Fraction Calculator
Free fraction calculator with steps. Type 1 1/2 x 3, 3/4 divided by 2 or 2.8 and get the simplified, mixed-number and decimal answer, with the LCM and GCD shown.
Whole numbers, decimals, fractions and mixed numbers all work, with + − x ÷ or the words plus, minus, times, divided by, of. Leave this empty to use the numerator and denominator boxes below.
Top number of the first fraction. Any whole number: positive, negative, or zero.
Bottom number of the first fraction. Must be a positive whole number greater than zero.
Top number of the second fraction. Any whole number: positive, negative, or zero.
Bottom number of the second fraction. Must be a positive whole number greater than zero.
Results
Addition
3/4 + 1/3
- Result (Fraction)
- 13/12
- As a Mixed Number
- 1 1/12
- Decimal Equivalent
- 1.08333333
- Exact Decimal (repeating digits in brackets)
- 1.08(3)
- Simplified
- 13/12
The formula
a/b + c/d = (ad + bc) / bd | a/b - c/d = (ad - bc) / bd | a/b × c/d = ac / bd | a/b ÷ c/d = ad / bc
- a/b, c/d
- Fractions
- LCM
- Least Common Multiple
- GCD
- Greatest Common Divisor
Worked example — First Numerator 3, First Denominator 4, Second Numerator 1, Second Denominator 3
Addition = 3/4 + 1/3
How Fraction Calculator Works
a/b + c/d = (ad + bc) / bd | a/b - c/d = (ad - bc) / bd | a/b × c/d = ac / bd | a/b ÷ c/d = ad / bc
Fraction arithmetic follows specific rules depending on the operation. For addition and subtraction, the fractions must share a common denominator — the least common multiple (LCM) of the denominators is used to find equivalent fractions with the same denominator. Once denominators match, the numerators are added or subtracted directly while the denominator stays the same. For multiplication, multiply numerators together and denominators together directly — no common denominator is needed.
For division, multiply by the reciprocal (flip the second fraction upside down and multiply). Every result is automatically simplified to lowest terms by dividing both numerator and denominator by their greatest common divisor (GCD). This ensures the answer is always in its simplest, most readable form.
- a/b, c/d
- Fractions — Two fractions with numerators and denominators. a and c are numerators (can be any integer). b and d are denominators (must be positive integers). The denominator determines the "size" of each part.
- LCM
- Least Common Multiple — The smallest positive integer that is divisible by both denominators. Used to find a common denominator for addition and subtraction. Computed as (a × b) / GCD(a, b).
- GCD
- Greatest Common Divisor — The largest positive integer that divides both the numerator and denominator evenly. Used to simplify the result to lowest terms through Euclidean algorithm.
Worked Examples
Sarah is baking cookies and needs to double the recipe. The original recipe calls for 3/4 cup of sugar. To double it, she needs to multiply 3/4 by 2 (which is 2/1 as a fraction).
mode = multiply · aNum = 3 · aDen = 4 · bNum = 2 · bDen = 1
3/4 × 2/1 = 6/4 = 3/2 (1.5)
3/4 × 2/1 = 6/4, which simplifies to 3/2 or 1.5 cups. Sarah needs one and a half cups of sugar. The calculator shows both the simplified fraction (3/2) and the decimal equivalent (1.5), making it easy to measure using standard kitchen tools.
A carpenter is cutting trim for a window frame. She needs to subtract a 3/8 inch notch from a 7/8 inch board. What is the remaining width?
mode = subtract · aNum = 7 · aDen = 8 · bNum = 3 · bDen = 8
7/8 − 3/8 = 4/8 = 1/2 (0.5)
7/8 − 3/8 = 4/8, which simplifies to 1/2 inch. Since both fractions already share the same denominator, the subtraction is straightforward: just subtract the numerators. The remaining board width is exactly 1/2 inch, which is a standard measurement.
A student is studying probability and needs to find P(A and B) for independent events. Event A has probability 1/3 and Event B has probability 1/4. For independent events, P(A and B) = P(A) × P(B).
mode = multiply · aNum = 1 · aDen = 3 · bNum = 1 · bDen = 4
1/3 × 1/4 = 1/12 (≈0.0833)
1/3 × 1/4 = 1/12. For independent events, multiply the probabilities directly. The calculator confirms the result is already in simplest form (1 and 12 share no common factors). The probability of both events occurring is 1/12 or approximately 0.0833 (8.33%).
3 4 divided by 2 (a student types the fraction 3/4 as "3 4"): what is three quarters shared between 2?
expr = 3 4 divided by 2
3/4 ÷ 2/1 = 3/4 × 1/2 = 3/8 (0.375)
Dividing by a whole number means multiplying by its reciprocal: 2 becomes 2/1, flips to 1/2, and 3/4 × 1/2 = 3/8. The answer is already in lowest terms because 3 and 8 share no factor.
1 1/2 x 3: a recipe that needs 1 1/2 cups of flour is made three times.
expr = 1 1/2 x 3
1 1/2 = 3/2, then 3/2 × 3/1 = 9/2 = 4 1/2 (4.5)
Convert the mixed number first (1×2 + 1 = 3, over 2), multiply across, then turn 9/2 back into the mixed number 4 1/2 by dividing 9 by 2.
2 divided by 3 as a fraction: any division of whole numbers is a fraction.
expr = 2 divided by 3 as a fraction
2 ÷ 3 = 2/3 (0.(6), which is 0.6666…)
Written as fractions, 2/1 ÷ 3/1 = 2/1 × 1/3 = 2/3. The decimal never ends, so the exact row shows the repeating digit in brackets.
1 1/3 as a fraction: turn a mixed number into an improper fraction.
expr = 1 1/3 as a fraction
1 1/3 = (1×3 + 1)/3 = 4/3
Whole number × denominator + numerator, over the same denominator. The result 4/3 is the improper fraction, 1 1/3 is the same value as a mixed number, and the decimal is 1.(3).
3 divided by 4/9: dividing a whole number by a fraction.
expr = 3 divided by 4/9
3/1 × 9/4 = 27/4 = 6 3/4 (6.75)
Flip the fraction you divide by (4/9 becomes 9/4) and multiply. The answer is larger than 3 because 4/9 is less than 1, so more than 3 of those fit.
1/3 plus 2/5: adding fractions with different denominators.
expr = 1/3 plus 2/5
LCM(3, 5) = 15, so 5/15 + 6/15 = 11/15 (≈ 0.7333)
The common denominator is the least common multiple of 3 and 5. Rewrite each fraction over 15, add the numerators (5 + 6) and keep the 15.
simplify 13/12: is this fraction already in lowest terms?
expr = simplify 13/12
13/12 (GCD = 1) = 1 1/12
The greatest common divisor of 13 and 12 is 1, so 13/12 cannot be reduced. It is an improper fraction; as a mixed number it is 1 1/12.
2.8 inches to fraction: a decimal measurement written as a fraction.
expr = 2.8 inches to fraction
2.8 = 28/10 = 14/5 = 2 4/5
One decimal place means over 10, then divide top and bottom by the GCD (2). On a tape measure marked in sixteenths, 2.8 is nearest to 2 13/16 (2.8125).
0.5 in fraction: the simplest decimal to fraction conversion.
expr = 0.5 in fraction
0.5 = 5/10 = 1/2
Write the digits over a power of 10 and reduce by the GCD (5). The same method turns 0.75 into 75/100 = 3/4.
-5 3/4 divided by 2 1/16: dividing negative mixed numbers.
expr = -5 3/4 divided by 2 1/16
-23/4 ÷ 33/16 = -23/4 × 16/33 = -368/132 = -92/33 = -2 26/33
The minus sign belongs to the whole mixed number (-5 3/4 means -(5 + 3/4)). Convert both to improper fractions, multiply by the reciprocal of 33/16, then divide top and bottom by their GCD, 4.
1/2 of 1/3: "of" between fractions means multiply.
expr = 1/2 of 1/3
1/2 × 1/3 = 1/6
Half of a third is a sixth. Taking a fraction "of" something is multiplication, so no common denominator is needed.
3/4 x 2 as a fraction: multiply a fraction by a whole number.
expr = 3/4 x 2 as a fraction
3/4 × 2/1 = 6/4 = 3/2 = 1 1/2
Write 2 as 2/1, multiply across to get 6/4, and reduce by the GCD (2) to 3/2, which is 1 1/2 as a mixed number.
1/2 + 1/3 + 1/4: more than two fractions in one line.
expr = 1/2 + 1/3 + 1/4
(1/2 + 1/3) = 5/6, then 5/6 + 1/4 = 13/12 = 1 1/12
Chains are worked left to right, with multiplication and division done before addition and subtraction. Each step shows its own common denominator.
Understanding the Result
A fraction represents a part of a whole and is one of the oldest mathematical concepts, dating back to Egyptian papyri from around 1800 BCE. The Rhind Mathematical Papyrus (c. 1550 BCE), a famous Egyptian document, contains extensive work with unit fractions (fractions with numerator 1). The word "fraction" comes from the Latin "fractio" meaning "to break," reflecting the idea of breaking a whole into equal parts. The denominator (bottom number) tells you how many equal parts the whole is divided into, while the numerator (top number) tells you how many of those parts you have.
Fractions are foundational to all of mathematics — they are the gateway to algebra, ratios, proportions, percentages, probability, and calculus. The key insight for fraction arithmetic is that you can only directly add or subtract fractions when they share the same denominator, because the denominator defines the "size" of each part. For addition and subtraction, the calculator finds the least common multiple (LCM) of the denominators to create equivalent fractions with a common denominator, then adds or subtracts the numerators.
For multiplication, you simply multiply numerators together and denominators together — no common denominator is needed because multiplication naturally combines the fractional parts. For division, you multiply by the reciprocal of the second fraction (flip it upside down), which is equivalent to the "keep-change-flip" mnemonic taught in schools. Every result is automatically simplified to lowest terms by dividing both the numerator and denominator by their greatest common divisor (GCD). Fractions appear everywhere in daily life: cooking uses cup and teaspoon fractions, construction uses inch fractions (1/16, 1/8, 1/4), music uses time signatures like 3/4 and 4/4, photography uses aperture fractions (f/2.8, f/4), and finance uses fractional shares and stock splits.
How to Use
- Type the problem your way in the first box: 1 1/2 x 3, 3/4 divided by 2, 1/3 plus 2/5, 2.8, or simplify 13/12. Whole numbers, decimals, fractions and mixed numbers can be mixed freely.
- Or leave that box empty, pick an operation (add, subtract, multiply, divide) and enter the numerator and denominator of each fraction. Denominators must be positive whole numbers.
- Read the answer as a simplified improper fraction, a mixed number, a decimal (with repeating digits in brackets) and, when it differs, the fraction before simplifying.
- Open the step-by-step solution: it names each rule used (common denominator = LCM, multiply by the reciprocal, convert a mixed number, divide by the GCD) with your own numbers.
- Use the reference table below for common fraction, decimal and percent equivalents.
Quick Reference
| Addition Rule | a/b + c/d = (ad + bc) / bd |
| Subtraction Rule | a/b − c/d = (ad − bc) / bd |
| Multiplication Rule | a/b × c/d = ac / bd |
| Division Rule | a/b ÷ c/d = ad / bc |
| Simplify | Divide numerator and denominator by GCD |
| LCM | Least Common Multiple for common denominator |
| Reciprocal | Flip fraction (swap numerator and denominator) |
| Improper Fraction | Numerator > Denominator (value > 1) |
| Mixed → Improper | Whole × Den + Num over original Den |
| Division by Zero | Undefined — denominator must never be zero |
| 1/2 as decimal and percent | 0.5 = 50% |
| 1/3 as decimal and percent | 0.(3) ≈ 33.33% |
| 2/3 as decimal and percent | 0.(6) ≈ 66.67% |
| 1/4 as decimal and percent | 0.25 = 25% |
| 3/4 as decimal and percent | 0.75 = 75% |
| 1/5 as decimal and percent | 0.2 = 20% |
| 2/5 as decimal and percent | 0.4 = 40% |
| 3/5 as decimal and percent | 0.6 = 60% |
| 4/5 as decimal and percent | 0.8 = 80% |
| 1/6 as decimal and percent | 0.1(6) ≈ 16.67% |
| 5/6 as decimal and percent | 0.8(3) ≈ 83.33% |
| 1/8 as decimal and percent | 0.125 = 12.5% |
| 3/8 as decimal and percent | 0.375 = 37.5% |
| 5/8 as decimal and percent | 0.625 = 62.5% |
| 7/8 as decimal and percent | 0.875 = 87.5% |
| 1/10 as decimal and percent | 0.1 = 10% |
| 1/12 as decimal and percent | 0.08(3) ≈ 8.33% |
| 1/16 as decimal and percent | 0.0625 = 6.25% |
Frequently Asked Questions
- Why do I need a common denominator for addition but not multiplication?
- Addition combines parts of potentially different-sized wholes. Think of adding 1/2 (half a dollar) + 1/4 (a quarter) — you need to express both in the same units before combining. Converting 1/2 to 2/4 lets you add: 2/4 + 1/4 = 3/4. Multiplication takes a fraction of a fraction: 1/2 multiplied by 1/4 means "take half of one quarter," which naturally gives 1/8 without needing a common denominator. Multiplication inherently accounts for the different unit sizes.
- Why do we simplify fractions?
- Simplifying (reducing to lowest terms) makes the result easier to understand, compare, and work with. The fractions 4/8 and 1/2 represent the same value, but 1/2 is immediately recognizable. Simplification also helps with comparing fractions (which is larger: 3/6 or 2/4? Both equal 1/2) and with performing further arithmetic — simpler numbers mean fewer chances for calculation errors.
- What is an improper fraction and when should I use it?
- An improper fraction has a numerator larger than the denominator (e.g., 7/4 or 3/2). It represents a value greater than 1. In higher mathematics, improper fractions are preferred over mixed numbers because they are easier to multiply, divide, and use in algebraic formulas. The calculator displays improper fractions in their simplified form — for example, 6/4 simplifies to 3/2, not 1 1/2. In algebra and calculus, always work with improper fractions rather than mixed numbers.
- How do I enter a mixed number like 2 1/3?
- Type it in the first box with a space between the whole number and the fraction: 2 1/3. Negative mixed numbers work too: -5 3/4 means -(5 + 3/4). The calculator converts it to the improper fraction 7/3 for you and shows that step. You can also convert by hand and use the numerator and denominator boxes: 2 × 3 + 1 = 7, so enter 7 over 3.
- How do you divide fractions?
- Keep the first fraction, change the division sign to multiplication, and flip the second fraction (use its reciprocal). Then multiply the numerators, multiply the denominators and simplify. For 3/4 ÷ 2/3: flip 2/3 to 3/2, then 3/4 × 3/2 = 9/8 = 1 1/8. To divide by a whole number such as 3/4 ÷ 2, write 2 as 2/1, flip it to 1/2, and 3/4 × 1/2 = 3/8. Dividing a whole number by a fraction, such as 3 ÷ 4/9, works the same way: 3/1 × 9/4 = 27/4 = 6 3/4.
- How do you turn a decimal into a fraction?
- Write the digits after the point over a power of 10 with the same number of zeros as decimal places, then simplify with the GCD. 0.5 = 5/10 = 1/2. 0.75 = 75/100 = 3/4. 2.8 = 28/10 = 14/5, which is the mixed number 2 4/5. Type the decimal in the first box and the calculator shows each of these steps.
- How do you convert a mixed number to an improper fraction?
- Multiply the whole number by the denominator, add the numerator, and put the total over the original denominator. 1 1/3 = (1 × 3 + 1)/3 = 4/3. To go back, divide the numerator by the denominator: the quotient is the whole number and the remainder goes over the denominator, so 13/12 = 1 1/12.
- How do you simplify a fraction?
- Find the greatest common divisor (GCD) of the numerator and denominator and divide both by it. 12/18 has GCD 6, so it simplifies to 2/3. If the GCD is 1, as with 13/12, the fraction is already in lowest terms. Mixed numbers are simplified by reducing the fractional part.
- How do you add fractions with different denominators?
- Find a common denominator (the least common multiple of the denominators), rewrite each fraction over it, add the numerators and keep the denominator. For 1/3 + 2/5 the LCM of 3 and 5 is 15, so 5/15 + 6/15 = 11/15. Subtraction works the same way.
- How do you write a division like 2 divided by 3 as a fraction?
- The division bar is the fraction bar: 2 ÷ 3 = 2/3. Any whole-number division becomes a fraction with the dividend on top and the divisor on the bottom, and the fraction is then reduced if it can be. In decimal form 2/3 is 0.6666… (0.(6) with the repeating digit in brackets).
- How do you multiply a fraction by a whole number?
- Write the whole number over 1 and multiply across: 3/4 × 2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2. "Of" means multiply, so 1/2 of 1/3 is 1/2 × 1/3 = 1/6.
- What happens when the result is a whole number?
- When the numerator is divisible by the denominator (e.g., 6/3 = 2), the calculator displays the result as a whole number rather than a fraction. The simplified fraction would be 2/1, which the calculator recognizes as an integer and displays as just "2". The decimal equivalent will also show the whole number without any fractional part.
- Can I use negative fractions and what do they mean?
- Yes, you can enter negative numerators. The calculator handles negative values correctly in all operations. By convention, the negative sign is kept on the numerator and the denominator stays positive. For example, −3/4 + 1/4 = −2/4, which simplifies to −1/2. The rules for adding and subtracting negative numbers follow standard integer arithmetic: adding a negative is the same as subtraction, and subtracting a negative is the same as addition. In real-world contexts, a negative fraction might represent a loss (finance), a deficit (temperature below zero), or a downward measurement (elevation below sea level).
- Why does dividing by a fraction give a larger number?
- Dividing by a fraction is equivalent to multiplying by its reciprocal. When you divide 6 by 1/2, you are asking "how many halves are in 6?" The answer is 12, which is larger than 6. This makes intuitive sense: a half is smaller than 1, so more halves fit into any number. When you divide by a fraction less than 1, the quotient is larger than the dividend. When you divide by a fraction greater than 1 (e.g., 3/2), the quotient is smaller. This is the same principle as: dividing by 0.5 is the same as multiplying by 2.
Common Uses
- •Cooking and baking: scaling recipes using fractional measurements — doubling 1/3 cup to 2/3 cup, or halving 3/4 teaspoon to 3/8 teaspoon
- •Construction and carpentry: working with inch fractions (1/16, 1/8, 1/4, 1/2) for precise measurements and material cuts
- •Mathematics education: teaching students the fundamentals of rational numbers, equivalent fractions, and arithmetic operations
- •Music theory: understanding time signatures (3/4, 4/4, 6/8) and note durations (whole notes, half notes, quarter notes, eighth notes)
- •Financial calculations: working with fractional interest rates, stock splits (3:2 split expressed as 3/2), and proportional ownership
Pro Tips
- →When adding or subtracting, first check if the denominators already match — if so, you can skip the LCM step and just add or subtract the numerators directly. This is much faster.
- →Remember the "keep-change-flip" mnemonic for division: Keep the first fraction, Change the division sign to multiplication, and Flip the second fraction (use its reciprocal). This transforms division into multiplication, which is much simpler.
- →Always check if your answer can be simplified. Even if the calculator does it automatically, understanding the simplification process helps you catch errors and build number sense.
- →For cooking and construction, you may want to convert the simplified fraction to a common inch fraction or cup measurement after seeing the result. The decimal equivalent helps with this conversion.
- →Negative numerators are mathematically valid — a negative fraction is just a fraction with a minus sign. The convention is to keep the denominator positive and put the sign on the numerator.
Limitations to Know
- •This calculator works with proper and improper fractions using integer numerators and denominators. Denominators must be positive integers greater than zero. Mixed numbers (2 1/3), decimals (2.8) and whole numbers can be typed in the first box; the numerator and denominator boxes take whole numbers only.
- •Arithmetic is exact for whole numbers of any practical size (it does not use floating point). The decimal equivalent is rounded to 8 decimal places; when the exact decimal is longer or repeats, a second row shows it with the repeating digits in brackets (for example 1/7 = 0.(142857)).
- •For comparing which fraction is larger (e.g., 3/7 vs 4/9), use the decimal equivalents or cross-multiply — this calculator performs arithmetic, not comparison.
- •This tool handles only finite rational arithmetic between two fractions at a time. For symbolic fractions involving variables, continued fractions, or infinite series, use a computer algebra system.
Cite this calculator
TheCalcUniverse. "Fraction Calculator: Add, Subtract, Multiply, Divide." TheCalcUniverse, 2026, https://thecalcuniverse.com/math/fraction-calculator/. Accessed October 25, 2026.
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