Decimal to fraction:
convert and simplify in three steps

How to convert a decimal to a fraction

A decimal is a fraction whose denominator is 10, 100, 1,000 and so on. To convert one, write the digits after the point as the numerator and put 1 followed by the same number of zeros underneath. Then simplify.

So 0.7 is 7/10, 0.07 is 7/100 and 0.007 is 7/1,000. The place value of the last digit decides the denominator, which is why 0.07 and 0.7 are very different fractions.

The method works because each place after the point is a tenth of the one before it. Tenths, hundredths and thousandths are fractions already; the decimal point is just a compact way of writing them. Converting is a matter of reading the last place value aloud: 0.375 is "three hundred seventy-five thousandths", so the denominator is 1,000.

1. 1Count the digits after the decimal point. That count is d.
2. 2Write those digits as the numerator and 10ᵈ as the denominator.
3. 3Find the greatest common factor of the two numbers.
4. 4Divide the numerator and the denominator by it.

Example: 0.375 as a fraction

Check the answer by dividing back: 3 ÷ 8 = 0.375. If the division returns your starting decimal, the fraction is right.

If you cannot spot the greatest common factor, simplify in small steps. Halve both numbers while they are even, then try 5, then 3. You will reach the same answer, just in more lines.

A quick test for whether you have finished: the numerator and denominator should share no factor except 1. 9/20 is finished because 9 and 20 have no common factor, whereas 18/40 still has a 2 in common.

It also helps to know which decimals will give a neat answer. Any decimal that ends, such as 0.45 or 0.0625, becomes a fraction whose simplified denominator contains only the factors 2 and 5. That is why 0.45 turns into 9/20 and 0.0625 into 1/16, while a denominator with a 3 or a 7 signals a recurring decimal instead.

Decimal to fraction chart for common values

Everyday weights follow the same pattern. A grocer's 0.125 kg is 1/8 kg, which is 125 g, and 0.75 kg is 3/4 kg, which is 750 g. Learn five or six of the fractions above and most everyday decimals need no working at all.

Money behaves the same way. ₹0.75 is 75/100 of a rupee, which is 3/4 of a rupee, or 75 paise. A price of ₹12.50 is 12 1/2 rupees, and ₹0.05 is 1/20 of a rupee. Seeing the fraction makes it easy to split amounts evenly without a calculator.

Decimals greater than 1: whole numbers and mixed fractions

Keep the whole part and convert only the digits after the point. So 3.6 is 3 and 6/10, which simplifies to 3 3/5. The same number as an improper fraction is (3 × 5 + 3) ÷ 5 = 18/5.

Another case: 2.35 is 2 and 35/100, which simplifies to 2 7/20, or 47/20 as an improper fraction. Pick whichever form the question asks for. Mixed numbers read better for measurements, while improper fractions are easier to multiply and divide.

Adding, subtracting and converting between those two forms is covered in the mixed fraction guide.

How to convert a recurring decimal to a fraction

A decimal that repeats forever, such as 0.666…, cannot be written over a power of 10 directly. Algebra solves it: multiply by 10 for each repeating digit, then subtract the original so the repeating tail cancels.

1. 1Let x = 0.666…
2. 2Multiply by 10: 10x = 6.666…
3. 3Subtract the first line from the second: 9x = 6.
4. 4Divide: x = 6/9, which simplifies to 2/3.

When a non-repeating digit sits before the repeat, as in 0.1666…, use two multiplications. 100x = 16.666… and 10x = 1.666…, so 90x = 15 and x = 15/90 = 1/6.

When two digits repeat, multiply by 100 instead of 10. For 0.2727…, let x = 0.2727…; then 100x = 27.2727…, so 99x = 27 and x = 27/99 = 3/11. The rule is simple: multiply by 10 raised to the length of the repeating block.

Try it yourself: five decimals with answers

Cover the answers and work each one through the three steps. Count the places, write the denominator, then simplify.

1. 10.8 becomes 8/10, which simplifies to 4/5.
2. 20.36 becomes 36/100, which simplifies to 9/25 after dividing by 4.
3. 30.004 becomes 4/1,000, which simplifies to 1/250 after dividing by 4.
4. 41.25 becomes 1 and 25/100, which is 1 1/4, or 5/4 as an improper fraction.
5. 50.0625 becomes 625/10,000, which simplifies to 1/16 after dividing by 625.

If the numerator and denominator are both even, always halve first. The remaining work is usually smaller than it looks.

Mistakes that turn up in decimal-to-fraction answers

• Treating 0.33 as 1/3. The decimal 0.33 is 33/100; only 0.333… repeating equals 1/3.
• Miscounting places: 0.05 has two decimal places, so it is 5/100 (1/20), not 5/10.
• Stopping before simplifying. 25/100 is correct but incomplete; 1/4 is the answer.
• Reading a calculator display as exact. A display of 0.142857 is rounded, so the fraction is 1/7 and not 142857/1,000,000.
• Forgetting the whole number. 4.5 is 4 1/2, not 1/2 or 5/10.
• Cancelling only one side. Dividing the numerator by 5 but not the denominator changes the value.

When a decimal has many places, such as 0.8125, the denominator looks frightening (10,000), but the simplification is the same. 8,125 and 10,000 share a factor of 625, giving 13/16. Dividing 13 by 16 returns 0.8125, which confirms it.

To go the other way, from a fraction to its decimal by division, see the fraction to decimal guide.