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# Multiplication & Division of Decimals

August 3, 2022

Arithmetic operations with decimal numbers find their place in many real-world problems. For example, if the cost of chocolate is ₹$15.65$ and you want to buy $6$ such chocolates. The amount that you will pay to the shop owner is a product of $15.65$ and $6$. Similarly, if you have a coloured sheet of paper of length $1.5$ m and want to share it among your $3$ other friends, then you will be dividing $1.5$ by $4$.

In this article, you will learn multiplication and division of decimal numbers that will help you solve problems similar to the ones mentioned above.

## Multiplication of Decimal Numbers

The process of multiplication of decimals is quite similar to the multiplication of whole numbers. The only difference is placing the decimal point at an appropriate place in the product.

The placement of a decimal point in the result follows a simple rule and it is – “Number of digits after the decimal point in the result is the sum of the number of digits in the numbers being multiplied after their respective decimal points”.

For example, in the case of $2.3 \times 3.75$, the number of digits after the decimal point will be $3 \left(1 + 2 \right)$.

### Multiplying Decimals with Whole Numbers

Multiplication of decimals with whole numbers is similar to the multiplication of whole numbers, the only difference being that the number of digits after a decimal point in the result will be the same as the number of digits after a decimal point in the decimal number being multiplied. (A whole number does not contain decimal places).

### Steps for Multiplying Decimals with Whole Numbers

These are the steps to multiply a decimal number with a whole number.

Step 1: Initially, ignore the decimal point and multiply the two numbers in a normal way

Step 2: After multiplication, count the number of decimal places in the decimal number. The product obtained after multiplication will have the same number of decimal places.

Step 3: Place the decimal point in the obtained product obtained in Step 1′

### Examples

Ex 1: Multiply $2.8$ and $4$.

Numbers after ignoring the decimal places are $28$ and $4$.

Product of $28$ and $4$ is $28 \times 4 = 112$.

The number of decimal places in $2.8$ is $1$.

Therefore, $2.8 \times 4 = 11.2$.

Ex 2: Find the product of $7$ and $3.96$.

Numbers after ignoring the decimal places are $7$ and $396$.

Product of $7$ and $396$ is $7 \times 396 = 2772$.

The number of decimal places in $3.96$ is $2$.

Therefore, $7 \times 3.96$ is $27.72$.

Ex 3: Find the product of $17.2$ and $5$.

Numbers after ignoring the decimal places are $172$ and $5$.

Product of $172$ and $5$ is $172 \times 5 = 860$.

The number of decimal places in $17.2$ is $1$.

Therefore, $17.2 \times 5$ is $86.0 = 86$. (Since $0$ in $86.0$ is a trailing zero).

Maths can be really interesting for kids

### Multiplying Decimals By 10, 100, 1000, …

When a decimal number is multiplied by $10$, $100$, $1000$ or any other power of $10$, we just simply shift the decimal point towards the right as many places as the number of zeroes in the power of $10$. If that many decimal places are not there then fill the remaining places by $0$s.

### Examples

Ex 1: $56.28 \times 10$

The number of zeroes in $10$ is $1$, so shift the decimal point by $1$ position towards the right.

$56.28 \times 10 = 562.8$

Ex 2: $49.32 \times 100$

The number of zeroes in $100$ is $2$, so shift the decimal point by $2$ positions towards the right.

$49.32 \times 100 = 4932$

Ex 3: $76.84 \times 1000$

The number of zeroes in $1000$ is $3$, but there are only $2$ decimal places, therefore add one $0$ to the last and remove the decimal point.

$76.84 \times 1000 = 76840$

Ex 4: $95.6 \times 1000$

The number of zeroes in $1000$ is $3$, but there is only a $1$ decimal place, therefore add two $0$s to the last and remove the decimal point.

$95.6 \times 1000 = 95600$

### Multiplying Two Decimals Numbers

The rules for multiplying two decimal numbers are the general rules and apply to the cases learned above also. Whenever you’re multiplying two decimal numbers, first of all, ignore the decimal point in both the numbers and use the method of multiplying two whole numbers to get the product. After that count the number of decimal places in both the numbers. The number of decimal places in the product is the sum of the decimal places of the two numbers.

### Steps For Multiplying Two Decimals Numbers

Step 1: Ignore the decimal point and multiply the two numbers in a normal way.

Step 2: After multiplication, count the total number of decimal places in both the numbers

Step 3: Place the decimal point after the number of places obtained in Step 2 in the product obtained in Step 1

### Examples

Ex 1: $65.2 \times 12.7$

Numbers after ignoring the decimal points are $652$ and $127$.

Product of $652$ and $127$ = $652 \times 127 = 82804$.

The number of decimal places in both the numbers $65.2$ and $12.7$ is $1$, therefore the number of decimal places in the product will be $1 + 1 = 2$.

Therefore, $65.2 \times 12.7 = 828.04$

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## Division of Decimal Numbers

The process of multiplication of decimals is quite similar to the multiplication of whole numbers. The only difference is placing the decimal point at an appropriate place in the quotient.

There can be two cases involved in the division.

• Dividing a whole number by a decimal number
• Dividing a decimal number by a decimal number

### Dividing Whole Numbers By Decimal Numbers

The steps for the process of division of a whole number by a decimal number are

Step 1: Count the number of decimal places in the decimal number

Step 2: Multiply the two numbers with the power of $10$ with power as the number obtained in step 1

Step 3: Divide the two numbers obtained in step 2

### Examples

Ex 1: Divide $28$ by $0.4$

$0.4$ is a decimal number with a number of decimal places equal to $1$, so multiply the two numbers by $10$.

$28 \times 10 = 280$ and $0.4 \times 10 = 4$

Now, divide $280$ by $4$, i.e.,  $280 \div 4$.

Remainder is $0$ means $280$ is completely divisible by $4$ and therefore, $28 \div 0.4 = 70$.

Ex 2: Divide $56$ by $3.2$

$3.2$ is a decimal number with a number of decimal places equal to $1$, so multiply the two numbers by $10$.

$56 \times 10 = 560$ and $3.2 \times 10 = 32$

The above figure shows the normal division(long division method),  you follow to divide two numbers. Here quotient is $17$ and the remainder is $16$. ($16 \lt 32$).

Next, put a decimal point in the quotient and one zero ($0$) in the divisor i.e., ($560$), and proceed with the division process.

Repeat the process, i.e., add one $0$ in the divisor and divide till the remainder becomes $0$ or you get the required number of decimal places.

Note:

When a number is divided by another number, the answer can be either of the three types:

• terminating decimal number
• non-terminating but recurring decimal number
• non-terminating and non-recurring decimal number

The remainder is $0$, therefore division process stops here and the quotient is $17.5$.

Ex 3: Divide $282$ by $4.2$

$4.2$ is a decimal number with a number of decimal places equal to $1$, so multiply the two numbers by $10$.

$282 \times 10 = 2820$ and $4.2 \times 10 = 42$

Now, divide $2820$ by $42$, i.e.,  $2820 \div 42$.

The above figure shows the normal division(long division method),  you follow to divide two numbers. Here quotient is $67$ and the remainder is $6$. ($6 \lt 42$).

Repeat the process.

Again repeat the process.

The quotient for $2820 \div 42$ is a non-terminating and non-repeating number and is equal to $67.1428…$.

### Dividing Decimal Numbers By Decimal Numbers

When you divide a decimal number by another decimal number, the first step is to count the number of decimal places in both decimal numbers and note down the highest number of decimal places.

Next, multiply both the numbers by the power of $10$ raise to the power by this number, and divide the two numbers.

To understand the process let’s consider the following examples.

### Examples

Ex 1: $4.2$ divided by $0.2$

The number of decimal places in both the numbers is $1$.

Multiply both the numbers by $10^{1} = 10$.

$4.2 \times 10 = 42$ and $0.2 \times 10 = 2$

Therefore, divide $42$ by $2$.

$4.2 \div 0.2 = 21$

Ex 2: Divide $5.6$ by $0.08$

The number of decimal places in $5.6$ is $1$ and the number of decimal places in $0.08$ is $2$. Therefore, multiply both the numbers by $10^{2} = 100$.

$5.6 \times 100 = 560$ and $0.08 \times 100 = 8$.

So, divide $560$ by $8$.

Therefore, $5.6 \div 0.08 = 70$.

## Conclusion

The process of multiplication and division of decimal numbers is very much the same as that with whole numbers. The only difference in the case of decimal numbers is that you need to take care of the decimal places in the result, i.e., quotient.

## Practice Problems

1. Multiply the following
1. $52$ by $0.3$
2. $2.7$ by $8$
3. $5.9$ by $3.8$
4. $2.75$ by $4.4$
5. $6.08$ by $0.02$
6. $12.56$ by $1.02$
2. Divide the following
1. $12$ by $0.4$
2. $12$ by $0.04$
3. $1.2$ by $4$
4. $1.2$ by $0.4$
5. $1.2$ by $0.04$
6. $1.2$ by $40$

## FAQs

### What is the formula of decimal division?

In the division of decimal numbers, we subtract the number of decimal places of the dividend and the divisor.
For example, $0.0024 \div 0.004 = 0.6$. In this case, $4 – 3 = 1$, so we will divide the numbers normally (ignoring the decimals) and then place the decimal point after $1$ place from the right.

### How do you multiply decimals by 10, 100, or 1000?

When a decimal number is multiplied by $10$, $100$, $1000$ or any other power of $10$, we just simply shift the decimal point towards the right as many places as the number of zeroes in the power of $10$. If that many decimal places are not there then fill the remaining places by $0$s.

For example, $2.56 \times 10 = 25.6$, $2.56 \times 100 = 256$, and $0.000256 \times 10000 = 2.56$.

### How do you divide decimals by 10, 100, or 1000?

When a decimal number is divided by $10$, $100$, $1000$ or any other power of $10$, we just simply shift the decimal point towards the left as many places as the number of zeroes in the power of $10$. If that many decimal places are not there then fill the remaining places by $0$s.

For example, $564.89 \div 10 = 56.489$, $564.89 \div 100 = 5.6489$, $564.89 \div 1000 = 0.56489$, and $564.89 \div 100000 = 0.0056489$.

### Can a decimal be divided by a whole number?

Yes, all whole numbers can be represented as decimal numbers with digits after the decimal point as $0$.

For example, $8 = 8.0 = 8.00 = 8.000$ or $564 = 564.0 = 564.00 = 564.000$.

### How to divide decimals by whole numbers?

The steps for the process of division of a whole number by a decimal number is
Step 1: Count the number of decimal places in the decimal number
Step 2: Multiply the two numbers with the power of $10$ with power as the number obtained in step $1$
Step 3: Divide the two numbers obtained in step $1$

For example, dividing $56$ by $0.08$
$0.08$ is a decimal number with a number of decimal places equal to $2$, so multiply the two numbers by $100$.
$0.08 \times 100 = 8$ and $56 \times 100 = 5600$

Now, divide $5600$ by $8$, i.e.,  $5600 \div 8 = 700$.

Therefore, $56 \div 0.08 = 700$.

### What is the difference between multiplying and dividing decimals?

While multiplying decimals, we add the number of decimal places of the two numbers and then place the decimal point in the product according to those number of places.

For example: $0.0014 \times 0.004 = 0.0000056$. In this case, $0.0014$ has $4$ digits after the decimal point and $0.004$ has $3$ digits after the decimal point. This gives us $3 + 4 = 7$. So, we will multiply the numbers normally and place the decimal point in the product after $7$ places from the right.

In division, we subtract the number of decimal places of the two numbers. For example, $0.00056 \div 0.004$ = 0.14. In this case, $5 – 3 = 2$, so we will divide the numbers normally (ignoring the decimals) and then place the decimal point after $2$ places from the right.

### How to divide decimals by decimals?

When you divide a decimal number by another decimal number, the first step is to count the number of decimal places in both decimal numbers and note down the highest number of decimal places.

Next, multiply both the numbers by the power of $10$ raise to the power by this number, and divide the two numbers.

To understand the process let’s consider this example..

$4.8$ divided by $0.04$
The number of decimal places in $4.8$ is $1$ and the number of decimal places in $0.04$ is $2$.

Since, $2 \gt 1$, therefore, multiply both the numbers by $100$.
$4.8 \times 100 = 480$ and $0.04 \times 100 = 4$

Now, divide $480$ by $4$.
$480 \div 4 = 120$, therefore, $4.8 \div 0.04 = 120$.