Convert Decimals to Fractions
Decimals can easily be represented as fractions with a denominator
of 10, 100, 1000, 10,000 etc. based on how many decimal places
the decimal number has. From here, we can try and simplify if we
wish.
There are 2 ways we will look at doing this:
Method 1: Work out the denominator first by counting how
many decimal places. Just saying the decimal gives away the
fraction
1 decimal place means we are working with tenths
Denominator is /10
2 decimal places means we are working with hundredths
Denominator is /100
3 decimal places means we are working with thousandths
Denominator is /1000
etc.
In the following examples we DO NOT SIMPLIFY
Example
0.345
Notice 3 decimal places tells us we are working with thousandths.
How many thousandths do we have? We have 345 of them.
So the answer is 345/1000.
Example
0.82
Notice 2 decimal places tells us we are working with hundredths.
How many hundredths do we have? We have 82 of them.
So the answer is 82/1000.
Example
0.05
Notice 2 decimal places tells us we are working with hundredths.
How many hundredths do we have? We have 5 of them.
So the answer is 5/100.
Example
4.7
Notice 1 decimal place tells us we are working with tenths.
How many tenths do we have? We have 47 of them.
So the answer is 47/10. (Note, this is an improper fraction and could also be
written as 4 and 7/10).
In the following examples we go on and simplify the answer
Example
0.005
Notice 3 decimal places tells us we are dealing with thousandths.
How many thousandths do we have? We have 5 of them.
So the answer is 5/1000.
Now, 5 goes into both 5 and 1000. So we simplify to 1/200. (5 goes into 1000
200 times. How did I get this? 5 goes into 10 twice. And 10 goes 1000 100 times.
So 2 x 100 = 200).
Example
0.75
Notice 2 decimal places tells us we are dealing with hundredths.
How many? 75 of them.
So the answer is 75/100.
Now 25 goes into both 75 and 100. So we simplify to ¾. Answer is ¾.
REMEMBER THIS:
You should know a few standard decimals off by heart
0.5 is ½
0.25 is ¼
0.75 is ¾
0.33333…. is 1/3
0.1 = 1/10
0.2 = 1/5
0.1111…. is 1/9
0.6666… is 1/6
0.125 = 1/8
On top of that, extra zeroes in front of any of these decimals just means the
denominator is 10 times bigger.
0.05 is 1/20
0.005 is 1/200
0.075 = 3/40
0.01 = 1/100
0.02 = 1/50
0.0125 = 1/80
0.00125 = 1/800
etc.
Knowing these allows you to SAVE time by not having to do calculations. Even if
you don’t know them, you can deduce them by working it out relatively quickly
using Method 1 or Method 2 as discussed in these notes.
Method 1: Write the decimal as a fraction over 1, then keep
multiplying by 10 until you get a whole number on the top…
then simplify
To convert a Decimal to a Fraction follow these steps:
Step 1: Write down the decimal divided by 1, like this:
decimal
/1
Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there
are two numbers after the decimal point, then use 100, if there are three then use 1000, etc.)
Step 3: Simplify (or reduce) the fraction
Example: Express 0.75 as a fraction
Step 1: Write down 0.75 divided by 1:
0.75
1
Step 2: Multiply both top and bottom by 100 (there were 2 digits after the
decimal point so that is 10×10=100):
× 100
0.75
1
=
75
100
× 100
(Do you see how it turns the top number
into a whole number?)
Step 3: Simplify the fraction (this took me two steps):
÷5
75
100
÷5
15
=
20
÷5
3
=
4
÷5
3
Answer = /4
Note: 75/100 is called a decimal fraction and 3/4 is called a common
fraction !
Example: Express 0.625 as a fraction
Step 1: write down:
0.625
1
Step 2: multiply both top and bottom by 1,000 (there were 3 digits after
the decimal point so that is 10×10×10=1,000)
625
1,000
Step 3: Simplify the fraction (it took me two steps here):
÷ 25
625
25
=
1,000
÷5
40
÷ 25
5
=
8
÷5
5
Answer = /8
Example: Express 0.333 as a fraction
Step 1: Write down:
0.333
1
Step 2: Multiply both top and bottom by 1,000 (there were 3 digits after
the decimal point so that is 10×10×10=1,000)
333
1,000
Step 3: Simplify Fraction:
Can't get any simpler!
333
Answer =
/1,000
But a Special Note:
If you really meant 0.333... (in other words 3s repeating forever which is called 3
recurring) then we need to follow a special argument. In this case we would write
down:
0.333...
1
Then MULTIPLY both top and bottom by 3:
×3
0.333...
1
=
0.999...
3
×3
And 0.999... = 1 (Does it? - see the 9 Recurring discussion for more if you are
interested), so:
Answer = 1/3