Decimal to Fraction Calculator

Type any decimal into the box above and this tool turns it into a fraction in its simplest form, including a mixed number when the value is larger than 1. Repeating decimals are supported too, so a number like 0.333... does not need to be rounded off before you get an answer. Every result comes with the working shown underneath it, so you can check the math yourself or use it for homework without guessing how the answer was reached.

How Does a Decimal Convert Into a Fraction?

Every decimal is really a fraction written in a different form, and the conversion just puts it back into that shape. Write the decimal over 1, then multiply the top and bottom by 10 for each digit that sits after the decimal point. This turns both numbers into whole numbers without changing the value of the fraction. Once that is done, the fraction gets reduced to its lowest terms, which is the last step and often the one people skip.

Take 0.75 as an example. Written over 1 it becomes 0.75/1. There are two digits after the decimal point, so both numbers are multiplied by 100, giving 75/100. Dividing both numbers by 25 brings it down to 3/4, which is the final answer.

What Are the Exact Steps to Convert a Terminating Decimal to a Fraction?

A terminating decimal is one that stops after a fixed number of digits, such as 0.125 or 0.6. The process has four parts: place the decimal over 1, count how many digits come after the decimal point, multiply the numerator and denominator by 10 raised to that count, then reduce the result.

For 0.125, there are three digits after the point, so both numbers are multiplied by 1000. That gives 125/1000, which reduces to 1/8 once both numbers are divided by 125.

How Do You Simplify a Fraction After Converting a Decimal?

Simplifying means dividing the numerator and denominator by the largest number that goes into both evenly, known as the greatest common factor. Continuing the 0.75 example, 75 and 100 both divide evenly by 25, so 75/100 becomes 3/4. A fraction left as 75/100 is not wrong mathematically, but it is not the answer most people expect when they convert a decimal, so this step should never be skipped.

How Do You Convert a Repeating Decimal to a Fraction?

A repeating decimal keeps going forever with the same digit or group of digits, like 0.333... or 0.454545... These cannot be handled with the multiply by 10 and simplify method above, since there is no fixed number of decimal places to count. Instead, set the decimal equal to a variable, multiply both sides by a power of 10 that matches the length of the repeating part, then subtract the original equation from the new one. This cancels the repeating digits and leaves a normal equation to solve.

For 0.333..., let x equal 0.333... Multiplying by 10 gives 10x = 3.333... Subtracting the first equation from the second leaves 9x = 3, so x = 3/9, which reduces to 1/3.

What Is the Formula for a Repeating Decimal With No Non-Repeating Digits?

When every digit after the decimal point repeats, there is a shortcut. Put the repeating digits over a denominator made of that many 9s. A single repeating digit goes over 9, two repeating digits go over 99, and so on. This gives the same answer as the full algebra method, just without writing out each step.

How Do You Convert a Mixed Repeating Decimal, Where Some Digits Do Not Repeat?

Some decimals have a part that does not repeat before the repeating block begins, like 0.41666... where the 6 repeats but the 41 does not. The same subtraction method still works here, but it needs two different powers of 10, one to shift the non-repeating digits in front of the decimal point and a second to match the length of the repeating block. Subtracting the two equations still cancels the repeating part. For 0.41666..., this works out to 5/12.

How Do You Convert a Decimal Greater Than 1 Into a Mixed Number?

When a decimal is larger than 1, the whole number part is set aside first, and only the part after the decimal point goes through the conversion. Once the decimal part becomes a fraction and is simplified, it gets attached back to the whole number to form a mixed number.

For 2.5, the whole number 2 stays as it is. The 0.5 becomes 1/2 using the same steps as any other decimal, so the final answer is 2 and 1/2.

Negative decimals follow the same rule. Convert the number as if the minus sign was not there, then put it back on the final fraction. Negative 0.5 becomes negative 1/2.

What Mistakes Cause a Wrong Answer When Converting Decimals to Fractions?

A few small oversights account for most wrong answers people run into, usually one of the following:

•     Not reducing the final fraction to its lowest terms, which leaves an answer that is technically correct but not in the form expected.

•     Rounding a repeating decimal instead of marking which digits actually repeat, which produces an approximate fraction rather than the exact one.

•     Mixing up which digits belong to the repeating block in a mixed repeating decimal, which throws off the whole calculation.

•     Forgetting to separate the whole number before converting a decimal larger than 1, which leads to the wrong mixed number.

How Does Decimal to Fraction Conversion Apply to Real Measurements Like Inches?

Outside the classroom, this kind of conversion shows up constantly in trades that measure in inches. A tape measure, a drill bit, or a set of woodworking plans rarely show a number like 0.4375 inches. Instead they use fractions with a denominator that is always a power of two, such as halves, quarters, eighths, sixteenths, thirty-seconds, and sixty-fourths, because that matches the marks stamped on a physical ruler.

The math is not exactly the same as converting a plain decimal, since the goal shifts from finding the exact fraction to finding the nearest one a ruler can actually show. A CAD drawing giving a measurement of 0.515 inches, for example, would round to 33/64 at 1/32 precision, close enough for most shop work but not identical to the unrounded value.

How Is Converting a Decimal to an Inch Fraction Different From a Plain Decimal to Fraction?

A plain decimal to fraction conversion always looks for the exact, fully reduced answer, no matter how unusual the denominator turns out to be. Inch fractions work differently, because a tape measure can only mark halves, quarters, eighths, sixteenths, thirty-seconds, and sixty-fourths. A decimal like 0.515 does not convert to its exact fraction here, it gets rounded to whichever of those denominators matches the precision needed. Anyone working from a CAD file or a caliper reading, and marking that measurement on a physical board or part, will get better use out of the inches to fraction calculator, since it is built around this rounding rule rather than exact simplification.

How Do You Convert a Fraction Back Into a Decimal?

The reverse process is simpler than converting a decimal in the first place. Divide the numerator by the denominator and the result is the decimal value. Taking 3/4 as an example, 3 divided by 4 gives 0.75, which matches the earlier example run in the other direction. For checking a fraction against a decimal measurement, or working a problem backwards, the fraction to decimal calculator handles this conversion directly.

How Do You Reduce Any Fraction to Its Lowest Terms?

Simplifying a fraction that was not produced from a decimal works the same way described earlier: find the greatest common factor of the numerator and denominator, then divide both numbers by it. This comes up often when a fraction is typed in directly rather than generated from a decimal, such as when checking whether 18/24 and 3/4 are actually the same value. The fraction calculator is built for exactly this, taking any fraction and reducing it in one step.

How Do You Convert a Percentage Into a Fraction?

Percentages and decimals are closely related, since a percentage is just a decimal multiplied by 100. To turn a percentage into a fraction, drop the percent sign, divide the number by 100, and simplify as usual. 75 percent becomes 75/100, which reduces to 3/4, the same answer as the 0.75 example above. For percentages specifically, the percent to fraction calculator skips the extra step of converting to a decimal first.

Frequently Asked Questions

Does 0.999 repeating really equal 1?

Yes. Using the same subtraction method shown earlier, letting x equal 0.999... and multiplying by 10 gives 9x = 9, so x = 1 exactly. It is not an approximation, both are the same number written differently.

Can every decimal be written as an exact fraction?

Terminating and repeating decimals can always be written as an exact fraction. Decimals that never terminate and never settle into a repeating pattern, such as pi, cannot be written as a fraction at all.

What is 0.625 as a fraction?

0.625 is 5/8. Written over 1000 it is 625/1000, and dividing both numbers by 125 brings it down to 5/8.

How do you convert a negative decimal to a fraction?

Convert the number as if the negative sign was not there, then put it back on the finished fraction. Negative 0.5 becomes negative 1/2.

What is the difference between a repeating and a terminating decimal?

A terminating decimal ends after a fixed number of digits, like 0.75. A repeating decimal continues forever with the same digit or group of digits, like 0.333... Each type needs a different conversion method, covered separately above.