Converting Between Decimals, Fractions, and Percents

Notes

Fraction to Decimal Remember that fractions are division. For example:

Try these examples: 3 = 4

2 = 7

Percent to Decimal 27 If percent means “out of one hundred” then 27% means 27 out of 100 or so 0.27. Divide the percent by one 100 hundred for the equivalent decimal. Try these examples:

104% =

0.5% =

Decimal to Percent Multiply the decimal by one hundred. For example: 0.23 = 0.23 × 100 = 23%. Try these examples:

2.34 =

0.0097 =

Terminating Decimal to Fraction Any terminating decimal can be converted to a fraction by counting the number of decimal places, and putting the 25 5 decimal's digits over a multiple of ten. For example 2.5 =  . 10 2

Try these examples:

1.5 =

10.2

0.0003 =

Percent to Fraction Use the fact that "percent" means "out of a hundred". Convert the percent to a decimal, and then to a fraction. For 40 40 4 2 example 40% = 0.40 = then reduce   . 100 100 10 5

©Dr Barbara Boschmans 1/5

Notes

Try these examples:

104% =

0.5% =

1 33 % = 3

1 12 % = 2

Non-terminating, Repeating Decimal to Fraction In the case of a non-terminating, repeating decimal, the following procedure is used. Suppose you have a number like 0.333333.... This number is equal to some fraction; call this fraction "x". That is:

Let x = 0.333333...

There is one repeating digit in this decimal, so multiply x by 10 to bring one repeating part in front of the decimal:

Then 10 x = 3.33333...

Subtract: 10 x = 3.33333... - x = 0.33333... 9 x = 3

3 1 So x =  9 3

1 You might have already known that 0.3  from previous experiences, but it is an example to show you the 3 procedure of converting a non-terminating, repeating decimal to a fraction.

Let’s do another example: Suppose you have a number like 0.5777777.... This number is equal to some fraction; call this fraction "x". That is:

Let x = 0.5777777...

There is one repeating digit in this decimal, so multiply x by 100 to bring the non-repeating part and the repeating part in front of the decimal:

Then 100 x = 57.77777...

Subtract: 100 x = 57.777777... - 10 x = 5.7777777... [Remember: your goal is to eliminate the repeating decimal part so 90 x = 52 subtracting x would not do this, but subtraction 10 x will!]

Then 90 x = 52 52 26 So x =  90 45

©Dr Barbara Boschmans 2/5

Notes

Try these examples:

0.777777….. =

0.2525252525…. =

2.345345345…. =

0.45666666…. =

0.00022222…. =

©Dr Barbara Boschmans 3/5

Notes

Fraction to Percent Convert to a decimal and then to a percent if you have a terminating decimal. For example:

Try these examples: 3 = 2

5 = 8

For non-terminating decimals you use a fraction inside the percent. For instance:

So 0.38888888… = 38.888888…%. The goal is to convert 0.888888… to a fraction, using the technique of converting non-terminating, repeating decimals to fractions.

Let x = 0.888888...

There is one repeating digit in this decimal, so multiply x by 10 to bring one repeating part in front of the decimal:

Then 10 x = 8.88888...

Subtract: 10 x = 8.88888... - x = 0.88888... 9 x = 8

8 So x = 9

7 8 So the final answer:  38 % 18 9

Here's a messier example:

This is non-terminating, so 0.5428571428571… = 54.28571428571% and you want to convert the 0.2857142857 to a fraction. You can also do this by decimal long division:

Note that the remainder is 10 and the divisor is 35, so the decimal answer is:

Notes

Try these examples:

89 = 37

297 = 81

421 = 23