Add, subtract, multiply, divide, simplify, compare, and convert fractions. See the simplified answer, mixed-number form, decimal value, formula, step-by-step work, and visual explanation.
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A fraction describes part of a whole, part of a group, or a number that has been divided into equal pieces. A fraction has two main numbers. Take 3/4. The top number, 3, is called the numerator. The bottom number, 4, is called the denominator. The denominator tells us how many equal parts make up one whole. The numerator tells us how many of those parts we have. So 3/4 means the whole was divided into 4 equal pieces and we have 3 of them.
Fractions become much easier once you stop looking at them as two random numbers stacked on top of each other. They are describing a relationship.
The denominator determines the size of the pieces. Imagine one pizza. Cut it into 2 equal pieces. Each piece is 1/2. Now take the same-size pizza and cut it into 8 equal pieces. Each piece is 1/8. A half is much larger than an eighth. This explains something that can look strange at first: a bigger denominator does NOT automatically mean a bigger fraction. For unit fractions such as 1/2, 1/4, 1/8, the pieces actually get smaller as the denominator becomes larger. That is because the same whole is being divided into more pieces.
A proper fraction has a numerator smaller than its denominator. Examples: 1/2, 3/4, 5/8. Each represents less than one whole.
An improper fraction has a numerator that is equal to or larger than its denominator. Examples: 5/4, 7/3, 12/5. An improper fraction can represent one whole or more. For example, 5/4 means four fourths make one whole plus one more fourth. So 5/4 = 1 1/4.
A mixed number contains a whole number plus a fraction. Example: 2 3/4. That means 2 whole units plus 3/4 of another. Mixed numbers and improper fractions can describe the same value.
Equivalent fractions look different but represent the same amount. For example, 1/2 = 2/4 = 4/8. Why? Because each fraction represents exactly half of the whole. If you multiply the numerator and denominator by the same nonzero number, the value of the fraction does not change. Start with 1/2. Multiply both numbers by 2: 1 × 2 = 2, 2 × 2 = 4. Now 1/2 = 2/4. Multiply both by 2 again: 2/4 = 4/8. The pieces may be divided differently, but the amount is still the same.
Simplifying means rewriting a fraction in its lowest terms without changing its value. Take 18/24. Both numbers can be divided by 2, 3, and 6. The greatest number that divides both evenly is 6. That is the greatest common factor, or GCF. Divide: 18 ÷ 6 = 3. 24 ÷ 6 = 4. So 18/24 simplifies to 3/4. The amount did not change. We simply wrote the same value using smaller numbers.
Because you are changing how the same amount is grouped. For example, 6/8 and 3/4 represent exactly the same portion of a whole. Dividing the numerator and denominator by 2 changes the labels on the pieces, but not the amount represented.
Then the fraction is already in lowest terms. Example: 5/7. The only positive whole number that divides both 5 and 7 is 1. So 5/7 cannot be simplified further.
When denominators already match, addition is easy. Example: 2/7 + 3/7. The pieces are already the same size. Add the numerators: 2 + 3 = 5. Keep the denominator: 7. Answer: 5/7. Do NOT add the denominators. 2/7 + 3/7 does not equal 5/14. Why? Because you still have sevenths. You simply have more of them.
Same rule. Example: 6/9 - 2/9. Subtract the numerators: 6 - 2 = 4. Keep 9. Result: 4/9. Then simplify if possible. In this case, 4/9 is already simplified.
This is where many students get stuck. Suppose 1/2 + 1/3. You cannot simply add 1 + 1 and 2 + 3. That would give 2/5, which is incorrect. Why? Because halves and thirds are different-sized pieces. Before adding them, we need to express both fractions using pieces of the same size. That means finding a common denominator. The least common denominator of 2 and 3 is 6. Convert 1/2 to 3/6. Convert 1/3 to 2/6. Now add: 3/6 + 2/6 = 5/6.
The least common denominator is the smallest denominator both fractions can be converted to evenly. It comes from the least common multiple of the denominators. For 1/4 and 1/6, multiples of 4 include 4, 8, 12, 16, 20, 24. Multiples of 6 include 6, 12, 18, 24. The first shared multiple is 12. So the LCD is 12. Convert 1/4 = 3/12. 1/6 = 2/12. Now the fractions can be added or subtracted easily.
Same principle. Example: 3/4 - 1/6. LCD of 4 and 6 is 12. Convert 3/4 = 9/12. 1/6 = 2/12. Subtract: 9/12 - 2/12 = 7/12. No further simplification is possible.
You technically can create a common denominator by multiplying the denominators. But that may produce numbers larger than necessary. For 1/4 and 1/6, multiplying gives 24. That works. But 12 is smaller and easier. That is why the least common denominator is usually preferred.
Multiplication is actually simpler than addition. You do NOT need a common denominator. Multiply the numerators. Then multiply the denominators. Example: 2/3 × 3/5. Numerators: 2 × 3 = 6. Denominators: 3 × 5 = 15. Result: 6/15. Simplify: 6/15 = 2/5. Formula: a/b × c/d = ac/bd.
Yes. Sometimes you can cross-cancel common factors before multiplying. Take 2/3 × 3/5. The 3 in the first denominator and the 3 in the second numerator cancel. That leaves 2/1 × 1/5 = 2/5. This gives the same result while keeping the numbers smaller.
Division uses the reciprocal of the second fraction. Example: 2/3 ÷ 4/5. Keep the first fraction: 2/3. Change division to multiplication. Flip the second fraction: 4/5 becomes 5/4. Now: 2/3 × 5/4 = 10/12. Simplify: 10/12 = 5/6. You may have heard "keep, change, flip." That is a useful memory shortcut. But it is also important to understand what is happening. Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.
A reciprocal is created by switching the numerator and denominator. 3/4 becomes 4/3. 2/7 becomes 7/2. A number multiplied by its reciprocal equals 1, as long as the original number is not zero. Example: 3/4 × 4/3 = 12/12 = 1. That is why reciprocals are useful in division.
Take 2 3/4. Multiply the whole number by the denominator: 2 × 4 = 8. Add the numerator: 8 + 3 = 11. Place that over the original denominator: 11/4. So 2 3/4 = 11/4. The general method is: Whole × Denominator + Numerator, then place the result over the original denominator.
Take 11/4. Divide: 11 ÷ 4. 4 fits into 11 two times, with 3 remaining. The whole-number part is 2. The remainder becomes the new numerator: 3. The denominator stays 4. So 11/4 = 2 3/4.
Divide the numerator by the denominator. Example: 3/8. 3 ÷ 8 = 0.375. Another example: 1/4. 1 ÷ 4 = 0.25. Some fractions create repeating decimals. Example: 1/3 = 0.3333… The decimal continues forever.
Look at the decimal places. Example: 0.375. There are 3 digits after the decimal. That means 0.375 = 375/1000. Now simplify. The GCF of 375 and 1000 is 125. 375 ÷ 125 = 3. 1000 ÷ 125 = 8. Therefore, 0.375 = 3/8. Another example: 0.25 = 25/100 = 1/4.
Suppose you need to decide which is larger: 3/4 or 5/8. One method is to find common denominators. 3/4 = 6/8. Now compare 6/8 and 5/8. Clearly 6/8 > 5/8. Therefore, 3/4 > 5/8. Another method is cross multiplication. 3 × 8 = 24. 5 × 4 = 20. Because 24 > 20, then 3/4 > 5/8. You can also convert to decimals. 3/4 = 0.75. 5/8 = 0.625. Again, 0.75 > 0.625.
A zero numerator is valid. Example: 0/5 = 0. You have 0 of the 5 possible pieces. Any fraction with numerator zero and a nonzero denominator equals 0.
A denominator tells us how many equal pieces the whole is divided into. Dividing by zero is undefined. So 3/0 is not a valid fraction value. The calculator must never return Infinity for a zero denominator. It should explain: "A fraction cannot have 0 as its denominator."
Yes. Examples: -1/2, 1/-2, and -(1/2) all describe the same numerical value. For clarity, this calculator displays the negative sign in front, such as -1/2, rather than leaving the denominator negative.
Fractions are everywhere. Cooking: 3/4 cup. Measurements: 1/2 inch. Time: 1/4 hour. Money: half of a bill. Construction: 5/8 inch. School: 3 out of 5 questions. Sports: a player makes 7 out of 10 attempts. Probability: 1 chance out of 6. Music: quarter notes, half notes, eighth notes. Fractions are one of the most practical ways to describe parts of a whole.
Usually the difficulty is not the arithmetic itself. The challenge is remembering whether denominators need to match, when to simplify, when to flip a fraction, and what the numerator and denominator are actually telling you. That is why this page should not simply display an answer. It should show the problem, the rule, the steps, the picture, and the answer. Once you can see what the fraction represents, many of the rules make more sense.
Adding denominators. Example: 1/2 + 1/3. A common mistake is 2/5. But halves and thirds are different-sized pieces. You cannot directly combine them. First convert them to equal-sized pieces. 1/2 = 3/6. 1/3 = 2/6. Then 3/6 + 2/6 = 5/6.
Suppose multiplication gives 6/15. That is mathematically correct. But 6/15 can be simplified. Divide both by 3. Result: 2/5. A complete fraction answer should generally be presented in lowest terms unless there is a reason to preserve the original form.
Keep-change-flip is used for dividing fractions, not multiplication. When multiplying, multiply straight across. When dividing, multiply by the reciprocal of the second fraction. Keeping those operations separate prevents many mistakes.
Do not leave knowing only that 1/2 + 1/3 = 5/6. Leave knowing:
And most importantly: I can see why the answer makes sense. That is what the CalculateThisWay Fraction Calculator should help someone do.
| Operation | Rule |
|---|---|
| Add | Find common denominator, then add numerators |
| Subtract | Find common denominator, then subtract numerators |
| Multiply | Multiply numerator × numerator and denominator × denominator |
| Divide | Multiply by the reciprocal of the second fraction |
| Simplify | Divide numerator and denominator by their GCF |
| Fraction → Decimal | Numerator ÷ Denominator |
| Decimal → Fraction | Write over a power of 10, then simplify |
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Open Calculator ›What is a fraction?
A fraction represents part of a whole, part of a group, or a division relationship between two numbers.
What is the numerator?
The numerator is the top number of a fraction. It tells how many parts are being represented.
What is the denominator?
The denominator is the bottom number. It tells how many equal parts make one whole.
Can a denominator be zero?
No. Division by zero is undefined, so a fraction cannot have zero as its denominator.
Can the numerator be zero?
Yes. A fraction such as 0/5 equals zero.
What is a proper fraction?
A proper fraction has a numerator smaller than its denominator, such as 3/4.
What is an improper fraction?
An improper fraction has a numerator equal to or greater than its denominator, such as 7/4.
What is a mixed number?
A mixed number combines a whole number with a proper fraction, such as 1 3/4.
How do I add fractions?
If the denominators match, add the numerators and keep the denominator. If they differ, first convert the fractions to a common denominator.
Do I add the denominators when adding fractions?
No. Once the pieces are the same size, the denominator stays the same.
How do I subtract fractions?
Find a common denominator when necessary, subtract the numerators, keep the denominator, and simplify.
How do I multiply fractions?
Multiply the numerators together and multiply the denominators together, then simplify.
Do I need a common denominator to multiply fractions?
No.
How do I divide fractions?
Multiply the first fraction by the reciprocal of the second fraction.
What does keep change flip mean?
Keep the first fraction, change division to multiplication, and flip the second fraction. It is a shortcut for multiplying by the reciprocal.
How do I simplify a fraction?
Divide the numerator and denominator by their greatest common factor.
What is the GCF?
The greatest common factor is the largest whole number that divides two numbers evenly.
What is the LCD?
The least common denominator is the smallest denominator that two or more fractions can share after conversion.
What are equivalent fractions?
Equivalent fractions have different numerators and denominators but represent the same value, such as 1/2 and 2/4.
Is 2/4 the same as 1/2?
Yes.
How do I convert a mixed number to an improper fraction?
Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.
How do I convert an improper fraction to a mixed number?
Divide the numerator by the denominator. The quotient is the whole number and the remainder becomes the new numerator.
How do I convert a fraction to a decimal?
Divide the numerator by the denominator.
What is 1/2 as a decimal?
0.5.
What is 1/4 as a decimal?
0.25.
What is 3/4 as a decimal?
0.75.
What is 1/3 as a decimal?
Approximately 0.3333…, with the 3 repeating.
How do I convert a decimal to a fraction?
Write the decimal as an integer over the appropriate power of 10, then simplify.
What is 0.5 as a fraction?
1/2.
What is 0.25 as a fraction?
1/4.
What is 0.75 as a fraction?
3/4.
How do I compare fractions?
Use a common denominator, cross multiplication, decimal conversion, or a number line.
Which is bigger, 3/4 or 5/8?
3/4 is larger.
Can fractions be negative?
Yes. A negative fraction can be displayed with the negative sign in front, such as -3/4.
CalculateThisWay performs fraction arithmetic using exact integer relationships where practical and simplifies results by dividing the numerator and denominator by their greatest common divisor. Addition and subtraction results may be explained using least common denominators even when an equivalent algebraic formula is used internally. Multiplication multiplies numerators and denominators directly, while division multiplies by the reciprocal of the second fraction. Mixed numbers are converted to improper fractions for arithmetic when necessary. Fraction-to-decimal results may require rounded decimal display when the decimal repeats or does not terminate.