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How to use this calculator#
- Enter each fraction, with a whole number if you have one2 3/4 goes in as whole 2, numerator 3, denominator 4. The calculator converts it to the improper fraction 11/4 internally, which is the only form the arithmetic actually works on.
- Pick the operationAddition and subtraction need a common denominator and the tool finds one for you. Multiplication and division do not — division simply flips the second fraction and multiplies.
- Read all three forms of the answerThe simplified fraction is the exact answer, the mixed number is easier to picture, and the decimal is what you need for a measurement or a spreadsheet. 23/20, 1 3/20 and 1.15 are the same number.
- Sanity-check the size before you use itA sum must be larger than either part, a difference smaller than the first. Dividing by a fraction below 1 makes the answer bigger, which surprises people every time — 2 ÷ 1/2 is 4, not 1.
The formula#
The four fraction operations
a/b + c/d = (a·d + c·b) ÷ (b·d) • a/b − c/d = (a·d − c·b) ÷ (b·d) • a/b × c/d = (a·c) ÷ (b·d) • a/b ÷ c/d = (a·d) ÷ (b·c)
- a
- Numerator of the first fraction
- b
- Denominator of the first fraction — never zero
- c
- Numerator of the second fraction
- d
- Denominator of the second fraction — never zero, and never zero as a divisor either
- GCD
- Greatest common divisor: the largest whole number dividing both parts, used to simplify the result
- Mixed number
- w n/d, equal to the improper fraction (w × d + n) ÷ d
Cross-multiplying with b·d always works but does not give the lowest common denominator: 1/4 + 1/6 comes out as 10/24, which then simplifies to 5/12. Using the LCD (12) directly gets you there in one step. The answer is identical either way — only the tidiness differs.
Adding and subtracting fractions#
Fractions can only be combined once they share a denominator. Cross-multiply: a/b + c/d = (a×d + c×b) ÷ (b×d). For 3/4 + 2/5 that is (3×5 + 2×4) ÷ 20 = 23/20, or 1 3/20 as a mixed number. Subtraction is identical with a minus sign. The classic error is adding straight across — 3/4 + 2/5 is not 5/9, which would be smaller than 3/4 alone.
Multiplying and dividing fractions#
Multiplication needs no common denominator: multiply the numerators together and the denominators together, so 2/3 × 5/7 = 10/21. Division means multiplying by the reciprocal — flip the second fraction, then multiply. 2/3 ÷ 4/9 becomes 2/3 × 9/4 = 18/12 = 3/2. Always flip the divisor and never the first fraction, and expect the answer to grow when you divide by a fraction smaller than 1.
Simplifying and converting to mixed numbers#
To reduce a fraction, divide the numerator and the denominator by their greatest common divisor. 18/24 has a GCD of 6, so it simplifies to 3/4. To turn an improper fraction into a mixed number, do the division: 23 ÷ 20 is 1 remainder 3, giving 1 3/20. The decimal equivalent is simply the numerator divided by the denominator, so 3 ÷ 4 = 0.75.
Worked examples#
Subtracting fractions with different denominators
5/6 − 3/8, a common step when scaling a recipe down.
- Cross-multiply: (5 × 8 − 3 × 6) ÷ (6 × 8)
- = (40 − 18) ÷ 48 = 22/48
- GCD(22, 48) = 2, so divide both: 22 ÷ 2 = 11, 48 ÷ 2 = 24
- As a decimal: 11 ÷ 24 = 0.4583
11/24, or about 0.4583 — smaller than 5/6 (0.8333), as a subtraction should be.
Dividing mixed numbers
2 3/4 ÷ 1 1/2 — how many one-and-a-half-cup scoops fit in two and three-quarter cups.
- Convert to improper fractions: 2 3/4 = (2 × 4 + 3) ÷ 4 = 11/4
- 1 1/2 = (1 × 2 + 1) ÷ 2 = 3/2
- Divide by multiplying by the reciprocal of the second: 11/4 × 2/3 = 22/12
- GCD(22, 12) = 2, so 22/12 = 11/6
- 11 ÷ 6 = 1 remainder 5, giving the mixed number 1 5/6
11/6 = 1 5/6 ≈ 1.8333 scoops. The divisor is greater than 1, so the answer is smaller than 2 3/4.
Reference tables#
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.3333… | 33.33% |
| 2/3 | 0.6667… | 66.67% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 4/5 | 0.8 | 80% |
| 1/6 | 0.1667… | 16.67% |
| 5/6 | 0.8333… | 83.33% |
| 1/7 | 0.142857… | 14.29% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 7/8 | 0.875 | 87.5% |
| 1/9 | 0.1111… | 11.11% |
| 1/10 | 0.1 | 10% |
| 1/12 | 0.0833… | 8.33% |
| 1/16 | 0.0625 | 6.25% |
| 1/32 | 0.03125 | 3.125% |
A fraction terminates as a decimal only when its simplified denominator has no prime factors other than 2 and 5. That is why 1/8 stops but 1/3, 1/6, 1/7 and 1/9 all repeat.
| Inch fraction | Decimal inch | Millimetres |
|---|---|---|
| 1/16 | 0.0625 | 1.5875 |
| 1/8 | 0.125 | 3.175 |
| 3/16 | 0.1875 | 4.7625 |
| 1/4 | 0.25 | 6.35 |
| 5/16 | 0.3125 | 7.9375 |
| 3/8 | 0.375 | 9.525 |
| 7/16 | 0.4375 | 11.1125 |
| 1/2 | 0.5 | 12.7 |
| 9/16 | 0.5625 | 14.2875 |
| 5/8 | 0.625 | 15.875 |
| 11/16 | 0.6875 | 17.4625 |
| 3/4 | 0.75 | 19.05 |
| 13/16 | 0.8125 | 20.6375 |
| 7/8 | 0.875 | 22.225 |
| 15/16 | 0.9375 | 23.8125 |
| 1 | 1.0 | 25.4 |
Fractions of an inch are already in lowest terms here — 8/16 is written 1/2, 4/16 is 1/4. Simplifying is what makes a fraction readable at a glance.
Common mistakes#
- Adding numerators and denominators straight across3/4 + 2/5 is not 5/9. That answer (0.556) is smaller than 3/4 on its own, which is impossible for a sum of two positive numbers. Denominators name the size of the pieces, so they have to match before the pieces can be counted together.
- Flipping the wrong fraction when dividingOnly the divisor is inverted. 2/3 ÷ 4/9 is 2/3 × 9/4 = 3/2, but flipping the first fraction instead gives 3/2 × 4/9 = 2/3 — a completely different answer that happens to look plausible.
- Cancelling across an additionIn (3 + x)/3 you cannot cancel the threes to leave x; cancelling is only valid on a factor of the whole numerator. The same rule blocks simplifying 5/6 − 3/8 by striking out anything before the fractions have been combined.
- Losing the sign on a negative mixed number−2 1/4 means −(2 + 1/4) = −9/4, not (−2 + 1/4) = −7/4. The minus applies to the whole quantity, and getting it wrong shifts the answer by twice the fractional part.
Frequently asked questions#
How do I add fractions with different denominators?
Multiply each numerator by the other fraction's denominator, add the results, and put the total over the product of both denominators. Then reduce by the greatest common divisor.
What is an improper fraction?
One where the numerator is larger than the denominator, such as 23/20. It is perfectly valid maths, and converts to the mixed number 1 3/20.
How do I turn a fraction into a decimal?
Divide the numerator by the denominator. 5/8 becomes 5 ÷ 8 = 0.625. Repeating decimals like 1/3 = 0.333… are rounded for display.
Can I use negative fractions?
Yes. Put the minus sign on the whole number if there is one, otherwise on the numerator, and the calculator carries the sign through every operation.
Key terms#
- Numerator
- The number on top — how many pieces you have.
- Denominator
- The number underneath — how many equal pieces the whole was split into. It can never be zero.
- Improper fraction
- A fraction whose numerator is at least as large as its denominator, such as 23/20. Perfectly valid, and the form arithmetic is done in.
- Mixed number
- A whole number written alongside a proper fraction, such as 1 3/20. Easier to read, harder to calculate with.
- Greatest common divisor (GCD)
- The largest whole number that divides both the numerator and denominator. Dividing by it reduces a fraction to lowest terms in a single step.
- Lowest common denominator (LCD)
- The smallest number both denominators divide into — 12 for quarters and sixths. Using it keeps the numbers small when adding.
- Reciprocal
- A fraction turned upside down. Dividing by a fraction is the same as multiplying by its reciprocal.
Sources#
- Prealgebra 2e — fractions — OpenStax, Rice University
- Arithmetic: adding and subtracting fractions — Khan Academy
- Guide for the Use of the International System of Units (SP 811) — inch-to-millimetre conversion — NIST
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