Fraction Calculator

Add, subtract, multiply, and divide fractions with the working shown. Enter whole numbers, fractions, or mixed numbers — the result comes back simplified to lowest terms, as a mixed number, and as a decimal, with every step explained so you can follow (or check) the maths.

What Is a Fraction?

A fraction is a part of a whole: the numerator (top number) counts how many parts you have, and the denominator (bottom number) says how many equal parts make the whole. 3/4 means three of four equal parts. Fractions are exact in a way decimals sometimes are not — 1/3 is exactly one third, while 0.333 is only an approximation. That exactness is why cooks, builders, and engineers keep fractions around long after decimal calculators arrived.

A proper fraction has a smaller numerator than denominator (3/4). An improper fraction has a numerator equal to or larger than the denominator (7/4). A mixed number pairs a whole number with a proper fraction (1 3/4). They are three ways of writing the same value: 7/4 = 1 3/4 = 1.75. The calculator converts between all three automatically.

The Four Operations, Step by Step

• Adding and subtracting: fractions need a common denominator first. Find the least common multiple (LCM) of the denominators, rewrite each fraction with it, then add or subtract the numerators. 1/4 + 1/6 becomes 3/12 + 2/12 = 5/12.
• Multiplying: multiply straight across — numerators together, denominators together — then simplify. 2/3 × 3/4 = 6/12 = 1/2.
• Dividing: flip the second fraction (its reciprocal) and multiply. 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
• Simplifying: divide numerator and denominator by their greatest common divisor (GCD), found by the Euclidean algorithm: repeatedly replace the larger number by its remainder until nothing remains. 24/36 shares GCD 12, so it reduces to 2/3.

The Khan Academy fraction courses teach these steps with visual models, and Wolfram MathWorld documents the GCD, LCM, and fraction arithmetic formally.

Common Fraction Mistakes

• Adding across the top and bottom. 1/2 + 1/3 is not 2/5 — denominators must match first. The answer is 5/6.
• Forgetting to simplify. 6/12 is correct but unfinished; always reduce to 1/2.
• Flipping the wrong fraction when dividing. Only the divisor (the second fraction) gets flipped.
• Mixing up mixed numbers. Convert 1 1/2 to 3/2 before operating — never add the whole parts and fraction parts as separate problems unless you track both carefully.

Related tools: the ratio calculator for proportions, the basic calculator for plain arithmetic, and the average calculator for lists of numbers.

Fractions in Everyday Life

Fractions hide in plain sight everywhere. Recipes are written in halves, thirds, and quarters: doubling a recipe that calls for 3/4 cup of flour means working out 3/4 × 2 = 6/4 = 1 1/2 cups, and halving it means 3/4 ÷ 2 = 3/8 cup. Builders read tape measures in sixteenths of an inch, where 10/16 is really 5/8 — simplifying on the fly is part of the job. Musicians live in fractions too: a whole note divides into halves, quarters, eighths, and sixteenths, and a 3/4 time signature means three quarter-note beats per bar.

Medicine is where fraction accuracy matters most. A child’s dose is often a fraction of an adult dose by weight, and a misplaced denominator changes the result. Money works the same way: a quarter of a $240 bill is $60, and a third of a pizza shared among three friends is fair only when the slices are genuinely equal. Even sports statistics lean on fractions — a batter hitting 3 out of 10 is batting .300, which is the fraction 3/10 expressed as a decimal. Whenever a whole is split into equal parts, fractions are the natural language, and simplifying them is what makes the answer readable at a glance.

Comparing Fractions Without a Calculator

Which is bigger, 3/7 or 4/9? Cross-multiplication answers it without any division. Multiply the numerator of each fraction by the denominator of the other: 3 × 9 = 27 against 4 × 7 = 28. Since 28 is bigger, 4/9 is the larger fraction. The method works because both sides are really being put over the same denominator (63): 3/7 = 27/63 and 4/9 = 28/63. It also settles the classic trap: 1/3 is bigger than 1/4, even though 4 is the bigger number, because a whole cut into fewer pieces gives bigger pieces.

A second shortcut is the benchmark test. Compare against 1/2: a fraction is more than a half exactly when its numerator is more than half its denominator. 5/8 beats 1/2 because half of 8 is 4 and 5 is more than 4. 7/12 beats 1/2 for the same reason, while 5/12 falls short. For quick everyday judgements — is this discount more than half price, is this tank more than half full — the benchmark test is faster than any exact method, and the calculator above confirms the precise values whenever they matter.

Simplifying before comparing makes everything easier. The fractions 6/10 and 9/15 look different, but both simplify to 3/5, so they are equal — a fact cross-multiplication confirms, since 6 × 15 = 90 and 9 × 10 = 90. Whenever two fractions look messy, reduce each to lowest terms first: equal fractions become identical, and genuinely different ones become smaller numbers that are far easier to compare. This is also why the calculator always shows its answers simplified rather than leaving them in the raw form the arithmetic produced.

Common Fractions as Decimals and Percentages

A small set of fractions shows up so often that memorising their decimal and percentage forms saves constant recalculation. Halves, quarters, fifths, eighths, and thirds cover most recipes, discounts, and test scores between them.

Notice which decimals terminate and which repeat: fractions whose denominators divide into powers of 10 (2, 4, 5, 8, 10) terminate, while thirds repeat forever. That is why keeping 1/3 as a fraction is more exact than writing 0.333 — and why the calculator shows both forms. To convert any fraction to a percentage, divide and multiply by 100, or use the percentage calculator for the full working.

Recipe Scaling: One Recipe, Three Sizes

Cooking is the most common place fractions are scaled in real life, and it shows why fraction arithmetic is worth doing properly. Take a cake recipe and halve it for a quiet evening, or double it for guests — every ingredient scales by the same factor, and the fractions must stay exact or the cake suffers.

The sugar row is the instructive one: half of 3/4 is 3/8, found by multiplying 3/4 × 1/2 = 3/8, and double is 3/4 × 2 = 6/4, simplified to 1 1/2. The same scaling logic powers the ratio calculator for paint mixes and proportions, and the basic calculator for plain doubling and halving.

How to Simplify Fractions to Lowest Terms

Simplifying means dividing the numerator and the denominator by the same number until no common factor is left except 1. The result has the same value, but it is shorter, clearer, and easier to use in the next step of a calculation. A fraction that cannot be reduced any further is in lowest terms. For example, 12/18 and 2/3 name the same amount, but 2/3 is the finished answer, and it is the form this calculator always returns.

1. Find the greatest common divisor (GCD) of the numerator and the denominator. For 42/56, the GCD is 14.
2. Divide both numbers by the GCD. Here, 42 divided by 14 is 3, and 56 divided by 14 is 4, so 42/56 simplifies to 3/4.
3. Check the result. The numbers 3 and 4 share no common factor except 1, so 3/4 is in lowest terms and the work is done.

When the GCD is not obvious, prime factors help. Write each number as a product of primes and cancel the factors they share. For 18/24, the numerator is 2 times 3 times 3, and the denominator is 2 times 2 times 2 times 3. Cancelling one 2 and one 3 leaves 3/4. The same idea works for large numbers, where guessing a common divisor would take far longer.

• A fraction is already in lowest terms when the numerator and denominator are consecutive numbers, such as 7/8, because consecutive numbers share no common factor except 1.
• If both numbers are even, the fraction can always be reduced by at least 2. If both end in 0 or 5, both divide by 5.
• Simplifying first, before adding or multiplying, keeps the numbers small and makes the rest of the calculation easier to follow.
• A simplified answer is not a different answer. It is the same value written in its neatest form, which is why teachers and examiners ask for it.

Adding and Subtracting Fractions With Unlike Denominators

Unlike denominators are the main difficulty in fraction addition and subtraction. Quarters and thirds are different sizes of piece, so they cannot be added directly. The fix is to rewrite both fractions with a shared denominator, add or subtract the numerators, and then simplify. The smallest shared denominator is the least common multiple (LCM) of the two denominators, and using it keeps the numbers as small as possible.

The LCM Method, Worked Example

Work out 2/3 + 1/4. The denominators are 3 and 4, and their LCM is 12. Rewrite each fraction over 12: 2/3 becomes 8/12, because 2 times 4 is 8, and 1/4 becomes 3/12, because 1 times 3 is 3. Add the numerators: 8 + 3 = 11, so the answer is 11/12. It cannot be simplified, since 11 and 12 share no common factor except 1. Subtraction follows the same pattern: 5/6 minus 1/4 uses LCM 12, giving 10/12 minus 3/12, which is 7/12.

1. Find the LCM of the two denominators. For 3 and 4 it is 12; for 6 and 8 it is 24.
2. Multiply the numerator and denominator of each fraction by the number that turns its denominator into the LCM.
3. Add or subtract the new numerators, keeping the common denominator unchanged.
4. Simplify the answer, and convert it to a mixed number if it is an improper fraction.

Multiplying the two denominators together always gives a common denominator, but it is not always the smallest one. For 1/6 + 1/8, multiplying gives 48, while the LCM is 24, and the smaller denominator produces smaller numerators and a quicker simplification. When one denominator divides the other, the larger denominator is already the LCM: for 1/4 + 1/8, work over 8 directly. Plain addition and subtraction practice is also available on the basic calculator when the numbers involved are whole numbers or decimals.

Multiplying Fractions and Cancelling First

Multiplication is the most forgiving fraction operation because no common denominator is needed. Multiply the numerators together, multiply the denominators together, and simplify. The result is exact, and the method works unchanged for proper fractions, improper fractions, mixed numbers once converted, and whole numbers written as fractions over 1.

Cancelling first means dividing a numerator and a denominator by a shared factor before multiplying. It gives the same answer with smaller numbers, which reduces arithmetic slips. In 4/5 times 5/8, the 5 in the first numerator and the 5 in the second denominator cancel to 1, leaving 4/8, which simplifies to 1/2. In 3/4 times 8/9, cancelling 3 with 9 and 8 with 4 leaves 2/3 directly, instead of working through 24/36.

• Cancel any numerator with any denominator, not only the ones in the same fraction. Cross-cancelling is valid because multiplication is commutative.
• Cancel before multiplying whenever a shared factor is visible. It is optional, but it makes the final simplification much shorter.
• Never cancel a numerator with another numerator, or a denominator with another denominator. Those pairs are not divided against each other.
• Multiplying by a proper fraction makes the result smaller, and multiplying by a fraction greater than 1 makes it larger. That check catches many sign and setup errors.

When a whole number is involved, write it over 1 first so the rule stays uniform: 3 times 2/5 is 3/1 times 2/5, which is 6/5, or 1 1/5. The same step applies in reverse when a fraction multiplies a whole quantity, such as finding 2/3 of 6, which is 4. Keeping the whole number visible as a fraction also prevents the common error of multiplying only the numerator and forgetting the denominator entirely.

Dividing Fractions: Multiply by the Reciprocal

Division asks how many times one quantity fits into another. With fractions, the reliable method is to turn the division into a multiplication: flip the second fraction and multiply. This works for every pair of fractions, including cases where the answer is a whole number, a proper fraction, or a mixed number.

What Is a Reciprocal?

The reciprocal of a fraction is the same fraction turned upside down: numerator and denominator swap places. Multiplying a number by its reciprocal always gives 1, which is what makes the division rule work. The reciprocal of a whole number is 1 over that number, and the reciprocal of 1 is 1 itself. Zero has no reciprocal, because no number multiplied by zero gives 1, which is another way of seeing why division by zero is undefined.

• The reciprocal of 2/3 is 3/2.
• The reciprocal of 5/4 is 4/5.
• The reciprocal of 7 is 1/7.
• The reciprocal of 1/8 is 8.
• The reciprocal of 3/3 is 3/3, which is 1, and 1 is its own reciprocal.

Dividing by a Whole Number and Dividing a Whole Number

Dividing a fraction by a whole number makes it smaller: 3/4 divided by 2 is 3/4 times 1/2, which is 3/8. Each half of 3/4 is 3/8, which is sensible. Dividing a whole number by a fraction makes it larger when the fraction is less than 1: 2 divided by 1/4 is 2 times 4, which is 8, because four quarters fit into each whole. Similarly, 5 divided by 2/3 is 5 times 3/2, which is 15/2, or 7 1/2.

Only the divisor is flipped. Flipping the first fraction as well, or flipping both, changes the question and gives a wrong answer. A quick sense check helps: dividing by a fraction smaller than 1 must give an answer larger than the starting number, and dividing by a fraction larger than 1 must give a smaller answer. If the result moves in the wrong direction, the reciprocal was probably applied to the wrong fraction.

Mixed Numbers and Improper Fractions

Mixed numbers are convenient for reading a result, and improper fractions are convenient for calculating with it. The value 11/4 and the mixed number 2 3/4 are identical; the first form is easier to multiply, and the second form is easier to picture. Convert to an improper fraction before operating, then convert back to a mixed number for the final answer.

Converting a Mixed Number to an Improper Fraction

1. Multiply the whole number by the denominator. For 3 2/5, that is 3 times 5, which is 15.
2. Add the numerator. Here, 15 plus 2 is 17.
3. Put the total over the same denominator. The result is 17/5.

Converting an Improper Fraction to a Mixed Number

1. Divide the numerator by the denominator and keep the whole-number part. For 23/6, 6 goes into 23 three times.
2. The remainder becomes the new numerator. Here, 23 minus 18 is 5, so the fraction part is 5/6.
3. Write the whole number with the fraction part, and simplify the fraction part if possible. The result is 3 5/6.

• To add mixed numbers, convert both to improper fractions first, add with a common denominator, and convert the answer back. For example, 1 1/2 plus 2 1/4 is 3/2 plus 9/4, which is 6/4 plus 9/4, giving 15/4, or 3 3/4.
• Subtracting mixed numbers sometimes needs borrowing. Convert to improper fractions and the borrowing happens automatically in the subtraction.
• Keep the sign with the whole number for negative mixed numbers. Negative 2 1/2 is negative 5/2, not negative 2 plus positive 1/2.
• A mixed number is only used when the value is greater than 1. Proper fractions stay as simple fractions.

Equivalent Fractions: Same Value, Different Numbers

Equivalent fractions look different but name the same amount. Multiply the numerator and denominator by the same number, and the value does not change, because the fraction is being multiplied by 1 in disguise: 2/2, 3/3, and 4/4 are all equal to 1. Dividing both by the same number reverses the process, which is exactly what simplifying does. Equivalent fractions are the reason a common denominator can always be found.

• Equivalent fractions make addition possible: 1/3 and 1/4 meet at 4/12 and 3/12.
• They make comparison possible: 6/10 and 9/15 are both 3/5, so they are equal despite looking different.
• They make scaling possible: doubling a recipe uses equivalent fractions with larger numerators and denominators.
• They connect fractions to ratios. The ratio calculator uses the same idea when a ratio is scaled up or down while keeping its proportions unchanged.

To test whether two fractions are equivalent, cross-multiply. If the two products match, the fractions are equal: 6 times 15 is 90, and 9 times 10 is 90, so 6/10 equals 9/15. If the products differ, the fractions differ, and the larger product belongs to the larger fraction. This single test settles equality, comparison, and ordering questions alike.

Comparing and Ordering Fractions

Comparing two fractions is straightforward once they share a denominator or can be cross-multiplied. Ordering three or more takes one extra step: put every fraction over the same denominator, then read the numerators in order. The smallest numerator belongs to the smallest fraction, and the largest numerator to the largest, because all the pieces are now the same size.

Ordering Three or More Fractions

Order 1/3, 3/8, and 1/2 from smallest to largest. A common denominator is 24. The fractions become 8/24, 9/24, and 12/24, so the order is 1/3, then 3/8, then 1/2. The same method works for longer lists, for fractions mixed with whole numbers, and for fractions that first need simplifying: reduce each one, find the LCM of the denominators, convert, and sort the numerators.

• Same denominator: the fraction with the larger numerator is larger. 5/9 beats 4/9.
• Same numerator: the fraction with the larger denominator is smaller, because the whole is cut into more pieces. 3/7 beats 3/8.
• Benchmark against 1/2: a fraction exceeds a half exactly when its numerator is more than half its denominator. 9/16 beats 1/2, while 7/16 does not.
• Benchmark against 1: any proper fraction is below 1, any improper fraction is at or above 1, so mixed numbers always beat proper fractions.

Estimation is a useful backup when exact ordering is not needed. Rounding each fraction to the nearest half or whole often reveals the order at a glance: 7/12 is close to 1/2, 5/8 is a little above 1/2, and 2/3 is clearly above both. The exact method above confirms the estimate whenever the order is close or the stakes are higher.

Converting Fractions to Decimals and Percents

Division converts any fraction to a decimal, and multiplying that decimal by 100 converts it to a percent. The reverse is also true: a percent is a fraction over 100, and a terminating decimal can be written as a fraction over a power of 10 and then simplified. Keeping all three forms in view makes it easy to choose whichever is clearest for the job at hand.

Fractions With Repeating Decimals

Some denominators produce decimals that never end. A denominator whose only prime factors are 2 and 5 gives a terminating decimal, such as 3/8 = 0.375. A denominator with any other prime factor, such as 3, 6, 7, 9, or 11, gives a repeating decimal. The repeating part is shown with an ellipsis, and the fraction itself remains the exact value. For calculations that must stay exact, keep the fraction and convert only for the final display.

Fractions to Percents

1. Divide the numerator by the denominator to get the decimal form.
2. Multiply the decimal by 100, which moves the decimal point two places to the right.
3. Add the percent sign. The fraction 3/5 becomes 0.6, then 60%.

Worked percent conversions show the pattern clearly: 3/5 is 60%, 7/8 is 87.5%, 9/20 is 45%, and 1/6 is approximately 16.67% when rounded to two decimal places. A percent above 100% simply means a value greater than the whole: 5/4 is 125%. For discount, increase, and change problems, the percentage calculator carries the same conversion through with the full working shown.

Measuring, Building, and Sharing Fairly

Beyond the kitchen, fractions govern measurement and fair division. A tape measure, a cutting list, a shared bill, and a rota all split wholes into equal parts, and the arithmetic is identical in each case: find the size of one part, count the parts, and simplify the result so it can be read and used.

Reading a Tape Measure in Sixteenths

Imperial tape measures divide each inch into 16 equal parts, and most marks are read in their simplified form. A mark at 10/16 is called 5/8, and a mark at 12/16 is called 3/4. Learning the simplified names avoids measuring errors, because two marks that look different on paper can be the same physical length once reduced. Builders add and subtract these lengths constantly, often by converting to sixteenths, doing the arithmetic, and simplifying the answer back to a familiar fraction.

Sharing and Splitting Fairly

Fair sharing is division with fractions as the answer. Three friends sharing two pizzas get 2/3 of a pizza each, because 2 divided by 3 is 2/3. Four people splitting a bill of 120 units pay 30 units each, which is 1/4 of the bill. A worker who completes 3/4 of a task in 3 hours has 1/4 left, and at the same rate that remaining quarter takes 1 hour.

• Split the total by the number of people, and write the share as a fraction before simplifying. Equal shares must have equal denominators.
• When shares are unequal, name each share as a fraction of the whole and check that the fractions add to 1. Shares of 1/2, 1/3, and 1/6 add to 1, so nothing is missing.
• For time, a quarter of an hour is 15 minutes, a third of an hour is 20 minutes, and a sixth of an hour is 10 minutes. Fraction language makes rotas and schedules easy to state exactly.
• For money, a quarter of 240 is 60, a fifth of 250 is 50, and an eighth of 400 is 50. Checking the fraction against the money total catches division slips quickly.

Checking Your Answer: Does It Make Sense?

Fraction answers deserve a quick sense check before they are trusted. Estimation takes seconds and catches the most common errors: a flipped fraction, a missed common denominator, or a simplification that changed the value. If the estimate and the exact answer disagree wildly, redo the step that looks least certain rather than trusting the longer calculation.

• Adding two fractions below 1/2 must give an answer below 1. If 1/4 plus 1/3 comes out above 1, the denominators were handled wrongly.
• Adding fractions that are each close to 1 must give an answer close to 2. For example, 7/8 plus 1/8 is exactly 1, and 7/8 plus 7/8 is 1 3/4.
• Multiplying by a proper fraction shrinks the number. 2/3 times 3/4 is 1/2, which is correctly smaller than 2/3.
• Dividing by a proper fraction grows the number. 3/4 divided by 1/2 is 3/2, or 1 1/2, which is correctly larger than 3/4.
• Simplifying never changes the value. Test it with decimals: 6/10 and 3/5 are both 0.6, so the simplification is safe.
• Reversing the operation should return the starting number. If 3/4 divided by 1/2 is 3/2, then 3/2 times 1/2 should give 3/4 again.

Rounding to friendly benchmarks is the fastest estimation tool. Treat 1/3 as about 0.33, 2/3 as about 0.67, and 5/6 as about 0.83, and compare the rough total with the exact result. The scientific calculator can confirm the decimal check when a result needs a second opinion, and the calculator at the top of this page shows the exact fraction alongside its decimal form for the same purpose.

Frequently Asked Questions

Is 0/5 a valid fraction?

Yes. A numerator of zero is fine: 0/5 means none of five parts, which is simply 0. What is never allowed is a denominator of zero, because dividing by zero has no meaning. So 0/5 = 0 is valid, while 5/0 is undefined.

Can the denominator be negative?

It can, but convention puts the sign on the numerator: 3/−4 is written as −3/4. The value is the same, but keeping the denominator positive avoids confusion, especially when adding or comparing fractions. This calculator normalises signs the same way in its answers.

How do I simplify a fraction?

Divide the numerator and denominator by their greatest common divisor. For 24/36, the GCD is 12, so 24 ÷ 12 = 2 and 36 ÷ 12 = 3, giving 2/3.

What is the difference between a proper and improper fraction?

A proper fraction is less than 1 (numerator smaller than denominator). An improper fraction is 1 or more, and can always be rewritten as a mixed number.

How do I convert a mixed number to an improper fraction?

Multiply the whole number by the denominator, add the numerator, and put the result over the same denominator. 2 3/4 becomes (2 × 4 + 3)/4 = 11/4.

Why do I need a common denominator to add fractions?

Because you can only add like-sized parts. Quarters and sixths are different sizes; twelfths are a size both divide into, so convert first, then add.

How do I turn a fraction into a decimal?

Divide the numerator by the denominator. 3/8 = 0.375. Some fractions give repeating decimals (1/3 = 0.333…), which is where keeping the fraction is more exact.

How do I add a whole number to a fraction?

Turn the whole number into a fraction with the same denominator, then add. To work out 2 + 1/3, write 2 as 6/3, then add: 6/3 + 1/3 = 7/3, which is 2 1/3 as a mixed number. The calculator does this automatically when you enter a whole number alongside a fraction.

What is 1/2 of 3/4?

Multiply: 1/2 times 3/4 is 3/8. The word of in this context means multiply, and the answer is smaller than 3/4, as expected when taking a part of a part.

How do I add fractions with the same denominator?

Add the numerators and keep the denominator unchanged, then simplify. For 3/7 plus 2/7, the numerators add to 5, giving 5/7. No common denominator step is needed, because the pieces are already the same size.

What happens when I multiply a fraction by 1?

Nothing changes. Any number multiplied by 1 keeps its value, whether 1 is written as 1/1, 2/2, or 5/5. That fact is what makes equivalent fractions possible: multiplying by 2/2 or 3/3 changes the appearance without changing the value.

Can a fraction be greater than 1?

Yes. Any fraction whose numerator is larger than its denominator is greater than 1, and it can be written as a mixed number. For example, 9/4 is 2 1/4, and 7/3 is 2 1/3. Fractions equal to 1, such as 4/4, sit exactly on the boundary between proper and improper forms.

How do I compare fractions with the same numerator?

The fraction with the smaller denominator is larger, because the whole is divided into fewer, bigger pieces. So 3/5 beats 3/8, and 5/6 beats 5/9. When the numerators differ as well, use a common denominator or cross-multiplication instead of guessing from one number alone.

Disclaimer: This calculator is for education and everyday maths. Results are exact fraction arithmetic; decimals are rounded for display.