Concept of Multiplication in Whole Numbers and Fractions (Definition, Types and Examples) - BYJUS

# Concept of Multiplication in Whole Numbers and Fractions

Multiplication is one of the four basic operations in math. It can be used for solving problems involving repeated addition. Whole numbers are a set of numbers that include 0,1, 2, 3, and so on, and fractions are the numbers that exist between whole numbers. Let’s learn how to multiply whole numbers with fractions....Read MoreRead Less

## Whole numbers

A set of numbers without fractions, decimals, or even negative integers is known as a whole number. It consists of a set of positive integers and zeros. A set of whole numbers comprises natural numbers, including zero (0). In math, this set of whole numbers is given as {0, 1, 2, 3, …} which is represented by the letter W.

W = {0, 1, 2, 3, 4, …}

## Fractions

Fractions are parts of a whole or a collection that is equal in size. Each part of a whole divided into equal parts is a fraction of the whole. A fraction has two parts. The numerator is the number at the top of the line. It specifies how many equal parts have been taken from the entire collection or whole. The number below the line is the denominator. It displays the total number of equal parts that can be divided into the whole or the total number of equal parts in a collection.

For example, an orange is divided into two parts that are equal in size. As a result, each part is one-half $$(\frac{1}{2})$$.

## Multiplying a fraction by a whole number

To multiply a fraction by a whole number, follow these steps:

1. The whole has one as the denominator, so write it in a fraction form.

2. Multiply the whole by the numerator.

3. If necessary, simplify. Consider writing your answer as a mixed number if it is greater than one.

For example, $$\frac{1}{5}\times 3 = ?$$

To begin, convert the whole number to a fraction with one as the denominator.

$$3 = \frac{3}{1}$$

Now, multiply the numerators and the denominators.

$$\frac{1}{5}\times \frac{3}{1} = \frac{1\times 3}{5\times 1} = \frac{3}{5}$$

Lastly, simplify. Since $$\frac{3}{5}$$ is not greater than 1, do not change it to a mixed number.

So, $$\frac{1}{5}\times 3=\frac{3}{5}$$

• When you multiply a fraction by a whole number, it is similar to repeated addition.

Look at the above example, $$\frac{1}{5}\times 3$$. It’s the same as $$\frac{1}{5}+\frac{1}{5}+\frac{1}{5}$$.

Remember that $$\frac{1}{5}\times \frac{3}{1}=\frac{1\times 3}{5\times 1}=\frac{3}{5}$$. There are three pieces, each of which is a fifth of the whole.

Examples of multiplying whole numbers and fractions using the number line:

1. You have four gallons of paint on hand and you use 34 gallons from it. How much paint did you use?

Solution:

Rewriting the question is a good way to think about it.

What is $$\frac{3}{4}\times 4?$$ (or) $$\frac{3}{4}\times 4=?$$

Step 1: Using the number line, we have to multiply the fraction $$\frac{3}{4}$$ by the whole number 4. First, draw a line and make hash marks. Let’s say that each hash mark represents a $$\frac{3}{4}$$. So, the partitions will be $$\frac{3}{4}$$$$\frac{6}{4}$$$$\frac{9}{4}$$$$\frac{12}{4}$$, and $$\frac{15}{4}$$.

Step 2: Now, $$\frac{3}{4}$$ times of 1 means we have to make one jump to get to $$\frac{3}{4}$$ from 0 on the number line so that $$\frac{3}{4}$$ times 1 is going to be $$\frac{3}{4}$$.

And $$\frac{3}{4}$$ times of 2 means we have to make two jumps to get to $$\frac{6}{4}$$ from 0 on the number line so that $$\frac{3}{4}$$ times 2 is going to be $$\frac{6}{4}$$.

Also, $$\frac{3}{4}$$ times of 3 means we have to make three jumps to get to $$\frac{9}{4}$$ from 0 on the number line so that $$\frac{3}{4}$$ times 3 is going to be $$\frac{9}{4}$$.

Similarly, $$\frac{3}{4}$$ times of 4 means we have to make four jumps to get to $$\frac{12}{4}$$ from 0 on the number line so that $$\frac{3}{4}$$ times 4 is going to be $$\frac{12}{4}$$.

Step 3: Each part is $$\frac{3}{4}$$ gallons and you used 4 of them. Written as multiplication,

we have $$\frac{3}{4}\times4=\frac{12}{4}$$ or 3 (By observing the number line)

Therefore, you used $$\frac{12}{4}$$ or 3 gallons of paint.

## Multiplying fraction and whole number

Multiply the fraction’s numerator by the whole number. Then, write the product over the denominator. A common factor is a factor that two or more whole numbers have in common. For example, 2 and 4 share 2 as the common factor.

Algebraic formula: $$\text{whole number}\times \frac{numerator}{denominator}=\frac{\text{whole number}\times \text{numerator}}{denominator}$$, where the denominator is never equal to 0.

Examples of multiplying fractions and whole numbers:

Example 1 :

Find: $$2\times\frac{7}{8}=?$$

Solution:

$$2\times\frac{7}{8}$$

$$=\frac{2\times 7}{8}$$( Multiply the numerator and the whole number, then write the product over the denominator)

$$=\frac{14}{8}$$

$$=\frac{7}{4}$$ or $$1\frac{3}{4}$$ (simplified)

So, the product is $$1\frac{6}{8}$$

Example 2 :

Find: $$9\times\frac{2}{3}=?$$

Solution:

$$9\times\frac{2}{3}$$

$$=\frac{9\times 2}{3}$$ ( Multiply the numerator and the whole number, then write the product over the denominator)

$$=\frac{18}{3}$$

$$=6$$         (simplified)

So, the product is $$6$$.

Example 3 : Water makes up about $$\frac{6}{10}$$ of the weight of a watermelon. A watermelon weighs 30 lbs. in total. How much water does the watermelon contain?

Solution:

To find $$\frac{6}{10}$$ of  30, multiply.

$$30\times\frac{6}{10}$$

$$=\frac{30\times 6}{10}$$ (Multiply the numerator and the whole number, write the product over the denominator)

$$= 18$$       (simplified)

So, the watermelon contains $$18$$ pounds of water.