Fractions, Decimals and Percentages: One Number, Three Forms

Three ways to write the same value

Three quarters can be written as the fraction 3/4, the decimal 0.75 or the percentage 75%. They are not approximately equal; they are exactly the same number in three different notations. Each form is good at something different.

Fractions are exact and easy to multiply. One third is exactly 1/3, while its decimal never ends.
Decimals are easy to compare, add and type into a calculator.
Percentages are easy to understand at a glance and to compare shares of different totals.

Moving between them comes down to two operations: dividing, and multiplying or dividing by 100.

Fraction to decimal: divide

A fraction is a division. 3/8 means 3 ÷ 8 = 0.375. Type it into the basic calculator, or do it by long division: 8 goes into 30 three times (24, remainder 6), into 60 seven times (56, remainder 4), into 40 five times exactly, giving 0.375.

5/16 = 5 ÷ 16 = 0.3125.
9/40 = 9 ÷ 40 = 0.225.
7/4 = 7 ÷ 4 = 1.75. Fractions larger than 1 give decimals larger than 1.

Decimal to fraction: use place value

Write the digits over 10, 100, 1,000 and so on, according to how many decimal places there are, then simplify.

0.625 has three decimal places, so it is 625/1,000. Divide both by 125: 5/8.
0.45 = 45/100. Divide both by 5: 9/20.
2.75 = 2 + 75/100 = 2 + 3/4 = 2 3/4, or 11/4 as an improper fraction.

Simplifying uses the greatest common factor, exactly as in the fractions guide. The fraction calculator shows the simplified result and its decimal side by side.

Moving to and from percentages

A percentage is a number of hundredths, so you only move the decimal point.

Decimal to percent: multiply by 100. 0.375 = 37.5%; 0.045 = 4.5%; 1.2 = 120%.
Percent to decimal: divide by 100. 8% = 0.08; 0.5% = 0.005.
Percent to fraction: put it over 100 and simplify. 150% = 150/100 = 3/2; 12.5% = 12.5/100 = 125/1,000 = 1/8.
Fraction to percent: divide, then multiply by 100. 7/20 = 0.35 = 35%.

The percentage guide shows what to do with a percentage once you have it.

Which fractions give decimals that stop?

Some fractions give a decimal that ends (a terminating decimal), while others repeat forever. There is a simple test.

Rule: write the fraction in simplest form. If the denominator has no prime factors other than 2 and 5, the decimal terminates. If it has any other prime factor (3, 7, 11 and so on), the decimal repeats.

Fraction	Denominator's prime factors	Decimal
7/20	2 × 2 × 5	0.35 (stops)
1/16	2 × 2 × 2 × 2	0.0625 (stops)
1/40	2 × 2 × 2 × 5	0.025 (stops)
1/6	2 × 3	0.1666… (repeats)
3/11	11	0.2727… (repeats)
1/7	7	0.142857142857… (repeats)

The reason is that our number system is built on 10 = 2 × 5. A denominator made only of 2s and 5s can always be scaled up to a power of 10: 7/20 = 35/100. A denominator containing 3 can never become 10, 100 or 1,000, because no power of 10 is divisible by 3.

Simplify first. 3/12 looks as if it should repeat because 12 contains 3, but it simplifies to 1/4 = 0.25, which stops. The number of decimal places in a terminating decimal equals the larger of the two counts of 2s and 5s: 1/16 has four 2s, so it has four decimal places (0.0625).

The length of the repeating block can be surprising. 1/3 repeats one digit, 1/11 repeats two (0.0909…), and both 1/7 and 1/13 repeat six digits: 1/13 = 0.076923076923…

Repeating decimals back to fractions

A repeating decimal can always be turned into an exact fraction with a little algebra. The idea is to shift the decimal point so that the repeating parts line up, then subtract to make them cancel.

Example 1: 0.333…

Let x = 0.333…
Multiply by 10: 10x = 3.333…
Subtract: 10x − x = 3.333… − 0.333…, so 9x = 3.
x = 3/9 = 1/3.

Example 2: 0.272727… The block is two digits long, so multiply by 100. 100x = 27.2727…, and 100x − x = 27, so 99x = 27 and x = 27/99 = 3/11.

Example 3: 0.1666… Here one digit (1) comes before the repeating part. Multiply by 10 and by 100 so that both copies start their repeat right after the point: 10x = 1.666… and 100x = 16.666… Subtracting gives 90x = 15, so x = 15/90 = 1/6.

Example 4: 0.8333… 100x = 83.333… and 10x = 8.333…, so 90x = 75 and x = 75/90 = 5/6.

A shortcut you can use without algebra

Repeating from the first digit: write the repeating block over the same number of 9s. 0.1212… = 12/99 = 4/33. 0.142857142857… = 142857/999999 = 1/7.
With digits before the repeat: take all the digits up to the end of the first repeat, subtract the digits that do not repeat, and divide by as many 9s as there are repeating digits followed by as many 0s as there are non-repeating digits. For 0.58333…: (583 − 58) ÷ 900 = 525/900 = 7/12. For 0.1666…: (16 − 1) ÷ 90 = 15/90 = 1/6.

The same logic shows that 0.999… = 9/9 = 1 exactly. It is not "just below" 1; it is another way of writing 1.

Table of common values

Fraction	Decimal	Percent
1/2	0.5	50%
1/3	0.333…	33.33%
2/3	0.666…	66.67%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/6	0.1666…	16.67%
1/7	0.142857…	14.29%
1/8	0.125	12.5%
3/8	0.375	37.5%
5/8	0.625	62.5%
7/8	0.875	87.5%
1/9	0.111…	11.11%
1/10	0.1	10%
1/12	0.08333…	8.33%
1/16	0.0625	6.25%
1/20	0.05	5%
1/25	0.04	4%

Percentages for repeating decimals are rounded to two decimal places.

Comparing and approximating

Decimals make comparison easy. Which is larger, 5/8 or 7/11? 5/8 = 0.625 and 7/11 = 0.6363…, so 7/11 is larger, though only slightly.

Fractions can also approximate awkward decimals. The well-known 22/7 = 3.142857… is close to π = 3.14159…, correct to two decimal places. 355/113 = 3.14159292… is correct to six decimal places, remarkable for such small numbers.

What your calculator shows

A calculator can only show a limited number of digits, so 2/3 appears as something like 0.666666666667, with the last digit rounded up. That display is an approximation. If you copy it and multiply by 3, you may get 2.000000000001 on some devices rather than exactly 2. When you need an exact answer, keep the fraction, or use the fraction calculator, which works with whole-number tops and bottoms and never rounds. More on rounded displays is in the rounding guide.

Common mistakes

Reading 1/4 as 0.4. 1/4 is 1 ÷ 4 = 0.25.
Moving the decimal point the wrong way. 0.5 is 50%, not 0.5%.
Judging termination before simplifying. 3/12 = 1/4 terminates.
Treating 0.33 as exactly one third. It is 33/100, slightly less than 1/3.
Using too few 9s. A two-digit repeating block needs 99, not 9.

FAQ

Is 0.1666… the same as 1/6?
Yes, exactly. The algebra above shows 90x = 15, which gives 1/6.

Can every decimal be written as a fraction?
Every terminating or repeating decimal can. Decimals that never end and never repeat, such as π or √2, cannot be written as a fraction of whole numbers.

How do I write 0.5% as a fraction?
0.5% = 0.005 = 5/1,000 = 1/200.

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