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Fractions: How to Multiply Fractions by a Whole Number

multiply fractions by a whole number

Multiply fractions by a whole number is a straightforward operation. It is one of the basic concepts taught in the lower grades. It is taught to enhance the arithmetic capacity of the students. Students are often confused while multiplying and dividing the fractions. This article will review the techniques to multiply a fraction by a whole number with some examples.

Before learning how to multiply a fraction by a whole number, let us look at some basic terminologies we shall use in multiplication. Are you well aware of what fractions are? Fractions, in general, are numbers that are presented in the form of p/q. For example, 2/3, 9/2, etc. You must understand that a fractional number has two parts. The part above the ‘dash’ is the numerator, and the number below the ‘dash’ is called the denominator. All types of mathematical operations can be applied to the fractions. 

If you divide the numerator with the denominator, you either get a whole number or decimal number. It doesn’t remain a fraction anymore. So a value can never be a fraction if the form isn’t p/q.

To first learn how to multiply a fraction by a whole number, we need to learn how to multiply a fraction by another fraction. Let us know about the same in the article.

How to Multiply Two Fractions

Consider we have two fractions, d/c and j/k. To multiply these fractions, you need to write them properly against each other. Then see the numerators and denominators of the fractions. Multiply the numerators together (d x j) = x. Write the result of the multiplication of the numerators in different fractions, say ‘x’.

Similarly, multiply the denominators of the fractions. Write the multiplicative result of the denominator below the resultant numerator, (c x k) = y. Thus the multiplication of two fractions can be represented as:

d/c x j/k = (d x j) / (c x k) = x/y

Let us look at an example for a better understanding. 

Example 1: Multiply the fractions ⅘ and 3/7.

Solution: Look at the stepwise solution below→

Step 1: Write the numbers together using the multiplication symbol. 

             4/5 x 3/7

Step 2: Multiply the numerators and write the result as a new fraction. 

           (4 x 3) = 12

Step 3: Now multiply the denominators and write them below the resultant numerator using a dash.

           (5 x 7) = 35

Step 4: Arrange the fraction properly: 12/35. 

The fraction after the multiplication of 4/5 with 3/7 is 12/35. If the numbers in the fractions are multiple of some smaller number, then you can easily reduce the fraction in a simpler form. The only condition is that if you divide the numerator by any number ‘a’, you need to divide the denominator by the same number ‘a’.

Example 2: Simplify the fraction 12/9.

Solution: We can see that both numerator = 12 and denominator = 9 are multiple of 3. Hence we shall divide the numerator by 3 and the denominator by 3. This gives us

12/ 3 = 4 

9/3 = 3 

Therefore the resultant fraction is 4/3, which is an improper function. 

How Do You Multiply a Fraction by a Whole Number 

Now we have learned how to multiply two fractions. In this section, you will learn how do you multiply a fraction by a whole number. Consider we have a fractions d/c and a whole number ‘k’. The first step to multiply a fraction and a whole number will always begin with writing them properly against each other with the multiplication sign. We don’t have a ‘p/q’ form in the case of whole numbers. Hence we will first change the number into fraction form. To do so, place a dash below the whole number with 1 as its denominator. Now we have the whole number as ‘p/q’ form. 

Now we follow similar steps that are mentioned in the multiplying fraction part. Now see the numerators and denominators of the fractions and the whole number. Multiply the numerators with the whole number ‘k’, thus (d x k) = x. Now multiply the fraction’s denominator with 1 (because the denominator of the whole number is 1). This gives (c x 1 = c). Thus the multiplication of two fractions can be represented as:

d/c x k/1 = (d x k) / (c x 1) = x/c

In the case of multiplying fractions with whole numbers, the fraction’s denominator is retained. Thus whenever you find such a type of multiplication, place the denominator as that of a fraction and multiply the numerator.

Example 1: Multiply the fraction 5/11 by 6. 

Solution: Look at the stepwise solution given below→

Step 1: Write the numbers together using the multiplication symbol. 

             5/11 x 6

Step 2: Convert the whole number into the fractional form.

             5/11 x 6/1

Step 3: Multiply the numerators and write the result

             5 x 6 = 30

Step 4: Now multiply the denominators and write them below the resultant numerator using a dash.

           (11 x 1) = 11

Step 5: Arrange the fraction in the proper form: 30/11.

Alternate method 

Step 1: Write the numbers as 5/11 x 6.

Step 2: Since we know that the denominator will remain retained in such cases, we need to multiply the numerators.

Step 3: Multiply the numerators = 5 x 6 = 30

Step 4: Write the results in fractional form = 30/11

Hope now the question of how I multiply fractions by a whole number is clear. Let us look at a very interesting idea next.

Example 2: Multiply 11 by 3/11

Solution: The first step will always be to mention numbers with the multiplication symbol properly. Hence 

11 x 3/11 

Following the steps as above, we get the resultant fraction as 33/11 = 3

But we can observe that if we eliminate 11 from the beginning, we still get the answer as 3.

Thus from this example, we incur a very important concept. Let us learn about that next.

Special Case

As mentioned in the above example. Suppose a fraction is multiplied by a whole number that is either a multiple of the denominator or equal to the denominator. In that case, you can eliminate or reduce the numbers before actual multiplication. Understand this with the examples below:

Example 1: Multiply 1/7 by 7.

Solution: First, we write the numbers given in the proper format.

1/7 x 7 

We can see that the whole number and denominators are the same. Hence we can eliminate them and get the fraction’s numerator is the answer. 

Here 7 eliminates 7, and the answer is 1. 

Example 2: Multiply 3/7 by 21.

Solution: Initially properly formatting the multiplication

3/7 x 21 

We observe that the denominator and the whole numbers are multiples of 7. Hence we reduce them to their lowest form. 7 divided by itself becomes 1 and 21 divided by 7 becomes 3. Now we have 3 x 3 remaining in the numerator. Hence the answer to this question is 9.

Multiplying a Fraction with Zero

As we all know, the whole numbers are the set of real numbers starting from zero and stretching to positive infinity. We have already seen all the cases of what happens when a fraction is multiplied by a whole number. If you multiply a fraction by 1, it yields the number itself. But, what happens when you multiply a fraction by zero.

We know that anything multiplied by zero is 0. Hence when a fraction is multiplied by zero, the resulting fraction is zero or 0/1.

Example: Multiply 22/17 by 0.

Solution: Writing the fractions in the proper form we get 

22/7 x 0 

Since we know that a number multiplied by zero is 0. Hence,

22/7 x 0 = 0

What happens when we multiply the number by the reciprocal of zero?

The reciprocal of a number is when the numerators and denominators interchange their positions. Since 0 = 0/1 hence its reciprocal is 1/0. Assuming we have a fraction ‘a/b’. Using the steps we learnt up till now, a x 1 = a and b x 0 = 0. Hence the resulting fraction is a/0. In mathematics, 1/0 or anything divided by zero is not defined. Hence we cannot multiply the number with the reciprocal of zero.

Example of Multiplying a Fraction by a Whole Number

Word problem: Jill is making homemade hot chocolate. She uses one-fourth teaspoon of the hot chocolate mix to make 1 cup of hot chocolate. Calculate the number of teaspoons required to form 10 cups of hot chocolate?

Solution: Teaspoons required for 1 cup = ¼

                 Teaspoons required for 10 cups = ¼ x 10

                 → 10/4

 We can reduce it by 2, 

                 → 5/2

Hence the number of teaspoons required to make 10 cups of hot chocolate is 5/2.

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