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Multiplying and dividing decimals

Tutor: Alice

# Multiplying and dividing decimals

## ​​In a nutshell

A decimal can be expressed as a whole number divided by a power of $10$. Multiplication and division with decimals can be made simpler by using whole numbers and then applying this same division at the end.

## Multiplication with decimals

There are two methods that can be used to multiply using decimals.

### Method 1

#### procedure

 ​​1. Count how many digits are to the right of the decimal point in each number and add to get the total.​ 2. Remove the decimal and multiply the numbers as if they were whole numbers.​ 3. Move the decimal point to the left of the units digit of the answer by the same number.

##### Example 1

What is $0.53 \times 2.21$?

Count how many digits are to the right of the decimal point in total.

$0.\underline{5}\space\underline{3}\rightarrow 2 \\2.\underline{2}\space\underline{1}\rightarrow2\\2+2=4$

Remove the decimals from $0.53$ and $2.21$ and multiply as if they were whole numbers.

$53\times221=11\space713$​​

Move the decimal point four spaces to the left.

$\\1\enspace1 \enspace7\enspace1\enspace3.\\\enspace\space\curvearrowleft \curvearrowleft\curvearrowleft\curvearrowleft\enspace\\1.\space1 \enspace7\enspace1\enspace3\\$​

$0.53\times2.21=\underline{1.1713}$​​

### Method 2

#### ​​Procedure

 1 Rewrite each decimal as a whole number divided by a $10,100$​ etc. 2 Multiply the whole numbers together. 3 Divide the result by the same powers of $10$​ that the whole numbers were originally divided by.

Note: You can use any method to multiply whole numbers, whether it be using a calculator, using the column method, or using the grid method.

##### Example 2

What is $0.5 \times 42.1$?

Rewrite the decimals as whole numbers divided by powers of $10$.

$0.5=5\div10\\42.1=421\div10$

Multiply the whole numbers together.

$421\times5=2105$

Divide by $10$ and $10$.

$2105\div10\div10=\underline{21.05}$​​

Note: When multiplying decimals, your final answer should always have the same total number of decimal places as the sum of of the number of decimal places you started with.

## Division with decimals

Dividing using decimals also involves two methods: one being a very similar method to multiplication.

### Method 1

#### ​​Procedure

 1 Rewrite each decimal as a whole number divided by a power of $10$. 2 If the divisor is a decimal then the division of the power of $10$​ turns into multiplication.​ 2 Complete the division of the whole numbers by using a calculator or short division (where required). 4 Divide/multiply the final answer accordingly.

##### Example 3

What is $0.132 \div 1.1$?

Rewrite each decimal as a whole number divided by a power of $10$.

$0.132=132\div1000\\1.1=11\div10$​​

​​

The divisor in this case is $1.1$ so turn its $\div10$ into $\times 10$ and divide.

$132\div11=12$

Divide by $1000$ and multiply by $10$.

$12\div1000\times10=0.12$​​

​​

$\underline{0.132\div1.1=0.12}$​​

### Method 2

Simply multiply the dividend and divisor by the same power of $10$ until both are whole numbers and divide as usual.

##### Example 4

What is $0.99\div0.9$?

Multiply by the same powers of $10$ until both are whole.

$0.99\times100=99\\0.9\times100=90$

Divide $99$​ by $90$.

$90\overset{\,\,\,\,\,\,\,\,\,\,1\,.\,\,\,1\,\,}{\overline{\smash{)}\cancel9\space^99.^90\,\,\,}} \\$​​

$\underline{0.99\div0.9=1.1}$​​

## FAQs - Frequently Asked Questions

### What are the rules for multiplying and dividing decimals?

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