Fractions

Fraction to Decimal Chart

This fraction to decimal chart converts every common fraction from halves through sixteenths, with the decimal and the percentage beside it and repeating decimals marked. Below the chart are the methods for converting in either direction by hand, and the rule that tells you which fractions terminate.

The fraction to decimal chart

A fraction is a division waiting to happen. The line between the two numbers means “divided by”, so 3/8 is 3 divided by 8, which is 0.375. That single fact converts any fraction to a decimal, and the chart below has already done the division for the fractions you meet most often.

Everything is in lowest terms. If the fraction you want is not listed, reduce it first and it will be: 6/16 is 3/8, so it is 0.375, and 4/12 is 1/3, so it is 0.3333… The percentage column is the decimal moved two places right, which is why 0.375 and 37.5% are the same quantity written two ways.

Repeating decimals are marked. Their percentages are rounded to two places for readability, with the exact value in parentheses — 1/3 is 33.33% rounded, but exactly 33 1/3%. For fractions above 1, split off the whole part: 11/8 is 1 + 3/8, so 1.375. The improper fraction calculator handles that split when the numbers are less friendly.

How to convert a fraction to a decimal by hand

Divide the numerator by the denominator using long division, and keep going past the decimal point by adding zeros to the number you are dividing into.

The remainder is the thing to watch. Once it hits zero the decimal has finished. If a remainder you have already seen comes back, the digits from that point on repeat forever.

There is a hard limit on how long this can go on. Dividing by 12, the only possible remainders are 0 through 11, so within twelve steps a remainder must either be zero or one you have seen before. That is the reason every fraction is either a terminating decimal or a repeating one, and never something random. Dividing by 7 is the classic case: 1/7 = 0.142857142857…, a six-digit block that cycles because all six non-zero remainders appear before one comes back around.

The long division calculator shows each of these steps written out, and the remainder calculator gives just the remainder if that is the part you need. To skip the working entirely, the fraction to decimal calculator converts any fraction and flags the repeating block.

The decimal to fraction chart, and how to go backwards

Going the other way is a matter of counting decimal places. Put the digits over 1 followed by that many zeros, then reduce. Two decimal places means hundredths, three means thousandths.

Repeating decimals need a different move, because there is no last digit to count to. Give the decimal a name, multiply it by a power of ten large enough to shift one full repeating block, and subtract the original from the result. The repeating tails cancel exactly.

The same procedure on 0.333… gives 10x − x = 3, so x = 3/9 = 1/3. The decimal to fraction calculator does both cases, including the repeating one, and reduces the result for you.

What is a terminating decimal?

A terminating decimal is one that stops. 0.375 terminates after three digits. 0.3333… never does, so it is called a repeating or recurring decimal. Every fraction of whole numbers is one or the other, and you can tell which without doing any division at all.

The reason is that a terminating decimal is really a fraction over a power of ten. 0.375 is 375/1000, and 1000 is 2³ × 5³. For a fraction to be rewritten with a denominator of 10, 100 or 1000, its own denominator has to divide one of those numbers, and the only prime factors available are 2 and 5.

Reducing first is not optional. 6/12 looks like it has a 3 in the denominator, but it reduces to 1/2 and terminates at 0.5. Only the denominator in lowest terms decides.

The rule even tells you how many decimal places you will get: count the 2s and the 5s in the denominator and take whichever there are more of. 1/16 has four 2s, so 0.0625 has four places. 3/20 has two 2s and one 5, so 0.15 has two places. 7/40 has three 2s and one 5, so 0.175 has three.

How do you divide decimals?

Move the decimal point in the divisor until it is a whole number, move the point in the other number by exactly the same number of places, then divide normally. Shifting both by the same amount does not change the answer, because you have multiplied top and bottom by the same power of ten.

Check the first one by multiplying back: 12.15 × 0.4 = 4.86. That reversal catches a misplaced decimal point faster than re-reading the division.

When the divisor is already whole, you only need to keep the decimal point in the answer lined up above the point in the number you are dividing into, then continue adding zeros for as long as you want digits. This is exactly the same procedure as converting a fraction to a decimal, because that is exactly what it is.

Numerator and denominator, proper and improper

The words are worth getting right, because every instruction you will read uses them.

• Numerator is the top number. It counts how many parts you have. In 3/8, that is 3.
• Denominator is the bottom number. It names how many equal parts one whole was cut into. In 3/8, that is 8, so the parts are eighths. The bigger the denominator, the smaller each individual piece.

A memory hook that holds up: the denominator is down, and it denominates — it names the unit, the way “dollars” names what a 5 refers to.

Proper, improper and mixed

• A proper fraction has a numerator smaller than its denominator, so its value is less than 1. 3/8, 5/6 and 15/16 are all proper, and every one of them converts to a decimal between 0 and 1.
• An improper fraction has a numerator equal to or larger than its denominator, so it is worth 1 or more. 9/8 is 1.125, and 8/8 is exactly 1. Nothing is wrong with an improper fraction; the name is historical, not a warning.
• A mixed number writes the same quantity as a whole part and a proper fraction: 9/8 becomes 1 1/8. The improper fraction calculator converts between the two forms and shows the division behind it.
• A unit fraction has 1 on top. 1/2, 1/3 and 1/16 are the fractions the whole chart is built from, since 5/16 is just five copies of 1/16.

That last point is a shortcut worth using. If you know 1/16 is 0.0625, then 5/16 is 5 × 0.0625 = 0.3125, and 11/16 is 11 × 0.0625 = 0.6875. Learn the unit fraction and the rest of its family comes free.

Common denominator meaning, and the least common denominator

A common denominator is a single bottom number that two or more fractions can share. You need one to add or subtract fractions, because you cannot combine sixths and eighths any more than you can add a length in feet to a length in meters. Rewrite both in the same unit first.

The least common denominator is the smallest number that works. It is the least common multiple of the denominators: list the multiples of the larger one and stop at the first that the smaller one divides into.

Multiplying the denominators together also works and needs no searching: 6 × 8 = 48 gives 8/48 + 18/48 = 26/48, which reduces to the same 13/24. The least common denominator just saves you the reducing at the end. For 4 and 6 it is 12, not 24, because 12 is the first multiple of 6 that 4 divides into.

Decimals sidestep the problem completely, which is one of the reasons the chart is useful. 1/6 is about 0.1667 and 3/8 is 0.375, and adding those gives about 0.5417. The exact answer, 13/24, is 0.541666… The decimal route is quicker and the fraction route is exact, and the gap between them is entirely down to where you rounded 1/6.

Common mistakes

• Dividing the wrong way round. 3/8 is 3 ÷ 8 = 0.375. Doing 8 ÷ 3 gives 2.666…, which is the reciprocal, not the fraction. A proper fraction must land between 0 and 1.
• Treating a rounded repeating decimal as exact. 1/3 is not 0.33. Multiplied by 300 the true value gives 100 and the rounded one gives 99, and that gap grows with the numbers.
• Forgetting to reduce after converting. 0.64 becomes 64/100, which is correct but not finished. It is 16/25.
• Miscounting decimal places. 0.375 is 375 thousandths, not 375 hundredths. Count the digits after the point and match the zeros.
• Confusing 0.3 with 0.03. The first is 3/10, the second is 3/100. One is ten times the other.
• Checking the unreduced denominator for terminating. 6/12 has a 3 on the bottom but reduces to 1/2 and terminates at 0.5. Reduce first, then look at the factors.
• Assuming a decimal must be neat. Denominators of 3, 7, 9, 11 and 12 never produce a terminating decimal, no matter what sits on top.
• Adding fractions by adding tops and bottoms. 1/6 + 3/8 is not 4/14. Convert to a common denominator first, then add only the numerators.

Frequently asked questions

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