This mixed fraction calculator adds, subtracts, multiplies and divides mixed numbers, fractions, whole numbers and decimals, and shows every step. Type a mixed number with a space, such as 2 1/2, or an improper fraction such as 7/3. You can also enter a negative, a whole number or a decimal like 0.75. The answer comes out as a simplified mixed number, an improper fraction, a decimal and a percent. The working shows the common denominator, the reciprocal and Euclid's greatest common divisor steps. Switch to convert mode to simplify one fraction or change a decimal to a fraction.
How to use the mixed number calculator
- Enter the first number. A mixed number is the whole number, a space and the fraction:
3 1/4. Put a minus sign in front for a negative:-1 3/4. - Choose +, −, × or ÷ and enter the second number. Whole numbers (
5) and decimals (0.2) work too. - Read the result as a mixed number, an improper fraction, a decimal and a percent, and follow the numbered steps.
- The fraction bars draw the answer: each full bar is one whole, and the last bar shows the leftover fraction.
- For a single number, choose "Convert or simplify". It turns decimals into fractions, improper fractions into mixed numbers and reduces any fraction to lowest terms.
What is a mixed fraction?
OpenStax Prealgebra 2e defines a mixed number as a whole number and a fraction written together, such as 3 1/4, meaning 3 + 1/4. A fraction a/b is proper when the numerator is smaller than the denominator, so its value is less than one. It is improper when the numerator is at least as large as the denominator, so its value is one or more. Every improper fraction can be written as a mixed number and back again, and the calculator shows both.
Mixed number to improper fraction
Multiply the whole number by the denominator, add the numerator, and write the sum over the original denominator. For 2 1/3: 2 × 3 = 6, 6 + 1 = 7, so 2 1/3 = 7/3. For 5 3/8: 5 × 8 + 3 = 43, so 43/8.
Improper fraction to mixed number
Divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator and the divisor stays as the denominator. 11/6 is 1 remainder 5, so 1 5/6; 43/8 is 5 remainder 3, so 5 3/8.
Adding and subtracting mixed numbers
Fractions can only be added when they share a denominator. OpenStax's method is to rewrite each fraction over the least common denominator (LCD), add or subtract the fractions and the whole numbers, then simplify. The calculator converts both numbers to improper fractions first. That gives the same answer and never needs borrowing, which is where most mistakes happen.
Example: 3 5/6 + 2 3/4. Improper fractions: 23/6 and 11/4. The LCD of 6 and 4 is 12, so 23/6 = 46/12 and 11/4 = 33/12. 46 + 33 = 79, giving 79/12 = 6 7/12.
Example: 5 1/4 − 2 2/3. 21/4 − 8/3 = 63/12 − 32/12 = 31/12 = 2 7/12. By the borrowing method you would rewrite 5 3/12 as 4 15/12 and subtract 2 8/12. The result is the same 2 7/12.
If the second number is larger, the answer is negative: 1 1/2 − 2 3/4 = −1 1/4.
Multiplying and dividing mixed numbers
Never multiply the whole parts and fraction parts separately: 2 1/2 × 2 1/2 is not 4 1/4. OpenStax's procedure is to convert both mixed numbers to improper fractions, apply the fraction rule, and simplify. To multiply, multiply the numerators together and the denominators together: 5/2 × 5/2 = 25/4 = 6 1/4.
To divide, multiply by the reciprocal of the divisor. The reciprocal of a/b is b/a with the same sign. Example: 3 3/4 ÷ 1 1/2 = 15/4 ÷ 3/2 = 15/4 × 2/3 = 30/12 = 2 1/2. Dividing by zero is undefined, and the calculator says so instead of giving a number.
Worked examples people search for
2/3 × 2 1/2
2/3 × 2 1/2 = 5/3 = 1 2/3. Convert the mixed number first: 2 1/2 = (2 × 2 + 1)/2 = 5/2. Multiply straight across: (2 × 5)/(3 × 2) = 10/6. The GCD of 10 and 6 is 2, so 10/6 = 5/3, which is 1 2/3 as a mixed number and 1.666… as a decimal. You can also cancel before multiplying: the 2 on top of 2/3 and the 2 under 5/2 cancel, leaving 5/3 directly.
2/3 divided by 3/4
2/3 ÷ 3/4 = 8/9. OpenStax's rule is to multiply the first fraction by the reciprocal of the second. The reciprocal of 3/4 is 4/3, so 2/3 × 4/3 = 8/9, already in lowest terms because 8 and 9 share no factor. The answer is below 1 because you are asking how many 3/4s fit into 2/3, and 2/3 is the smaller amount. The reverse, 3/4 ÷ 2/3, is 1 1/8.
1/2 of 3/4
1/2 of 3/4 = 3/8. "Of" between two fractions means multiply. OpenStax Prealgebra introduces fraction multiplication with this very example: to multiply 1/2 and 3/4, think of 1/2 of 3/4, and the model shows it equals 3/8. So 1/2 × 3/4 = (1 × 3)/(2 × 4) = 3/8. The same reading works for percentages: 20% of 50 means 0.20 × 50; the percentage calculator handles those.
| Problem | Working | Answer |
|---|---|---|
| 1/2 of 3/4 | Multiply numerators and denominators: (1 × 3) / (2 × 4) = 3/8 | 3/8 |
| 1/3 of 3/4 | Multiply numerators and denominators: (1 × 3) / (3 × 4) = 3/12 | 1/4 |
| 3/4 of 2/3 | Multiply numerators and denominators: (3 × 2) / (4 × 3) = 6/12 | 1/2 |
| 2/3 × 2 1/2 | Multiply numerators and denominators: (2 × 5) / (3 × 2) = 10/6 | 1 2/3 |
| 2/3 × 1 1/2 | Multiply numerators and denominators: (2 × 3) / (3 × 2) = 6/6 | 1 |
| 1/2 of 2 1/4 | Multiply numerators and denominators: (1 × 9) / (2 × 4) = 9/8 | 1 1/8 |
| 2/3 ÷ 3/4 | Dividing is multiplying by the reciprocal: 2/3 × 4/3 = 8/9 | 8/9 |
| 3/4 ÷ 2/3 | Dividing is multiplying by the reciprocal: 3/4 × 3/2 = 9/8 | 1 1/8 |
| 1/2 ÷ 3/4 | Dividing is multiplying by the reciprocal: 1/2 × 4/3 = 4/6 | 2/3 |
| 2 1/2 ÷ 2/3 | Dividing is multiplying by the reciprocal: 5/2 × 3/2 = 15/4 | 3 3/4 |
Simplifying fractions with the GCD
A fraction is in lowest terms when the numerator and denominator share no factor except 1. To reduce it, divide both by their greatest common divisor (GCD). The equivalent fractions property, a/b = (a·c)/(b·c), guarantees the value doesn't change. The calculator finds the GCD with Euclid's algorithm, as described in the NIST Dictionary of Algorithms and Data Structures: divide the larger number by the smaller, replace the pair with the smaller number and the remainder, and repeat until the remainder is 0. For 1,071 and 462: 1,071 = 2 × 462 + 147, 462 = 3 × 147 + 21 and 147 = 7 × 21 + 0, so the GCD is 21. This is much faster than listing factors when the numbers are large.
| Fraction | GCD | Simplified | Mixed number | Decimal |
|---|---|---|---|---|
| 30/180 | 30 | 1/6 | 1/6 | 0.1(6) |
| 45/180 | 45 | 1/4 | 1/4 | 0.25 |
| 60/180 | 60 | 1/3 | 1/3 | 0.(3) |
| 90/180 | 90 | 1/2 | 1/2 | 0.5 |
| 100/180 | 20 | 5/9 | 5/9 | 0.(5) |
| 120/180 | 60 | 2/3 | 2/3 | 0.(6) |
| 135/180 | 45 | 3/4 | 3/4 | 0.75 |
| 150/180 | 30 | 5/6 | 5/6 | 0.8(3) |
| 210/180 | 30 | 7/6 | 1 1/6 | 1.1(6) |
| 270/180 | 90 | 3/2 | 1 1/2 | 1.5 |
| 280/180 | 20 | 14/9 | 1 5/9 | 1.(5) |
| 310/180 | 10 | 31/18 | 1 13/18 | 1.7(2) |
| 350/180 | 10 | 35/18 | 1 17/18 | 1.9(4) |
Fractions over 180 appear when converting angles in degrees to radians: 180° is π radians, so 210° is 210/180 π = 7π/6 radians.
Decimal to fraction
OpenStax's method for a terminating decimal is to write the digits after the point over the place value of the last digit, keep any whole number in front, and simplify. 0.1 has its last digit in the tenths place, so it is 1/10. 0.375 ends in the thousandths, so 375/1000 = 3/8. 4.09 is 4 9/100.
Repeating decimals need a different trick. If x = 0.1666…, then 10x = 1.666… and 100x = 16.666…, so 90x = 15 and x = 15/90 = 1/6. The calculator does this automatically when you put the repeating block in brackets, such as 0.1(6). In the other direction it marks a repeating result the same way: 1/7 shows as 0.(142857).
| Decimal | Fraction | Mixed number |
|---|---|---|
| 0.1 | 1/10 | 1/10 |
| 0.125 | 1/8 | 1/8 |
| 0.2 | 1/5 | 1/5 |
| 0.25 | 1/4 | 1/4 |
| 0.3 | 3/10 | 3/10 |
| 0.33… | 1/3 | 1/3 |
| 0.375 | 3/8 | 3/8 |
| 0.4 | 2/5 | 2/5 |
| 0.5 | 1/2 | 1/2 |
| 0.6 | 3/5 | 3/5 |
| 0.66… | 2/3 | 2/3 |
| 0.625 | 5/8 | 5/8 |
| 0.7 | 7/10 | 7/10 |
| 0.75 | 3/4 | 3/4 |
| 0.8 | 4/5 | 4/5 |
| 0.875 | 7/8 | 7/8 |
| 0.9 | 9/10 | 9/10 |
| 1.15 | 23/20 | 1 3/20 |
| 1.25 | 5/4 | 1 1/4 |
| 1.5 | 3/2 | 1 1/2 |
| 2.75 | 11/4 | 2 3/4 |
Fractions with whole numbers
A whole number is a fraction with denominator 1, so 3 = 3/1. That is all you need for "2/3 times 2" (2/3 × 2/1 = 4/3) or "5 − 3/8" (40/8 − 3/8 = 37/8 = 4 5/8). Type the whole number on its own and the calculator treats it that way.
Negative mixed numbers
A minus sign in front of a mixed number negates the whole value: −2 1/4 = −(2 + 1/4) = −9/4. A common slip is to read it as −2 + 1/4, which is −1 3/4. After converting, the ordinary sign rules apply. The product or quotient of two negatives is positive, and of a negative and a positive is negative. OpenStax notes that the reciprocal keeps the sign, so the reciprocal of −9/4 is −4/9.
Common mistakes
- Adding denominators. 1/2 + 1/3 is not 2/5. Use a common denominator: 3/6 + 2/6 = 5/6.
- Multiplying mixed numbers part by part. Convert to improper fractions first.
- Forgetting to flip the divisor. Only the second fraction is inverted when dividing.
- Leaving an answer unsimplified. 50/24 is correct but not finished; divide by the GCD for 25/12, or 2 1/12.
- Losing the sign of a negative mixed number. Convert −1 3/4 to −7/4, not −1/4.
For grades expressed as fractions of a total, try the grade calculator. The mm to inches converter gives inch fractions such as 3/8 in and 5/16 in, and the percent off calculator handles discounts. For percentages of a number or a percent change, use the percentage calculator and the percentage increase calculator.
Frequently asked questions
How do you add mixed fractions?
Either add the whole numbers and the fractions separately, or convert both to improper fractions first. For 2 1/2 + 5 2/3, OpenStax rewrites the fractions over the least common denominator 6: 2 3/6 + 5 4/6 = 7 7/6, which becomes 8 1/6.
How do you multiply mixed numbers?
Convert each mixed number to an improper fraction, multiply the numerators and the denominators, then simplify. OpenStax's example 3 1/3 × 5/8 becomes 10/3 × 5/8 = 50/24 = 25/12, or 2 1/12.
How do you divide mixed fractions?
Convert to improper fractions and multiply by the reciprocal of the second one. 2 1/2 ÷ 1 1/4 = 5/2 × 4/5 = 20/10 = 2.
What is 2 1/3 as a fraction?
7/3. Multiply the whole number by the denominator (2 × 3 = 6), add the numerator (6 + 1 = 7) and keep the denominator: 7/3.
What is .1 (or o.1) as a fraction?
0.1 is one tenth, 1/10. The last digit is in the tenths place, so the denominator is 10. People often type "o.1" with a letter o; the calculator reads it as 0.1.
What is 1.15 as a fraction?
1.15 = 115/100, which simplifies to 23/20, or 1 3/20 as a mixed number. The GCD of 115 and 100 is 5.
What is 1/2 + 1/3 as a fraction?
5/6. The least common denominator is 6, so 1/2 = 3/6 and 1/3 = 2/6, and 3/6 + 2/6 = 5/6.
What is 2/3 times 2 1/2?
1 2/3, or 5/3 as an improper fraction. Change 2 1/2 to 5/2, then 2/3 × 5/2 = 10/6 = 5/3. Times a plain 2 it would be 2/3 × 2/1 = 4/3 = 1 1/3.
How do you simplify 100/180?
Divide the top and bottom by their greatest common divisor, 20: 100/180 = 5/9. The table on this page lists other fractions over 180.
How do you subtract mixed numbers when the top fraction is smaller?
Borrow 1 from the whole number. In 4 3/4 − 2 7/8, rewrite as 4 6/8 − 2 7/8. 6/8 is smaller than 7/8, so borrow: 3 14/8 − 2 7/8 = 1 7/8. Converting both to improper fractions avoids borrowing altogether.
What is 2/3 divided by 3/4?
8/9. Flip 3/4 to 4/3 and multiply: 2/3 × 4/3 = 8/9. It is less than 1 because 2/3 is smaller than 3/4.
What is 1/2 of 3/4?
3/8. In fraction problems "of" means multiply, so 1/2 of 3/4 = 1/2 × 3/4 = (1 × 3)/(2 × 4) = 3/8. OpenStax Prealgebra uses this exact example to introduce fraction multiplication.
Sources & method
- OpenStax Prealgebra 2e, §4.1 Visualize Fractions — mixed numbers, improper fractions and conversions
- OpenStax Prealgebra 2e, §4.2 Multiply and Divide Fractions — simplifying, multiplication and reciprocal rules
- OpenStax Prealgebra 2e, §4.3 Multiply and Divide Mixed Numbers and Complex Fractions
- OpenStax Prealgebra 2e, §4.6 Add and Subtract Mixed Numbers — common denominators and borrowing
- OpenStax Prealgebra 2e, §5.1 Decimals — converting a decimal to a fraction or mixed number
- NIST Dictionary of Algorithms and Data Structures — Euclid's algorithm for the greatest common divisor
Results are estimates for general information. Found an error? It helps everyone — see our methodology.