How to convert a decimal to a fraction (including repeating decimals)

Every decimal that ends, and every decimal that repeats forever, can be written as a fraction. Converting a decimal to a fraction takes three short steps for most numbers, plus one extra trick for repeating decimals such as 0.333…. Below you will find both methods, a list of the decimals worth knowing by heart, and the mistakes that trip people up most often.

How do you turn a decimal into a fraction?

To turn a decimal into a fraction, write the digits after the decimal point over 10, 100 or 1,000, depending on how many decimal places there are. One decimal place goes over 10, two go over 100, three go over 1,000. Then simplify the fraction by dividing the top and bottom by the same number.

Example

Convert 0.75 to a fraction.

1. Count the decimal places. 0.75 has two, so the bottom number is 100.
2. Write the digits over it. 0.75 = 75/100.
3. Simplify. Both 75 and 100 can be divided by 25, which gives 3/4.

So 0.75 = 3/4. A quick way to check is to divide 3 by 4 on paper or a calculator. You get 0.75 again.

What is 2.5 as a fraction?

2.5 has one decimal place, so it becomes 25/10. Both numbers divide by 5, which gives 5/2. You can also keep the whole number separate. The 2 stays as it is and the 0.5 becomes 5/10, which simplifies to 1/2. So 2.5 = 5/2 = 2 1/2.

What is 0.125 as a fraction?

0.125 has three decimal places, so it becomes 125/1,000. Both numbers divide by 125, which gives 1/8. This one is worth remembering, because eighths come up often in recipes, measurements and school exercises.

Should the answer be a mixed number or an improper fraction?

Both are correct. 1.25 can be written as 5/4 (an improper fraction, with a top number bigger than the bottom) or as 1 1/4 (a mixed number, with a whole part and a fraction). Teachers in some places prefer one form, so check what the exercise asks for. To switch, 5/4 means four quarters (one whole) plus one more quarter.

How do you simplify the fraction?

To simplify a fraction, divide the top and bottom by the largest number that goes into both, called the greatest common divisor. If you can't see it straight away, divide by small numbers such as 2, 5 or 10 again and again until nothing else divides both. A fraction is fully simplified when the only number that divides both is 1.

Example

Convert 0.36. It has two decimal places, so it starts as 36/100. Both numbers divide by 4, which gives 9/25. Now 9 and 25 have no common divisor except 1, so 9/25 is the simplest form.

You could also take smaller steps. 36/100 divided by 2 is 18/50, divided by 2 again is 9/25. You end up in the same place. Finding the greatest common divisor quickly is a skill of its own, and the GCD game turns it into a quick challenge. It is a good warm-up before longer fraction exercises.

Which decimals are worth knowing by heart?

A small set of decimals comes up again and again, and knowing their fractions saves time. These include halves, quarters, fifths and eighths, plus the repeating decimals for thirds. With these memorized, you can often convert a decimal in your head without writing anything down.

• 0.1 = 1/10
• 0.2 = 1/5
• 0.25 = 1/4
• 0.4 = 2/5
• 0.5 = 1/2
• 0.6 = 3/5
• 0.75 = 3/4
• 0.8 = 4/5
• 0.125 = 1/8
• 0.375 = 3/8
• 0.625 = 5/8
• 0.875 = 7/8
• 0.333… = 1/3
• 0.666… = 2/3
• 0.1666… = 1/6
• 0.111… = 1/9

Example

What is 0.625? If you know that 0.125 = 1/8, then 0.625 is five times as much, so it is 5/8. Known facts like this let you build new ones. Moving back and forth between the two forms becomes much faster with regular use, and short rounds of the decimals game or the fractions game are a playful way to build that speed. Even a few minutes a day makes the list above feel familiar.

How do you convert a repeating decimal to a fraction?

For a decimal where the same digits repeat right after the decimal point, write the repeating part over the same number of nines. One repeating digit goes over 9, two go over 99, three go over 999. Then simplify. So 0.777… = 7/9, and 0.272727… = 27/99, which simplifies to 3/11.

Example

Convert 0.454545… to a fraction. The repeating part is 45, which has two digits, so it goes over 99. That gives 45/99. Both numbers divide by 9, so the answer is 5/11. You can check it by dividing 5 by 11, which gives 0.454545… again.

When the repetition starts later, such as 0.1666…, the nines rule needs one extra step. A method that always works uses a little algebra:

1. Call the number x. So x = 0.1666…
2. Multiply by 10 to move the non-repeating digit in front of the point. 10x = 1.666…
3. Multiply by 100 to move one full repeat as well. 100x = 16.666…
4. Subtract the second line from the third. The repeating parts cancel out. 90x = 15.
5. Divide. x = 15/90, which simplifies to 1/6.

Is 0.333 the same as 1/3?

No. 0.333 with exactly three digits is 333/1,000, which is a tiny bit less than 1/3. Only 0.333…, where the 3s go on forever, is exactly 1/3. When you write a repeating decimal, show that it repeats with three dots or a line over the repeating digits. Otherwise it is a different number.

Why does 0.999… equal 1?

This surprises many people. Start from 1/3 = 0.333…. Multiply both sides by 3. On the left you get 3/3, which is 1. On the right you get 0.999…. So 0.999… = 1. The nines rule agrees, because 0.999… = 9/9 = 1. It is not a number just below 1, it is another way of writing 1.

What mistakes do people make when converting decimals to fractions?

The most common mistakes are using the wrong number of zeros on the bottom, forgetting to simplify, losing the whole number part, and treating a cut-off decimal as if it repeats. Each one is easy to catch if you check your answer by dividing the top by the bottom.

Example

A learner converts 0.05 and writes 5/10. That is 0.5, ten times too big. 0.05 has two decimal places, so it goes over 100. The correct answer is 5/100, which simplifies to 1/20.

Watch for these:

1. The wrong power of 10. Count every digit after the point, including zeros. 0.05 has two places, not one.
2. Not simplifying. 75/100 is correct but unfinished. The expected answer is 3/4.
3. Dropping the whole number. 3.2 is 3 2/10, which simplifies to 3 1/5, or 16/5. It is not 2/10.
4. Cutting off a repeating decimal. 0.66 is 66/100 = 33/50, while 0.666… is 2/3.

Which fractions turn into repeating decimals?

A fraction in its simplest form gives a decimal that ends if the bottom number has no prime factors other than 2 and 5. If the bottom number has any other prime factor, such as 3, 7 or 11, the decimal repeats forever. This is a quick way to know what to expect before you divide.

Example

Look at 3/8. The bottom number 8 is 2 × 2 × 2, so the decimal ends. 3/8 = 0.375. Now look at 1/6. The bottom number 6 is 2 × 3, and the 3 means the decimal repeats. 1/6 = 0.1666…. In the same way, 1/7 = 0.142857142857…, with the six digits repeating forever, because 7 is a prime other than 2 or 5.

This rule also explains why the two methods above cover every case. Decimals that end come from fractions with only 2s and 5s at the bottom, and decimals that repeat come from all the others. Either way, a decimal that ends or repeats is a fraction in disguise.