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Multiplying and Dividing Fractions

Multiplying and Dividing Fractions

Author: Colleen Atakpu

In this lesson, students will learn how to multiply and divide fractions.

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[MUSIC PLAYING] Let's look at our objectives for today. We will start with a review of PEMDAS, the order of operations. We'll then look at how to multiply fractions, how to divide fractions. We'll do some examples simplifying fractions. And then we'll see how to use order of operations with fractions.

Here's our review of the order of operations. PEMDAS is the acronym that we use to remember the order of operations. PEMDAS stands for parentheses, exponents, multiplication, division, addition, and subtraction. The order of operations applies to all numbers, including fractions. Multiplication and division involving fractions is useful when converting between measurements, which is common in many scientific field, such as chemistry, physics, and nursing.

Let's look at an example of how we multiply fractions. We want to multiply 3/4 times 1/5. In this problem, the 3 and the 1 are the numerators of the fractions. And the 4 and the 5 are the denominators.

When we multiply fractions, we multiply the numerators together to find the numerator of our answer. So 3 times 1 is 3. And then we multiply the denominators together to get the denominator of our answer. So 4 times 5 is 20. Our answer is 3/20.

Now, let's look at an example of how we divide fractions. We want to divide 3/8 by 1/2. Dividing fractions is equivalent to multiplying by the reciprocal.

Finding the reciprocal of a fraction means flipping it, or changing the numerator and the denominator. So our division problem becomes 3/8 times 2/1. Again, we change the division sign to multiplication. And we flipped our second fraction.

Now we can multiply our fractions in the same way we did in our first example. So we multiply our numerators together. 3 times 2 equals 6. And then multiply our denominators together. 8 times 1 is 8. So our answer is 6/8.

We need to make sure that the fraction is simplified, which means we need to cancel out any common factors of both the numerator and the denominator, 6 and 8. Let's look a little bit more closely at how we simplify a fraction.

A simplified fraction is a fraction where the numerator and the denominator have no common factors other than 1. We always want to write fractions in simplest form so they are easier to compare and calculate with.

For example, 50/100 can be simplified to 1/2. And it's much easier to say, you ate half of your dinner rather than 50 hundredths of your dinner.

So let's go back to our previous example and see how we simplify fractions. Our answer was 6/8. We need to see if the numerator 6 and the denominator 8 have any common factors.

To do this we can expand both numbers into their prime factors, the prime numbers that can be multiplied together to result in the original number. So the prime factors of 6 are 2 and 3. And the prime factors of 8 are 2, 2, and 2.

Since there is at least one 2 in both the numerator and the denominator, we can cancel them out. This leaves us with a 3 in the numerator and 2 times 2 in the denominator, which gives us a simplified fraction 3/4.

For our last example, let's see how we use order of operations, or PEMDAS, with fractions. We want to simplify the expression 2/9 divided by 1/3 squared times 1/4.

We start with our exponent operation and do 1/3 squared, which means 1/3 times 1/3. Multiplying the numerators and the denominators together gives us 1/9. So our expression becomes 2/9 divided by 1/9 times 1/4. We then perform multiplication and division from left to right in the order that they appear.

So we start by dividing 2/9 by 1/9. We know that when we divide fractions it's the same as multiplying by the reciprocal. So this becomes 2/9 multiplied by 9/1. So now our expression is 2/9 times 9/1 times 1/4.

Since we only have multiplication remaining, we can multiply all three numerators together-- 2 times 9 times 1, which will give us 18-- and all three denominators together-- 9 times 1 times 4, which will give us 36. We can then simplify 18/36 by writing the numerator and denominator as prime factors.

The prime factors of 18 are 2, 3, and 3. And the prime factors of 36 are 2, 2, 3, and 3. We can then cross out one 2 and two 3s in both the numerator and the denominator.

Since we have no more factors remaining in the numerator, this means our numerator is 1. And our denominator has one 2 remaining, so our final answer is 1/2.

Let's review our important points from today. Make sure you get these in your notes, so that you can refer to them later.

We use order of operations, or PEMDAS, with all numbers, including fractions. To multiply fractions, you multiply the numerators and the denominators together straight across to find the numerator and denominator of your answer.

To divide fractions, you keep the first fraction the same, change the division sign to multiplication, and then flip the second fraction.

To simplify fractions, you can cancel out any common factors of both the numerator and denominator.

So I hope that these key points and examples helped you understand a little bit more about multiplying and dividing fractions. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.

Terms to Know
Reciprocal (of a Fraction)

A fraction in which the numerator and denominator have been switched.