5th Grade Math

Adding, Subtracting, Multiplying, and Dividing Fractions

Unit 3


Adding, Subtracting, Multiplying, and Dividing Fractions

What is a Fraction?

You probably already know what a fraction is, but we are going to learn another way to view a fraction!

  • A fraction is a division of the numerator by the denominator (a/b = a ÷ b)

  • 3/5 can be interpreted as “3 divided by 5 and as 3 shared by 5” since 3 parts are being shared between a total of 5 people or objects

    • So, 3/5 isn't just "three-fifths", it can also be interpreted as 3 divided by 5!

Here is an example:

  1. Ten team members are sharing 3 boxes of cookies. How much of a box will each student get?

  • When working this problem, we recognize that the 3 boxes are being divided into 10 groups, so we see the solution to the following equation, 10 x n = 3 (10 groups of some amount is 3 boxes) which can also be written as n = 3 ÷ 10. Using models or diagram, we divide each box into 10 groups, resulting in each team member getting 3/10 of a box.

Adding Fractions

To add fractions there are Three Simple Steps:

  • Step 1: Make sure the bottom numbers (the denominators) are the same

  • Step 2: Add the top numbers (the numerators), put that answer over the denominator

  • Step 3: Simplify the fraction (if needed)

We'll look at each step in detail!

First, how do we find the least common denominator (what is that, anyway?)?

Finding a Common Denominator

But what should the new denominator be?

One simple answer is to multiply the current denominators together:

3 x 6 = 18

So instead of having 3 or 6 slices, we will make both of them have 18 slices.

The pizzas now look like this:

Least Common Denominator

That is correct... however, it is important to note that while multiplying the denominators will always give a common denominator but may not result in the smallest denominator.



Now, let's try using the least common denominator, also known as the LCD!

We want both fractions to have 6 slices:

  • When we multiply top and bottom of 1/3 by 2 we get 2/6

  • 1/6 already has a denominator of 6


A good model to use when adding fractions is a clock! This clock is showing 1/3 + 1/6. The clock hand moving from 12 o'clock to 4 o'clock is showing 1/3rd of the clock while the clock hand moving from 4 o'clock to 6 o'clock is showing 1/6th of the clock. (If you don't understand this, think about a clock as having 12 parts, so each part is 1/12th of the clock.)

Practice makes perfect! Practice adding two fractions with different denominators! Hint: You can use equivalent fractions to get two fractions with same denominators!

Subtracting Fractions

There are 3 simple steps to subtract fractions

  • Step 1. Make sure the bottom numbers (the denominators) are the same

  • Step 2. Subtract the top numbers (the numerators). Put the answer over the same denominator.

  • Step 3. Simplify the fraction (if needed).

Example:

  • 1/2 - 1/6

Step 1. The bottom numbers are different. See how the slices are different sizes? We need to make them the same before we can continue, because we can't subtract them like this:

To make the bottom numbers the same, multiply the top and bottom of the first fraction (1/2) by 3 like this:

And now our question looks like this:

The bottom numbers (the denominators) are the same, so we can go to step 2.

Step 2. Subtract the top numbers and put the answer over the same denominator:

Step 3. Simplify the fraction: 2/6 = 1/3

Still confused? Watch this video for a tutorial on how to add fractions with unlike denominators!

A little stuck? Watch this video to gain a better understanding of how to subtract fraction with unlike denominators!

Practice subtraction of fractions with different denominators! Hint: You can use equivalent fractions to get two fractions with same denominators.

Practice subtraction with mixed numbers! Remember, you can convert mixed numbers to fractions before subtracting.

Word Problems with Fractions

Fractions often show up in word problems, so it's important that we learn how to solve them! Let's learn from the examples and sample responses given below:

Example 1: Your teacher gave you 1/7 of the bag of candy. She also gave your friend 1/3 of the bag of candy. If you and your friend combined your candy, what fraction of the bag would you have? Estimate your answer and then calculate. How reasonable was your estimate?

Here are two student responses, both are correct! Your response may have been similar to one of these. Another example is given below. This time we are using a bar diagram!

Example 2: Sonia had 2 and 1/3 candy bars. She promised her brother that she would give him 1/2 of a candy bar. How much will she have left after she gives her brother the amount she promised?

In the next examples, we will use area models! Using different models can help us to visualize the problem and understand it better.

Example 3 and 4:

Good job for learning all the way to the point! We are on our last example, and hopefully you have a better understanding of how to solve word problems with fractions :)

Example 5: Elli drank 3 /5 quart of milk and Javier drank 1 /10 of a quart less than Ellie. How much milk did they drink all together?

You try! Josiah was making two different types of cookies. One recipe needed 3/4 cup of sugar and the other needed 2/3 cup of sugar. How much sugar did he need to make both recipes? Try to do a mental estimation if you can!

Mental estimation: You may have said that Josiah needs more than 1 cup of sugar but less than 2 cups. Your explanation may compare both fractions to 1/2 and state that both are larger than 1/2 so the total must be more than 1. In addition, both fractions are slightly less than 1 so the sum cannot be more than 2.

This graphic show that when multiplying a fraction by a whole number, we can convert the whole number into a fraction by putting it over 1 (like how the 3 become 3/1). This is because dividing a number by 1 still equals to the original number, so we are not changing the number itself. In this case, we are not changing the value of the 3 by putting it over 1.

Let's watch how to multiply fractions with fraction models!

Still confused about multiplying fractions? Follow along with this video!

Multiplying Fractions

One important thing to remember is that multiplication of a fraction by a whole number is the same as repeated addition of a unit fraction For example, 2 x (1/4) = 1/4 + 1/4.

Let's learn how to solve multiplying fractions from a worked-out example and sample student responses!

Example 1: Three-fourths of the class is boys. Two-thirds of the boys are wearing tennis shoes. What fraction of the class are boys wearing tennis shoes?

This question is asking what is 2/3 of 3/4 what is 2/3 x 3/4? In this case, we have 2/3 groups of size 3/4. (A way to think about it in terms of the language for whole numbers is by using an example such as 4 x 5, which means you have 4 groups of size 5.)

Below is an example of solving the problem using a bar diagram!

Here are a few other ways to solve this question! As you can see, there are many ways to multiply fractions, and you can see which way works best for you.

Another important concept about multiplying fractions is using it to find the area of a rectangle by tiling it with unit squares. Don't worry if you don't know what that means, we will learn all about it!

Example: The home builder needs to cover a small storage room floor with carpet. The storage room is 4 meters long and half of a meter wide. How much carpet do you need to cover the floor of the storage room? Use a grid to show your work and explain your answer.

In the grid below we shaded the top half of 4 boxes. When we added them together, we added ½ four times, which equals 2. We could also think about this with multiplication ½ x 4 is equal to 4/2 which is equal to 2.

Ready to have some fun? Play this game to practice multiplying fractions!

Multiplication is Scaling!

To better understand multiplication of fractions, we can compare the size of a product to the size of one factor on the basis of the size of the other factor, without actually solving the multiplication.

Let's look at an example below!


Now, let's think about a couple of questions:

  1. Why is it that when we are multiplying by a fraction greater than 1, the number increases?

  2. Why is it that when we are when multiplying by a fraction less the one, the number decreases?

To answer these questions, we will review the example problem below!

Example: Mrs. Bennett is planting two flower beds. The first flower bed is 5 meters long and 6/5 meters wide. The second flower bed is 5 meters long and 5 /6 meters wide. How do the areas of these two flower beds compare? Is the value of the area larger or smaller than 5 square meters? Draw pictures to prove your answer.

We can answer this question since we know 2 and 2/3 x 8 must be more than 8 because 2 groups of 8 is 16 and 2 and 2/3 is almost 3 groups of 8. So, the answer must be close to, but less than 24 (since 3 x 8 = 24 but 2 and 2/3 is less than 3). 3/4 = (5 x 3)/(5 x 4) because multiplying 3/4 by 5/5 is the same as multiplying by 1 (the 5 in the numerator and the 5 in the denominator cancel out).


When multiplying a number by a decimal less than one, the product will be smaller than the number being multiplied. This is because we are finding a fractional amount of a quantity. For example, 0.1 x 0.8 = 0.08, because the question is asking us to find one tenth of eight tenths. A tenth of a tenth (or a tenth multiplied by a tenth) is a hundredth, thus one tenth of eight tenths is eight hundredths.

You try! How many times greater is 4 x 10 than 2 x 10?

It is 2 times as large! We know this without even having to calculate it since 4 is two times greater than 2, while both expressions have 10.

Play this game to better understand multiplication as scaling! Practice finding a fraction of a whole, for example 3/4 of 20 is the same 3/4 x 20 = 5!

Up for a challenge? Play this game to practice multiplying two mixed numbers!

Dividing Fractions

When we are solving fraction division problems, it is important to create a story context for the problem. This just means if you are given a fraction division problem, you can create a scenario for the numbers. For example, if you are given 1/2 divided by 4, a story context could be we have half of a cake left, and we want to share with 3 friends including ourselves (so 4 people).

There are three parts of fraction division that we will look at with examples!

  1. We can divide a fraction by a whole number:

  • Example: You have 1/8 of a bag of pens and you need to share them among 3 people. How much of the bag does each person get?

  • Here are 3 ways that you can solve this question:

2. We can divide a whole number by a fraction.

  • Example: Create a story context for 5 ÷ 1 /6. Find your answer and then draw a picture to prove your answer and use multiplication to reason about whether your answer makes sense. How many 1 /6 are there in 5?

  • Let's look at how to solve this question!

3. The third thing we have to learn for division fraction is using division fraction in real-world problems! Let's look at an example!

  • Example: Angelo has 4 lbs. of peanuts. He wants to give each of his friends 1 /5 lb. How many friends can receive 1/5 lb. of peanuts?

Each box represents 1 lb of peanuts. Since there are five fifths in one whole, there must be 20 fifths in 4 lbs! Therefore, he can give 20 friends 1/5 lb of peanuts.

Still confused? Watch these videos to learn more about the step-by-step approach to divide fractions!

Dividing a unit fraction is like dividing a whole into smaller parts. Practice dividing unit fractions by a whole number!

Interpret the meaning of division of a unit fraction by a whole number with the help of visual models! Use a written method to perform the division!