One of the most crucial operations to learning fractions is knowing how to multiply fractions. Multiplying fractions results in either another fraction or a whole number. It is indeed very easy to multiply fractions. Before we dive deep into multiplying fractions, let us learn a few terminologies related to fractions.
What is a fraction? A fraction is any number that can be represented in the form m/n. The number above the ‘/’ sign is known as the numerator, while the number below the ‘/’ sign is known as the denominator. In the above case, ‘m’ is the numerator while ‘n’ is the denominator. Fractions are further divided into 3 types:
Proper fractions: The fractions in which the numerator is always smaller than the denominator are known as proper fractions.
Example: 1/2,3/4, 7/21, etc.
Improper fractions: The fractions where the numerator is always greater than the denominator are called improper fractions.
Example: 21/3, 99/4, 5/2, etc.
Mixed fractions: The fractions which have a combination of a whole number and a fractional number are known as mixed fractions.Example:3 4
We can see that improper fractions can be written as mixed fractions and vice versa.
In this article, we will learn a lot about multiplying fractions and multiplying fractions with whole numbers.
How to Multiply Fractions?
Multiplication of fractions differs a lot from addition and subtraction. While summing or subtracting two or more fractions, the denominator ought to be the same, but while multiplying fractions, the denominator need not be the same. Any two fractions with different denominators may be conveniently multiplied in this manner. The key thing to remember is that mixed fractions cannot be multiplied; instead, the fractions should either be proper or improper to get multiplied.
Multiplying two fractions is as easy as multiplying two whole numbers. Follow the steps given below to learn how to multiply two fractions.
- Step 1: First, multiply the numerator of the given fractions.
- Step 2: Now, multiply the denominators together.
- Step 3: Write them in the form m / n.
- Step 4: If you find that the fractions can be reduced further to smaller numbers, go for it.
Let us take an example to understand the steps mentioned above:
Example: Multiply 2/7 and 3/4.
Solution: Do your solution stepwise:
- Multiply 2 and 3 = 2 x 3 = 6
- Multiply 7 and 4 = 4 x 7 = 28
- Write them in the fractional form = 6/28
- If possible, reduce the value of the fraction. Here we can see that the fraction is a factor of 2. Thus we can divide the fraction by two; therefore, 6/28 = 3/14.
Therefore, the result of multiplying 2/7 and 3/4 is 3/14.
How do you Multiply Fractions?
Always keep the below-listed rules in your mind whenever you are multiplying fractions:
- Rule 1: If there are mixed fractions, turn them into improper fractions. The numerators of the given fractions should then be multiplied.
- Rule 2: Divide the numerators by the denominators individually.
- Rule 3: Reduce the acquired value to its simplest form.
These three principles may be used to get the product of any two fractions. Let us now look at the many multiplying fractions with various fractions.
Multiplying Fractions with the Same Denominator
Like fractions are fractions that have the same denominator. For instance, 5/7, 1/7, and 3/7 are like fractions, while 5/7 and 4/3 are unlike fractions. While the addition and subtraction of like fractions differ from the addition and subtraction of unlike fractions, the approach for multiplication and division is the same. The rule of fraction multiplication is unaffected by multiplying fractions with the same denominator. Let us take an example to understand this:
Example: Multiply 11/4 × 5/4
Solution: Let us recall the steps learned in the previous section to solve this
- Step 1: Multiply 11 and 5, 11 × 5 = 55.
- Step 2: Multiply 4 and 4, 4 × 4 = 16.
- Step 3: Arrange the result in the fractional form gives 55/16. Since the result isn’t the factor of any number, the resultant fraction is 55/16.
Multiplying Fractions with Different Denominators
Multiplying fractions with unlike denominators is the same as multiplying fractions with like denominators. Let’s look at an example to grasp this better.
Example: Multiply 2/7 × 9/11
Solution: Again, let us follow the same steps as the previous example
- Step 1: Multiply 2 and 9, 2 × 9 = 18.
- Step 2: Multiply 7 and 11, 7 × 11 = 77.
- Step 3: Arrange the result in the fractional form gives 18/77. Since the result isn’t the factor of any number, the resultant fraction is 18/77.
We can even use an alternative method to solve problems related to the multiplication of fractions.
This method involves writing the fractions given to us together using the multiplication sign ‘x’. Then we check whether the numbers which are present opposite to each other are factors of each other or some other number or not. We then simplify the equation and multiply the numerator and denominator respectively to get the result. You can refer to the example below to ease your understanding of this method.
Example: Multiply 8/6 × 3/24.
Solution: Write the fractions as 8/6 x 3/24. We can see that 24 can be divided by 8 three times and 6 can be divided by 3 two times. Therefore the fraction becomes 12 x 13. Now we have simplified the equation and can easily multiply them to get the answer as ⅙.
Wasn’t this easy? This method was developed to ease the solution because people get confused when dealing with larger numbers.
Multiplying Fractions by Whole Numbers
Before diving deep into the concept of multiplying fractions with whole numbers. Let us understand a very small concept but extremely important for this section. as we all are aware that whole numbers are 0, 1, 2, 3,………, infinity. But do you know how we can write a whole number into fractional form? We can easily write any whole number into fractions by making the whole number the numerator and then taking the denominator as 1. In this way, we get 0 as 0/1, 1 as 1/1, 2 and 2/1, and so on.
Now that we know this concept, it has become convenient to solve multiplication problems by whole numbers. From the example given below, you will understand this section easily.
Example: Multiply: 13 × 2/9.
Solution: Follow the steps given below :
- Step 1: Write 13 in fractional form as 13/1
- Step 2: The equation becomes 13/1 x 2/9
- Step 3: Since we cannot simply reduce this expression any further, multiply 13 and 2 = 13 x 2 = 26
- Step 4: Now, multiply 1 and 9. 1 x 9 = 9
- Step 5: The result of the multiplication of the given factor is 26/9, which is an improper fraction.
Multiplying Fractions with Mixed Numbers
We are aware that a fraction, a combination of a whole number and a fraction, is called a mixed fraction. It is impossible to multiply fractions if they are in mixed form. Thus, we must first change the mixed fractions into their improper fraction form. This can be done by multiplying the denominator with the whole number and adding the result to the numerator.For instance, change 6 into an improper fraction.
Solution: Multiply 6 by 5 = 6 x 5 = 30.
Add the result to the numerator. 30 + 2 = 32.
Therefore, the improper fraction = 32/5
The rest of the steps while multiplying the fractions remain the same as discussed many times in this article.Example: Multiply 4 and 7
- Step 1: Convert all the mixed fractions into improper fractions. 4 = 30/7 and 7 = 88/3
- Step 2: Write the fractions as 30/7 x 88/3. We can see that 30 is divided by 3 ten times therefore the equation changes to 10/7 x 88/1.
- Step 3: Multiply the expression and the result of the multiplication is 880/7.
- Step 4: Again change the improper fraction into a mixed fraction. 880/7 = 125 is the required answer.
If you multiply the fraction by its reciprocal the answer is always 1.
For example, let the fraction be ⅞. The reciprocal of the fraction is 8/7. Now, if we multiply them,
7/8 x 8/7 we see that 8 divided each other and 7 divides each other, leaving us with 1 as the answer.
Multiplying fractions is way too easy with these simple steps mentioned above. All you have to do is grasp the concepts quickly and practice to master the concept.