Introduction
It is possible to solve many division problems involving fractions and whole numbers visually using picture models.
This lesson will go through the process of representing the division of whole numbers and fractions using models.
Modeling Dividing Whole Numbers by Fractions
In general, the process for representing the division of whole numbers by fractions can be broken down into the following steps:
Step 1  Use a shape such as a rectangle or circle to represent a whole.
Step 2  Split this shape up into as many pieces as the denominator of the fraction.
Step 3  Count in groups of the fraction we are dividing by for as many wholes as we are dividing by.
This process will make more sense once we take a look at a few examples.
Modeling Whole Numbers Divided by Fractions with Numerators of 1
Let's take a look at dividing 2 by 1/3.
2 ÷ 1/3 = ?
Our first step is to use a shape to represent our whole. In this case, we will use a rectangle.
Step 2  Next, we will split our shape up into as many pieces as the denominator of our fraction. In this case, our denominator is 3 so we will split our shape up into 3 pieces.
This tells us that 1 divided by 1/3 is equal to 3.
1 ÷ 1/3 = 3
Step 3  Since we are trying to solve for 2 divided by 1/3, we need to multiply our result from the previous step by 2.
This tells us that 2 divided by 1/3 is equal to 6.
2 ÷ 1/3 = 6
3 × 2 = 6
Let's take a look at a more complex problem.
Modeling Whole Numbers Divided by Fractions with Complex Numerators
Let's take a look at dividing 3 by 3/8.
3 ÷ 3/8 = ?
Our first step is to use a shape to represent our whole. In this case, we will use a rectangle.
Step 2  Next, we will split our shape up into as many pieces as the denominator of our fraction. In this case, our denominator is 8 so we will split our shape up into 8 pieces.
Step 3  Since we are trying to solve for 3 divided by 3/8, we need to figure out how many times 3/8 goes into our whole. We so this by circling every 3/8.
Since we are dividing 3 by 3/8, we will need to do this for 3 wholes.
Step 4  Continue circling every groups of 3/8 until we reach the end of our wholes.
Step 5  Count the number of groups circled. This tells us that 3 divided by 3/8 is equal to 8.
Sometimes, we won't be able to circle perfect groups like you see above. In this situation, we will have a remainder.
Modeling Whole Numbers Divided by Fractions with Remainders
Let's take a look at dividing 2 by 3/5.
2 ÷ 3/5 = ?
Our first step is to use a shape to represent our whole. In this case, we will use a rectangle.
Step 2  Next, we will split our shape up into as many pieces as the denominator of our fraction. In this case, our denominator is 5 so we will split our shape up into 5 pieces.
Step 3  Since we are trying to solve for 2 divided by 3/5, we need to figure out how many times 3/5 goes into our whole. We can do this by circling groups of 3/5.
Since we are dividing 2 by 3/5, we will need to do this for 2 wholes.
Step 4  After circling groups of 3/5, we have 3 groups of 3/5 and are left over with 1/5.
Step 5  Calculate the remainder. Since we have 1/5 left over, we figure out what fraction of 3/5 our 1/5 is equal to. Visually, you can see that 1/5 is a third of 3/5.
This tells us that 2 divided by 3/5 gives us the mixed number 3 and 1/3.
2 ÷ 3/5 = 3 1/3
As you have seen, there are 3 tiers of difficulties when we divide whole numbers by fractions. It is important to remember how to deal with each situation.
Modeling Dividing Fractions by Whole Numbers
The process for dividing fractions by whole numbers is similar, but has some key differences.
Step 1  We will always begin these problems by creating a model to represent our original fraction.
Step 2  Then, we will split our model into as many pieces as the whole number we are dividing by.
Our numerator will never change when we divide a fraction by a whole number.
Alternately, we can multiply the denominator by the number we are dividing by to find our new denominator.
Example: Modeling Fractions Divided by Whole Numbers
Let's take a look at dividing 1/4 by 3.
1/4 ÷ 3 = ?
Our first step is to create a model to represent our fraction.
Next, we will split our shape into as many pieces as the whole number we are dividing by.
In this case, we will split our shape up into 3 since we are dividing by 3.
When we divide a fraction by a whole number, our numerator will never change.
The only part that will change is going to be our denominator.
Now that we have split up our shape we can count the total number of rectangles. This will be our new denominator.
In this case we have 12 rectangles, so our denominator will be 12.
1/4 ÷ 3 = 1/12
For bigger numbers, it does not make sense to count each individual rectangle.
Instead, we could multiply 3 by 4 to get our new denominator.
Let's take a look at solving a different problem.
Modeling Fractions Divided by Whole Numbers
Let's take a look at dividing 2/5 by 4.
2/5 ÷ 4 = ?
Our first step is to create a model to represent our fraction.
Next, we will split our shape into as many pieces as the whole number we are dividing by.
In this case, we will split our shape up into 4 since we are dividing by 4.
As we established last time, our numerator will never change when we divide a fraction by a whole number.
We can now count the number of rectangles to find our new denominator, or we can multiply 5 by 4 to find the value of our denominator.
5 × 4 = 20
Our numerator will remain 2 as it is in the original fraction, and our new denominator is 20.
2/20 can be reduced down to equal 1/10.
2/5 ÷ 4 = 1/10
We have now seen multiple situations involving the division of fractions and whole numbers.
Keep in mind the techniques we have seen as they can help you quickly solve otherwise difficult division problems.

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