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 $90$ .

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

$\underline{0.99\div0.9=1.1}$