Lesson Progress:

# -Fractions: Part 2

## Overview

Understanding what a fraction is and looks like is the first step to performing addition, subtraction, multiplication, and division with them. Doing these operations allows us to form relationships between fractions and answer questions about how fractions work with each other. These problems might be a source of frustration for some, but looking at the problem with a strong understanding of fractions will help make sense of the operations. In this lesson, you will simplify, multiply, divide, add, and subtract proper and improper fractions.

## Big Question

How can I add, subtract, multiply, and divide fractions?

## Overview

By the end of this lesson, we want to add, subtract, multiply, and divide proper and improper fractions. Keep in mind: methods for proper fractions work for improper fractions, and we will call them both “fractions” to make things easier. If you can simplify fractions, it will make multiplying and dividing them much easier. We’ll start our lesson here. Later, we will go over how to add and subtract mixed numbers (numbers with a whole and fraction part).

## Simplify Fractions Using GCF

A simplified fraction is a fraction that has no common factors in the numerator and denominator. If you find the GCF (greatest common factor) between the numerator and denominator, you can divide the top and bottom number by it. This is called reducing, or simplifying, a fraction. This simplifies the fraction into an equivalent fraction with smaller numbers.

For example:

• 2/3 is simplified because there are no common factors of 2 and 3.
• 10/15 is not simplified because 5 is a common factor of 10 and 15.

Example 1: Simplify 10/15

To simplify the fraction, we look for any common factors in the numerator and the denominator.

Since 5 is a common factor of 10 and 15, we can divide 10 and 15 by 5. This changes the fraction10/15 into 2/3. Since there are no more common factors, this fraction is simplified.

Example 2: Simplify 18/12

Let’s see what happens when we don’t simplify a fraction by its GCF.

2 is a common factor of 18 and 12. If we divide them both by 2, we have the fraction 9/6. This fraction is not simplified because 9 and 6 have a common factor of 3. If we divide out the 3 (or divide 6 in the original fraction), we will have the simplified fraction of 3/2 or 1 ½ (some fractions are more useful as mixed numbers).

## Modeling Multiplying Fractions

Next, we will multiply fractions. A model may help you understand multiplication of fractions.

Example 3: Use fraction tiles to model 1/2 ⋅ 3/4.

To multiply 1/2 and 3/4, think of “one half of three-fourths.”

Start with fraction tiles for three-fourths. To find one half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three 1/4 tiles evenly into two parts, we exchange them for smaller 1/8 tiles. We see 6/8 is equivalent to 3/4. Taking half of the six 18 tiles gives us three 1/8 tiles, which is 3/8.

Therefore, 1/2 ⋅ 3/4 = 3/8.

## Multiplying Fractions with Simplification

We can also multiply across numerators and denominators. This will get us an equivalent fraction that we can simplify to get our answer. We can simplify before or after the multiplication.

Example 4: Multiply 3/5 • 2/3

To multiply these fractions, we will multiply 3 and 2 for the numerator and 5 and 3 for the denominator. This gets us the fraction 6/15. Dividing out the common factor of 3, we can get the simplified fraction 2/5.

We can simplify our fractions before using both fractions at the same time.

If we reuse Example 4, we can see if the numerators of both fractions have any common factors with the denominators of both fractions. We can see that 3 is the common factor between the numerator of 3/5 and the denominator of 2/3. This allows us to simplify the problem 3/5 • 2/3 into 1/5 • 2/1. If we multiply them across, we will get the answer 2/5 (the same as before).

Example 5: Multiply 3/2 • 8/9

If we multiply across, we will get the fraction 24/18. If we divide out the GCF of 6, we will have the fraction 4/3.

If we try to simplify both fractions, we can divide a 3 out of 3 (numerator of the 1st fraction) and 9 (denominator of the 2nd fraction) and a 2 out of 2 (denominator of the 1st fraction) and 8 (numerator of the 2nd fraction). This changes 3/2 • 8/9 into 1/1 • 4/3 = 4/3, or 1 1/3.

## Dividing Proper and Improper Fractions

We need to use the reciprocal of a fraction to divide fractions. The reciprocal of a fraction is the fraction you get by flipping the numerator and denominator.

Example 6: Divide 1/2 ÷ 1/6 by modeling and changing the problem.

We need to figure out how many 1/6s there are in 1/2. Notice, there are three 1/6 tiles in 1/2, so 1/2 ÷ 1/6 = 3.

We can rewrite Example 6 into a multiplication problem by changing the sign and replacing the 2nd fraction with its reciprocal.

This changes 1/2 ÷ 1/6 into 1/2 • 6/1. This gives us 6/2 or 3 (same as before).

Example 7: Divide 3/8 ÷ 9/4 by changing it into a multiplication problem.

If we change this problem by changing the sign and replacing the 2nd fraction with its reciprocal, we have an equivalent problem of 3/8 • 4/9. From here, you can multiply across or simplify. The simplified solution would be 1/6.

This means 9/4 will fit into 3/8 one-sixth times. This may sound strange, but notice that 9/4 is larger than 3/8. It makes sense that you can only fit part of 9/4 into 3/8.

## Add or Subtract Fractions with Common Denominators

If your fraction has common denominators, adding or subtracting fractions is straightforward.
Let’s model fractions with common denominators.

Example 8: How many quarters are pictured, and what fraction of a dollar is this? One quarter plus 2 quarters equals 3 quarters. Since a quarter is 1/4 of a dollar, this model shows that 1/4 + 2/4 = 3/4 or 3 quarters.

Example 9: Subtract 4/5 – 1/5.

The same idea for addition works for subtraction, too. 4 fifths with 1 fifth taken away leaves 3 fifths or 3/5.

Example 10: Add 4/9 + 8/9.

If you add 4 ninths to 8 ninths, you will have 12 ninths or 12/9. We have an extra 3 ninths over 1, so we can simplify this fraction into 1 1/3 or 4/3 (sometimes an improper fraction is more useful for understanding).

## Discover

With limited understanding, adding or subtracting fractions with uncommon denominators can become difficult; however, we can find and use the lowest common denominator (or LCD) to turn our fractions with uncommon denominators into equivalent fractions with common denominators. The LCD of two fractions is the least common multiple (LCM) of their denominators.

## Equivalent Fractions Using LCD

Example 1: Change 1/4 and 1/6 to equivalent fractions with the LCD of 1/2. Since the LCD is 1/2, we want to find out how many 1 1/2 pieces fit into 1/4 and 1/6.

3-twelfths fit into 1-fourth, and 2-twelfths fit into 1-sixth. This means 1/4 = 3/12 and 1/6 = 2/12.

We can get this equivalent fraction by asking ourselves, “How many times does my denominator fit into the LCD?”

For 4, it fits into 12 three times; so, we need to break 1/4 into three equal pieces. This means we should multiply the numerator and denominator by 3.

For 6, it fits into 12 two times; so, we need to break 1/6 into two equal pieces. This means we should multiply the numerator and denominator by 2.

This will get us the same results as before: 1/4 • 3/3 = 3/12 and 1/6 • 2/2 = 2/12.

Example 2: Change 2/3 and 1/15 to equivalent fractions with the LCD of 15.

Notice that 1/15 already has the denominator of 15. This means we only need to change 2/3.

Since 3 fits into 15 five times, we need to multiply 2/3 by 5/5 to get the equivalent fraction of 10/15.

## Changing Fractions

Let’s get some practice adding and subtracting fractions with uncommon denominators.

Steps to add or subtract fractions with different denominators:

1. Find the LCD.
2. Convert each fraction to an equivalent form with the LCD as the denominator.
3. Add or subtract the fractions.
4. If needed, write the result in simplified form.

Example 3: 1/2 + 1/3

First, we must get the LCD of the fractions which is the LCM of 2 and 3. Since 2 and 3 don’t have any common factors, we can multiply 2 and 3 together to get the LCD of 6.

Next, we will convert the fractions into their equivalent form using the LCD of 6.

2 fits into 6 three times, so 1/2 • 3/3 = 3/6.

3 fits into 6 twice, so 1/3 • 2/2 = 2/6.

Then, we will add the fractions. 3/6 + 2/6 = 5/6. This means 1/2 + 1/3 = 5/6.

We do not need to simplify 5/6, but we should be careful of situations where we can.

Example 4: 9/10 – 13/20.

First, the LCD of the fractions is the LCM of 10 and 20. Using your preferred method, we find out that the LCD of the fractions is 20. This means that we do not need to change 13/20.

Next, we will covert 9/10 using the LCD of 20.

10 fits into 20 twice, so 9/10 • 2/2 = 18/20.

Then, we will subtract our fractions 18/20 – 13/20 to get the fraction 5/20.

Notice that 5 and 20 have a common factor of 5. This means we can (and should) simplify 5/20 into 1/4.

After our work, we found that 9/10 – 13/20 = 1/4.

## Common Pitfalls

Pitfall 1: Not finding a common denominator

If you had 1 quarter and 1 dime, you could say that you have 2 coins; but, when naming the coins, would it make sense to call them quarters? Dimes? Maybe quar-imes? Of course, the last is a joke, but that is what happens when students try to add 1/4 + 1/10 and get 2/14.

We can convert the value of a quarter and a dime into the common denomination of cents, or change the denominator to 100 (because there are one hundred cents in a dollar). This why 1/4 + 1/10 can be changed into 25/100 + 10/100 (meaning 25 cents + 10 cents). These add together to get 35/100 (meaning 35 cents).

Pitfall 2: Changing the denominator but not changing the numerator (when necessary)

Once you find the LCD, do not forget to multiply the top AND bottom of the fraction. This will make sure that our new fraction is equivalent to our old one. In the picture, we clearly see that we can break 1 half into 2 fourths.

A common mistake would be to find out 2 goes into 4 twice, then multiply 12 just the numerator or just the denominator by 2. This will get us the fractions 22 or 14. Both of these fractions are not the same size as 12.

Use the quiz below to check your understanding of this lesson’s content. You can take this quiz as many times as you like. Once you are finished taking the quiz, click on the “View questions” button to review the correct answers.

## Lesson Resources

##### Lesson Toolbox

Triplets – the equivalent fraction game

A game that practices modeling common fractions and equivalent fractions in circles. You must match all equivalent forms of a common fraction in a circle.

Matching Math Equivalent Fractions

A visual game that has you match pictures of fractions with their equivalent ones. There are three levels of difficulty and a timed mode for added challenge.

Equivalent Fraction BINGO

A game that flashes a picture and name of a fraction and you must select the correct square with the simplified fraction. You can increase the difficulty by making the grids larger or doing challenge mode.

## Terms • LCD
lowest common denominator
• LCD of two fractions
the least common multiple (LCM) of two fractions’ denominators
• reciprocal of a fraction
the fraction you get by flipping the numerator and denominator of another fraction
• simplified fraction
a fraction that has no common factors in the numerator and denominator

#### Lesson Content:

Authored and curated by Kashuan Hopkins for The TEL Library. CC BY NC SA 4.0

Title: Prealgebra; 4.2 Multiply and Divide Fractions; Rice University, OpenStax CNX. License: CC BY 4.0

Title: Prealgebra; 4.4 Add and Subtract Fractions with Common Denominators; Rice University, OpenStax CNX. License: CC BY 4.0

Title: Prealgebra; 4.5 Add and Subtract Fractions with Different Denominators; Rice University, OpenStax CNX. License: CC BY 4.0

## Media Sources Show solution 4/5−1/OpenStaxOpenStaxCC BY 4.0 ¼OpenStaxOpenStaxCC BY 4.0 Pizza Food ItalianigorovsyannykovPixabayCC 0 Convert Fractions to Equivalent FractionsOpenStaxOpenStaxCC BY 4.0 QuarterOpenStaxOpenStaxCC BY 4.0 Divide FractionsOpenStaxOpenStaxCC BY 4.0 Multiply fractionsOpenStaxOpenStaxCC BY 4.0 Measuring Cup BakenaturalpastelsPixabayCC 0