals 10, but what exactly does that mean? You already know that division means you are splitting your original number into equal parts, so how is it possible that your 5 is being split into more than 5 parts? It's easy when you have, say, 6 divided by 3. That's easy to visualize. You can see six candies, and you can visually move them into groups of 3. You end up with 2, so 6 divided by 3 is 2. But when you bring fractions into the mix, it gets a little bit more complicated.

A Visual Representation

This is where a visual representation can help you make sense of it. In this visual representation, you'll use boxes to represent your numbers. For the 5, you'll draw five boxes stacked on top of each other.


interpret the quotient


Then when you divide by the 1/2, you'll separate each of your five boxes in half, so you can easily find your answer. The rule here is, if you are dividing by a fraction, you'll want to separate your unit boxes into the number shown in the denominator. So for 1/2, the denominator is 2, so you separate each unit box into two boxes.


interpret the quotient


After you've divided each of your five boxes in half, all you need to do now is to count how many 1/2 boxes you have. This is because 1/2 is one of those squares, so to figure out how many halves are in 5, you count how many half boxes there are. You count, and you get 10. The rule here is you count by the number that is the numerator of the fraction you are dividing by. So, for 1/2, the numerator is 1, so you count by ones.

Example

Let's look at another example.


interpret the quotient


This time you are dividing 1 by 3/4. Is this something you can wrap your head around? Probably not. But if you use the block visualization method, you may be able to see it better.

You begin by drawing a single block for your 1.


interpret the quotient


Then you divide your single block into quarters since the denominator of the number you are dividing by is 4.


interpret the quotient


Now you can count your boxes to find out your answer. This time, though, you'll be counting by three's since the numerator of the fraction you are dividing by is 3.


interpret the quotient


The first set of 3 gets you to 1. Now, you only have one block left. Because you are counting by threes, this one leftover block is a one-third, 1/3. So your answer is 1 1/3.

Decimals

If your problem has decimals, you can first change all your numbers into whole numbers, and then you can continue dividing the way you know how with whole numbers. Take, for example, this problem.


interpret the quotient


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