-
A fraction represents a number that consists of a numerator and a denominator.
Let's look at the fraction one-fourth:
.png)
.png)
Let’s Explore How to Represent Fractions
Let’s divide this circle into 8 equal parts with 2 parts shaded in red. What fraction of the circle is red?
What Fraction of the Rectangle is Red?
Let’s divide this rectangle into 5 equal parts with 2 of the parts shaded in red. What fraction of the rectangle is shaded red?
Video: How to Read and Write Fractions
Some fractions can be simplified.
For example, the fraction 2/4 can be simplified to 1/2.
A simplified fraction is a fraction that is already in its simplest form. If you can’t find an equivalent fraction with a smaller numerator and denominator then you’ve got a simplified fraction.
BASIC CONCEPTS
-
Any number multiplied or divided by 1 doesn’t change its value.
-
Similarly, dividing the numerator and the denominator by the same number won’t change the value of the fraction.
-
The simplest form of a fraction has no common factors in the numerator and denominator.
-
Factors are numbers that are multiplied to give a product.
What Fraction of Fish can be Shaded Blue
-
6 blue fish
-
12 total fish
-
6/12 of the fish are blue
-
We can simplify this fraction further because 6 and 12 both have common factors.
1/2 is the simplest form of 6/12
How to Simplify Fractions
A fraction can be simplified by dividing both the numerator and denominator by a common factor.
6/8 = 6/8 ÷ 2/2 = 3/4
Let’s look at the fraction :
-
The numerator and denominator are both multiples of 2.
-
If we divide the numerator and denominator by 2, we’ll get an equivalent fraction.
-
6/8 can be simplified to 3/4
How to Simplify Fractions
-
If you divide the numerator and denominator of a fraction by the same number, you get an equivalent fraction. In the previous example we divided both 6 and 8 by 2.
-
6 and 8 are both divisible by 2. This means 6 and 8 can be divided by 2 without a remainder.
-
Since 6 and 8 are both divisible by 2, we can also say that 2 is a factor of 6 and 8.
-
2 as a factor of 6 can be written as: 2 x 3 = 6
-
2 as a factor of 8 can be written as: 4 x 2 = 8
Therefore 2 is a factor of both 6 and 8. This is also why we could divide the numerator, 6, and the denominator, 8, by 2 to get 3/4.
Finding Factors
Let’s review factors!
-
Factors are numbers that multiply together to make a larger number.
Example: What are the factors of 10?
Example Factors
Find the Simplest Form of the Fraction
We can see above that 2 and 4 are both divisible by 2 because
2÷2 = 1
4÷2 = 2
Since 2 and 4 are both divisible by 2, it means that 2 is a factor of these numbers.
Divide the numerator and denominator both by 2 to get the simplest fraction 1/2.
Find the Simplest Form of the Fraction
We can see above that 4 and 10 are both divisible by 2 because
4÷2 = 2
10÷2 = 5
Since 4 and 10 are both divisible by 2, it means that 2 is a factor of these numbers.
Divide the numerator and denominator both by 2 to get the simplest fraction 2/5 .
Quick Review!
5/10 ÷ 5/5 = 1/2
5/5 = 1
Notice: The value of the fraction doesn’t change.
Dividing the numerator and denominator by the same number is the same as dividing by 1.
Find the Simplest Form of the fraction
4/16
-
First, find the common factors of the numerator and denominator.
Factors of 4: 1 , 2 , 4
(1 × 4 = 4; 2 × 2 = 4; 4 × 1 = 4)
Factors of 16: 1 , 2 , 4 , 8 , 16
(1 × 16 = 16; 2 × 8 = 16; 4 × 4 = 16;
8 × 2 = 16; 16 × 1 = 16)
Find the Simplest Form of the fraction
-
Greatest Common Factor (GCF)
-
GCF is the greatest number that is both a factor for the numerator and denominator.
-
In the previous example, it is 4.
-
-
Divide the numerator and denominator by the GCF.
4 ÷ 4 = 1
16 ÷ 4 = 4
-
So, the simplest form of 4/16 is 1/4 .
Find the Simplest Form of the fraction
-
Greatest Common Factor (GCF)
-
GCF is the greatest number that is both a factor for the numerator and denominator.
-
In the previous example, it is 4.
-
-
Divide the numerator and denominator by the GCF.
4 ÷ 4 = 1
16 ÷ 4 = 4
• So, the simplest form of 4/16 is 1/4 .
Find the Simplest Form of the Fraction
Common factors:
16 : 1 , 2 , 4 , 8 , 16
84 : 1 , 2 , 4 , 6 , 7, 12 , 14 , 21 , 42 , 84
-
Once past the maximum value of the numerator, there won’t be a common factor.
-
Greatest Common Factor = 4
16/84 ÷ 4/4 = 4/21
Find the Simplest Form of the Fraction
Toy Cars
Alex has six toy cars. Three of the cars are blue. How can you write a fraction for the blue cars in simplest form?
Fraction 3/6 Blue cars (numerator)
Total cars (denominator)
-
Common factors
3 : 1, 3
6 : 1,2, 3, 6
3/6 ÷ 3/3 = 1/2
Video: Simplifying Fractions
Video: Finding Factors
Video: Simplifying Fractions
Common Denominators
Why Do We Find the Common Denominator?
-
In order to add, subtract or compare two fractions, we need to first find a common denominator.
Example: The fractions 1/3 and 1/2 have different denominators.
-
The fraction 1/3 has a denominator of 3.
-
The fraction 1/2 has a denominator of 2.
-
The way to find a common denominator is to find a pair of fractions that are equivalent to the fractions 1/3 and 1/2.
How to Find Common Denominators
Here’s how we’ll find a common denominator of
1/3 and 1/2 :
-
We want to multiply the numerator and denominator of 1/2 by a different number to make a new fraction that is equivalent to 1/2.
-
We want to multiply the numerator and denominator of 1/3 by a different number to make a new fraction that is equivalent to 1/3.
-
We’ll choose the number for each case such that the two new fractions have the same denominator.
1/3 1/2
Let’s Think: What number is a multiple of both 2 and 3?
-
The number 6 is the smallest multiple of both 2 and 3.
-
We’ll make a common denominator of 6.
-
Multiply the numerator and denominator by 2 to make an equivalent fraction with a denominator of 6.
1/3 = 1/3 X 2/2 = 2/6
Multiply the numerator and denominator of 1/2 by 3 to make an equivalent fraction with a denominator of 6.
1/2 = 1/2 X 3/3 = 3/6
-
The fraction 2/6 is equivalent to the original fraction 1/3.
-
The fraction 3/6 is equivalent to the original fraction 1/2.
-
Since 2/6 and 3/6 now have the same denominator, we have successfully found a common denominator and we can add, subtract, or compare these fractions!
We have also found the lowest common denominator because 6 is the smallest multiple of the original denominators (3 and 2).
Looking Ahead
The examples in this chapter show you how to find the lowest common denominator. We’ll practice the skill in this chapter, and then learn how to apply it in the next chapters when we compare, add, and subtract fractions.
In the previous example, we used the smallest multiple of both denominators as the least common denominator.
What Are Multiples?
-
The multiples of a number are the products that you can make from multiplication. They are whole numbers.
Multiples Cheat Sheet
Let’s review the first ten multiples of numbers 2 through 10:
Least Common Multiples (LCM)
-
The least common multiple (LCM) is the lowest shared multiple between two numbers.
-
To find the common denominator, we’ll use the LCM.
What is the Lowest Common Denominator of These Two Fractions?
Consider the fractions 1/4and 1/3. What is the lowest common denominator of these two fractions?
Step 1:
Find the LCM: The least common multiple of 3 and 4 is the number 12.
Step 2:
Multiply the numerator and denominator of 1/4 by 3 (because 4 x 3 = 12) to make an equivalent fraction with a denominator of 12.
1/4 = 1/4 X 3/3 = 3/12
Step 3:
Multiply the numerator and denominator of 1/3 by 4 (because 3 x 4 = 12) to make an equivalent fraction with a denominator of 12.
1/3 = 1/3 X 4/4 = 4/12
Step 4:
3/12 is an equivalent fraction of 1/4
1/3 is an equivalent fraction of 4/12
The lowest common denominator of 1/3 and 1/4 is 12.
What is the Lowest Common Denominator of These Two Fractions?
Consider the fractions 3/8 and 2/3. What is the lowest common denominator of these two fractions?
Step 1:
Find the LCM: The least common multiple of 8 and 3 is 24.
Step 2:
Multiply the numerator and denominator of 3/8 by 3 (because 8 x 3 = 24) to make an equivalent fraction with a denominator of 24.
3/8 = 3/8 X 3/3 + 9/24
Step 3:
Multiply the numerator and denominator of 2/3 by 8 (because 3 x 8 = 24) to make an equivalent fraction with a denominator of 24.
2/3 = 2/3 X 8/8 = 16/24
Step 4:
9/24 is an equivalent fraction of 3/8
16/24 is an equivalent fraction of 2/3
The lowest common denominator of 3/8 and 2/3 is 24.
Video: Common Denominators
Video: Common Denominators
Video: Least Common Multiples
Why did the fraction feel down?
Because it couldn't find its common denominator! 😄📊
Equivalent Fractions
We need to understand equivalent fractions before we can compare fractions. Let’s take a look at equivalent fractions.
Look at the fractions below. Do you notice any similarities?
2/3 = 4/6 = 8/12 = 16/24
-
These are equivalent fractions because they have the same value.
-
The simplest form of all these fractions is 2/3.
Equivalent Fractions
2/3 = 4/6 = 8/12 = 16/24
-
To find equivalent fractions multiply the numerator and denominator by the same non-zero whole number.
Note: Multiplying the numerator and the denominator by the same non-zero whole number will change that fraction into an equivalent fraction, but it will not change the value.
Which fraction is larger 1/2 or 3/4?
We can use what we learned in the previous chapter about common denominators to compare fractions.
Step 1:
Let’s find the common denominator for 1/2 and 3/4 .
-
We find the common denominator by finding a pair of fractions that are equivalent to the fractions 1/2 and 3/4 ..
-
Multiply the numerator and the denominator of the fraction by the same number to get an equivalent fraction.
-
We’ll choose the numbers in each case so that the two new fractions (which are equivalent to 1/2 and 3/4 .) have the same denominator.
-
The denominators of these fractions are 2 and 4. What number is a multiple of both 2 and 4?
The number 4 is the smallest multiple of both 2 and 4. Therefore, we’ll make a common denominator of 4.
Step 2:
-
Multiply the numerator and denominator of 1/2 by 2 to make an equivalent fraction with a denominator of 4.
Step 3:
-
Multiply the numerator and denominator of 3/4 by 1 to make an equivalent fraction with a denominator of 4. Multiplying by 1 doesn’t change the fraction so we still have 3/4
Now, the two fractions 2/4 and 3/4 have the same denominator.
Step 4:
We can compare the fractions now because they have the same denominator.
-
When we compare fractions with the same denominator, we look at each numerator.
-
In this case, the numerator 3 is larger than the numerator 2.
3/4 is greater than 2/4
Symbols to Compare Fractions
Instead of saying greater than, less than, or equal to, we can use symbols to compare fractions.
Greater Than
We use > for greater than.
Less Than
We use < for less than.
Equal To
We use = for equal to.
3/4 > 2/4
3/4 is greater than 2/4
2/3 x 2/2 = 4/6
4/6 x 2/2 = 8/12
4/6 x 2/2 = 8/12
8/12 x 2/2 = 16/24
Equivalent Fractions Examples
Showing Equivalent Fractions
Let’s Compare Fractions
Video: Comparing Fractions
Video: Equivalent Fractions
Adding/Subtracting Fractions
When you learned how to compare fractions in the previous chapter, you were well on your way to adding and subtracting fractions!
Remember how we first made the denominators the same. Then we compared the numerators to tell which fraction was bigger.
Similarly, we can add or subtract fractions by first finding a common denominator, using the strategy in Section 3. Then we simply add or subtract the numerators of the two fractions.
Your Turn! Add the Fractions
Step 1:
-
Find the common denominator of 3 and 4. We’ll use the method we used in Chapters 3 and 4.
-
To find the common denominator, we need to find equivalent fractions for the original fractions
The equivalent fractions we make from 2/3 and 1/4
-
need to have the same denominator.
-
We know that 12 is a multiple of both 3 and 4, so we can make 12 the common denominator.
Step 2:
-
Multiply the numerator and denominator of 2/3 by 4 (because 4 x 3 = 12) to make an equivalent fraction with a denominator of 12.
-
Now we have a denominator of 12, which is what we want.
Step 3:
-
Multiply the numerator and denominator of 1/4 by 3 (because 4 x 3 = 12).
-
Now, we have a denominator of 12, which is what we want.
Step 4:
We’re ready to add the fractions.
-
Now we can simply add the numerators of these two fractions.
-
Since 3/12 is an equivalent fraction of 1/4 is an equivalent fraction of 2/3, we can say that
Your Turn! Subtract the Fractions
Step 1:
-
Find the common denominator of 2 and 4. We’ll use the method we used in Chapters 3 and 4.
-
To find the common denominator, we need to find equivalent fractions for the original fractions 3/4 and 1/2
-
The equivalent fractions we make from 3/4 and 1/2 need to have the same denominator.
-
We know that 4 is a multiple of 2 and 4, so we can make 4 the common denominator.
Step 2:
-
Multiply the numerator and denominator of 3/4 by 1 (because 4 x 1 = 4) to make an equivalent fraction with a denominator of 4.
-
We can keep 3/4 the same because it already has a denominator of 4.
Step 3:
-
Multiply the numerator and denominator of 1/2 by 2 (because 2 x 2 = 4).
-
Now, we have a denominator of 4.
Step 4:
We’re ready to subtract the fractions.
-
Now, we can simply subtract the numerators of these two fractions.
-
Since 2/4 is an equivalent fraction of 1/2 we know that