How to convert between percents, fractions, and decimals

Percentages refer to fractions of a whole; that is, whatever you're looking at, the percentage is how much of the whole thing you have. For instance, "50%" means " 1/2 "; "25%" means " 1/4 "; "40%" means " 2/5 "; et cetera.

Often you will need to figure out what percentage of something another thing is. For instance, if a class has 26 students, and 14 are female, what percentage of the students are female?  It is 14 out of 26, or 14/26 = 0.538461538462..., or about 54%. (For more information on percent word problems, look at the Percent of lesson.)

"Percent" is actually "per cent", meaning "out of a hundred". (I think it's Latin.) You can use this fact, along with the fact that fractions mean division, to convert between fractions, percents, and decimals.


Percent to Decimal

Percent-to-decimal conversions are easy; you mostly just move the decimal point two places. The way I keep it straight is to remember that 50%, or one-half, of a dollar is $0.50. In other words, you have to move the decimal point two places to the left when you convert from a percent (50%) to a decimal (0.50). Some more examples are:

    27% = 0.27
    104% = 1.04

    0.5% = 0.005


Percent to Fraction

Percent-to-fraction conversions aren't too bad. This is where you use the fact that "percent" means "out of a hundred". Convert the percent to a decimal, and then to a fraction. For instance:

    40% = 0.40 = 40/100

Now you can reduce the fraction:

    40/100 = 4/10 = 2/5

Most conversions are simple like this, but some require a little extra care. The reason I converted to a decimal first is that the number of decimal places tells me how many zeroes to have underneath. Notice that "0.40" can also be written as "0.4". Then 0.4 = 4/10  =  2/5, which is the same answer as before. It works out because "0.40" has one decimal place and "10" has one zero. This concept helps in more complicated problems:

    104% = 1.04 = 104/100 = 26/25

Another example:

    0.5% = 0.005 = 1/200

    33 1/3% = 0.33 1/3 = (100/3)/(100/1) = 1/3

    12 1/2% = 12.5% = 0.125 = 125/1000 = 1/8


Decimal to Fraction

The technique I just demonstrated lets you convert any terminating decimal to a fraction.

("Terminating" means "it ends", unlike, say, the decimal for 1/3, which goes on forever. A non-terminating AND NON-REPEATING decimal CANNOT be converted to a fraction, because it is an irrational (non-fractional) number. You should probably just memorize some of the more basic repeating decimals, like 0.33333... = 1/3 and 0.666666... = 2/3. Check out the table at the bottom of this page.)

Any terminating decimal can be converted to a fraction by counting the number of decimal places, and putting the decimal's digits over 1 followed by the appropriate number of zeroes. For example:

    0.46 = 46/100 = 23/50

    1.5 = 15/10 = 3/2

    10.2 = 102/10 = 51/5

    0.0003 = 3/10000

In the case of a repeating decimal, the following procedure is often used. Suppose you have a number like 0.5777777.... This number is equal to some fraction; call this fraction "x". That is:

    x = 0.5777777...

There is one repeating digit in this decimal, so multiply x by "1" followed by one zero; that is, multiply by 10:

    10x = 5.777777...

Now subtract the former from the latter:

    10x - x = 9x = 5.2000000...

That is, 9x = 5.2 = 52/10 = 26/5. Solving this, we get x = 26/45. (You can verify this by plugging "26 45" into your calculator and seeing that you get "0.5777777..." for an answer.)

If there had been, say, three repeating digits (such as in 0.4123123123...), then you would multiply the x  by "1" followed by three zeroes; that is, you would multiply by 1000. Then subtract and solve, as in the above example. And don't worry if you have leading zeroes, as in "0.004444..."; the procedure will still work.


Decimal to Percent

Decimal-to-percent conversions are simple: just move the decimal point two places to the right. (Remember, $0.50 is one-half, or 50%, of a dollar.) For example:

    0.23 = 23%
    2.34 = 234%

    0.0097 = 0.97%

(Note that 0.97% is less than one percent. It should not be confused with 97%, which is 0.97 as a decimal.)


Fraction to Decimal

If you remember that fractions are division, then this is easy. The calculator can do the work for you, because you can just have it do the division. For example:

    3/5 = 0.6

    1/3 = 0.333333....

The bar is placed over the repeating digits, for convenience sake.

    3/4 = 0.75

    2/7 = 0.285714285714...

When converting fractions to decimals, you may be told to round to a certain place or to a certain number of decimal places. For instance, looking at that last example, 2/7 as a decimal rounded to the nearest tenth (rounded to one decimal place) is 0.3; to the nearest hundredth (to two decimal places) is 0.29; to the nearest thousandths (to three decimal places) is 0.286; to the nearest ten-thousandths (to four decimal places) is 0.2857; et cetera. If you're not sure how you should format your answer, then give the "exact" form and the rounded form:

    2/7 = 0.285714 285714 285714... or about 0.286

Note that the rounded form can be useful for word problems, where a final answer in rounded form may be more practical than a repeating decimal.


Fraction to Percent

This conversion starts the same as the previous one, but the final answer can come in a couple different formats sometimes. You always start by doing the division (fractions are division, remember!), and then (usually) you move the decimal point two places to the right. For example:

    3/4 = 0.75 = 75%

    3/2 = 1.5 = 150%

    5/8 = 0.625 = 62.5%

However, sometimes the "decimal expansion" doesn't end. This is where the answer can come in a couple different formats. You can either round the answer, or use a fraction inside the percent. For instance:

    7/18 = 0.38888888....

You can round this to, say, 0.389 = 38.9%. But if you aren't supposed to round, put out a sheet of paper and do the long division. You'll need to get TWO decimal places of answer across the top, and then look at the remainder at the bottom:

     7.00  18 = 0.38, with remainder 16

Fractions are division, so I took the 7and divided by the 18. I kept going until I had TWO decimal places (the ".38") across the top. At that point, the remainder is 16. If you think back to elementary school, you handle the remainder by putting it over the divisor (18, in this case), and tacking it on to the number across the top. In this case, I get:

    0.38 16/18 = 38 8/9%

So 7/18, expressed as an unrounded decimal, is 38 8/9%. This probably looks a little weird, so let's do a couple more examples. For instance, other than memorizing, how are you supposed to know that 0.333333... = 1/3? Here's how:

    1/3 = 0.333333...

This doesn't end, so do the long division by hand:

    1.00  3 = 0.33, with remainder 1

Note that the remainder is 1 and the divisor is 3, so you'll be tacking a " 1/3 " on to the "0.33" from the top:

    0.33 1/3 = 33 1/3%

Here's a messier example that you won't have memorized:

    19/35 = 0.5428571428571...

This doesn't end, so do the long division by hand:

    19.00  35 = 0.54, with remainder 10

Note that the remainder is 10 and the divisor is 35, so you'll be tacking a " 10/35 " on to the "0.54" from the top:

    0.54 10/35 = 54 2/7%

Top


Table of Common Fractions and Their Percentage Equivalents

1/2 = 50%      
1/3 = 33 1/3% 2/3 = 66 2/3%    
1/4 = 25% 3/4 = 75%    
1/5 = 20% 2/5 = 40% 3/5 = 60% 4/5 = 80%
1/6 = 16 2/3% 5/6 = 83 1/3%    
1/7 = 14 2/7% 2/7 = 28 4/7% 3/7 = 42 6/7% 4/7 = 57 1/7%
    5/7 = 71 3/7% 6/7 = 85 5/7%
1/8 = 12 1/2% 3/8 = 37 1/2% 5/8 = 62 1/2% 7/8 = 87 1/2%
1/9 = 11 1/9% 2/9 = 22 2/9% 4/9 = 44 4/9% 5/9 = 55 5/9%
    7/9 = 77 7/9% 8/9 = 88 8/9%
1/10 = 10% 3/10 = 30% 7/10 = 70% 9/10 = 90%
1/12 = 8 1/3%      

Table of Common Fractions and Their Decimal Equivalents or Approximations

1/2 = 0.5      
1/3 = 0.3333... 2/3 = 0.6666...    
1/4 = 0.25 3/4 = 0.75    
1/5 = 0.2 2/5 = 0.4 3/5 = 0.6 4/5 = 0.8
1/6 = 0.1666... 5/6 = 0.8333...    
1/7 = 0.142857142857... 2/7 = 0.285714285714...
3/7 = 0.428571428571... 4/7 = 0.571428571428...
5/7 = 0.714285714285... 6/7 = 0.8571428571428...
1/8 = 0.125 3/8 = 0.375 5/8 = 0.625 7/8 = 0.875
1/9 = 0.111... 2/9 = 0.222... 4/9 = 0.444... 5/9 = 0.555...
    7/9 = 0.777... 8/9 = 0.888...
1/10 = 0.1 3/10 = 0.3 7/10 = 0.7 9/10 = 0.9
1/12 = 0.08333...    

Top