Simplifying Fractions Lesson

In this lesson, we will learn how to simplify a fraction. This process is also known as reducing a fraction to its lowest terms. As we move forward, it is an expectation that any fraction is simplified before reporting an answer. Up to this point, when we saw different numbers they always represented different values. When working with fractions, we may have two or more fractions that look different, but represent the same value. Before we get into the math, let’s take a look at an example.
Let's revisit our pizza example from the introduction to fractions lesson. Let's again start with a whole pizza: Now, let's suppose we cut up the pizza into 8 equal slices: If we were to eat 4 out of 8 slices of pizza, this is represented as the fraction: $$\frac{4}{8}$$ What if we thought about a different scenario? Let’s suppose that the original pizza was cut into only 2 equal slices: If we were to eat 1 out of 2 slices of pizza, this is represented as the fraction: $$\frac{1}{2}$$ Now, let's compare the two scenarios side by side: We can see from our example that having 1/2 of a pizza is the same quantity or amount as having 4/8 of a pizza. This is just due to a whole quantity being split up into a different number of pieces. The same amount can be removed, but because of the numbers involved, it may look different. 1/2 and 4/8 are known as equivalent fractions. Equivalent fractions are fractions that have the same value.
How can we go from 1/2 to 4/8, without changing the value? The process relies on two earlier principles from multiplication and division. Recall from our properties of multiplication lesson, that multiplication by 1 leaves a number unchanged. Additionally, recall from our properties of division lesson, that a non-zero number divided by itself results in 1. We can put these two properties together to show how two fractions can look different, but be equivalent or have the same value. Let's walk through this example: $$\frac{4}{4}=1$$ $$\frac{1}{2}\cdot 1=\frac{1}{2}$$ $$\frac{1}{2}\cdot \frac{4}{4}=\frac{4}{8}$$ We will learn how to multiply fractions in the next lesson, for now, we can just state that we multiply numerator by numerator and place the result over the denominator times the denominator.
In our example above, we start with 4/4 and show the value to be 1. We then multiply 1/2 by 1 and show the value to be unchanged or 1/2. Finally, we put the two properties together and show a more complex form: 1/2 multiplied by 4/4 gives us a result of 4/8. We know that although 4/8 looks different than 1/2, it is the same value based on our two properties.
Let's look at another example. Suppose we saw the fractions: $$\frac{1}{3}$$ $$\frac{2}{6}$$ These two fractions are equivalent: $$\frac{1}{3}\cdot \frac{2}{2}=\frac{2}{6}$$ $$\frac{1}{3}=\frac{2}{6}$$ Now that we understand how two or more fractions can look different but have the same value, let's think about how we can simplify a fraction. A fraction is considered simplified if the numerator and denominator have no common factor other than 1. Earlier we saw that we could multiply by a complicated form of 1 and maintain an equivalent fraction. Now we will reverse this process and remove a complex form of 1.

Simplifying a Fraction - Reducing to its lowest terms

• Factor the numerator and denominator into the product of prime numbers
• Cancel all common factors between the numerator and denominator
  • When we cancel common factors, we slash out the numbers. The result of this operation is 1 (it will usually only be listed if all numbers are canceled).
• We can also simplify a fraction with simple division. We will divide the numerator and denominator by the GCF of the two numbers.

Example 1: Simplify each fraction. $$\frac{18}{20}$$

• Factor the numerator and denominator into the product of prime numbers
• 18 = 2 x 3 x 3
• 20 = 2 x 2 x 5
• Cancel all common factors between the numerator and denominator
• In this case, we only have a 2 that is common to both numerator and denominator

$$\require{cancel}$$ $$\frac{18}{20}=\frac{2 \cdot 3 \cdot 3}{2 \cdot 2 \cdot 5}$$ $$\frac{18}{20}=\frac{\cancel{2}\cdot 3 \cdot 3}{\cancel{2}\cdot 2 \cdot 5}$$ $$\frac{18}{20}=\frac{9}{10}$$ We can also use simple division. We know the GCF of 18 and 20 is 2. If we divide 18 by 2, we get 9. If we divide 20 by 2, we get 10. $$\frac{18}{20}=\frac{18 ÷ 2}{20 ÷ 2}=\frac{9}{10}$$ Example 2: Simplify each fraction. $$\frac{65}{390}$$

• Factor the numerator and denominator into the product of prime numbers
• 65 = 5 x 13
• 390 = 2 x 3 x 5 x 13
• Cancel all common factors between the numerator and denominator
• In this case, we have a 5 and a 13 that are common to both numerator and denominator

$$\frac{65}{390}=\frac{5 \cdot 13}{2 \cdot 3 \cdot 5 \cdot 13}$$ $$\frac{65}{390}=\frac{\cancel{5}\cdot \cancel{13}1}{2 \cdot 3 \cdot \cancel{5}\cdot \cancel{13}}$$ $$\frac{65}{390}=\frac{1}{6}$$ We can also use simple division. We know the GCF of 65 and 390 is 65. If we divide 65 by 65, we get 1. If we divide 390 by 65, we get 6. $$\frac{65}{390}=\frac{65 ÷ 65}{390 ÷ 65}=\frac{1}{6}$$ Example 3: Simplify each fraction. $$\frac{330}{420}$$

• Factor the numerator and denominator into the product of prime numbers
• 330 = 2 x 3 x 5 x 11
• 420 = 2 x 2 x 3 x 5 x 7
• Cancel all common factors between the numerator and denominator
• In this case, we have a 2, 3, and 5 that are common to both numerator and denominator

$$\require{cancel}$$ $$\frac{330}{420}=\frac{2 \cdot 3 \cdot 5 \cdot 11}{2 \cdot 2 \cdot 3 \cdot 5 \cdot 7}$$ $$\frac{330}{420}=\frac{\cancel{2}\cdot \cancel{3}\cdot \cancel{5}\cdot 11}{\cancel{2}\cdot 2 \cdot \cancel{3}\cdot \cancel{5}\cdot 7}$$ $$\frac{330}{420}=\frac{11}{14}$$ We can also use simple division. We know the GCF of 330 and 420 is 30. If we divide 330 by 30, we get 11. If we divide 420 by 30, we get 14. $$\frac{330}{420}=\frac{330 ÷ 30}{420 ÷ 30}=\frac{11}{14}$$

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