Adding fractions is one of the first places elementary math stops being obvious. Two-thirds plus one-half does not become three-fifths — you cannot just add the tops and add the bottoms — yet the rule that actually works trips up students and adults alike. This fraction addition calculator takes two fractions, finds their least common denominator (LCD), adds the numerators on a common base, reduces the result to lowest terms, and shows you both the mixed number and the decimal equivalent so you can check your work.
Type four numbers and the answer appears instantly. Use it for homework, cooking, carpentry, splitting a bill, checking a measurement, or converting a recipe that calls for half a cup plus three-quarters of a cup.
How to Use
- Enter the First Numerator — the top number of the first fraction.
- Enter the First Denominator — the bottom number of the first fraction. It cannot be zero.
- Enter the Second Numerator — the top number of the second fraction.
- Enter the Second Denominator — the bottom number of the second fraction. It also cannot be zero.
- Read the result card. The calculator shows the reduced fraction, the least common denominator it used, the decimal equivalent, and the mixed number (for results larger than one).
There is no need to find a common denominator yourself. The engine computes the LCD, rewrites both fractions against it, adds the numerators, and divides the result by the greatest common divisor so the answer is always in lowest terms.
The Fraction Addition Rule
Fractions can only be added once they share a denominator. The formal rule is:
a/b + c/d = (a·d + c·b) / (b·d)
Then the result is reduced by dividing the new numerator and denominator by their greatest common divisor (gcd). The least common denominator is simply the least common multiple of the two denominators:
LCD = lcm(b, d) = b · d / gcd(b, d)
Worked example on the default inputs, 1/4 + 1/6: gcd(4, 6) = 2, so LCD = 4·6/2 = 12. Rewriting: 1/4 = 3/12 and 1/6 = 2/12. Adding gives 3/12 + 2/12 = 5/12. The gcd of 5 and 12 is 1, so 5/12 is already reduced. The decimal equivalent is roughly 0.4167.
If the denominators are already equal — say 3/8 + 1/8 — the LCD is just 8, and you add the numerators directly: (3+1)/8 = 4/8 = 1/2.
Adding Mixed Numbers and Negative Fractions
The calculator treats every input as a signed rational number, so negative numerators work exactly the way you would expect. For example, -1/2 + 1/4 = -1/4: the engine finds the LCD (4), rewrites -1/2 as -2/4, adds +1/4, and lands on -1/4. The sign always lives on the numerator; denominators are kept positive.
Mixed numbers such as 2 1/2 are not entered directly. Convert them to improper fractions first: 2 1/2 = 5/2, then add as usual. The result card prints the mixed number form automatically once the result is greater than one, so 666667/2 comes back as 333333 1/2.
Why Reduce to Lowest Terms?
An unreduced answer like 4/8 is mathematically equal to 1/2, but graders, recipes, and calculators expect the simplified form. Reduction is always safe because dividing both numerator and denominator by their gcd does not change the value — it only makes the numbers smaller and easier to use. The Python fractions.Fraction type does this automatically, which is exactly the arithmetic this calculator uses under the hood.
Examples
Example 1 — Cooking. A bread recipe calls for 3/4 cup of warm milk and 1/3 cup of honey mixed into the same liquid measure. How much liquid is that in total? LCD(4, 3) = 12. 3/4 = 9/12, 1/3 = 4/12, so the total is 13/12, or 1 1/12 cups. That is just a little over one cup, so a one-cup measure plus a tablespoon will get you close.
Example 2 — Splitting a dinner bill. Three friends split a $120 restaurant tab. One person covered half up front, another covered one-quarter, and the third owes the rest. Together the first two have paid 1/2 + 1/4 = 3/4 of the bill, which is 75%. The third person therefore owes the remaining 1/4, or 25% — about $30. Checking on this page: 1/2 + 1/4 = 3/4, decimal 0.75, matches the expected 75% share.
Example 3 — Carpenter measuring trim. A finish nailer strip needs two offcuts: 5/8 inch and 7/16 inch. What is the combined length? LCD(8, 16) = 16. 5/8 = 10/16, plus 7/16 gives 17/16 = 1 1/16 inches. When you are laying out trim in fractions of an inch, keeping the sixteenths straight is the whole job.
Example 4 — Large numerators. Adding 1,000,000/3 + 1/6 uses exact integer arithmetic: LCD = 6, 1000000/3 = 2000000/6, plus 1/6 = 2000001/6, which reduces by gcd 3 to 666667/2 = 333333.5. Floating-point arithmetic would round this, but the exact fraction form keeps every digit.
FAQ
How do you add fractions with different denominators?
Find the least common denominator (LCD) of the two denominators, rewrite each fraction so it uses that denominator, then add the numerators. For 1/4 + 1/6, the LCD is 12; rewriting gives 3/12 + 2/12 = 5/12. The calculator on this page does all three steps automatically.
Can I add fractions with negative numbers?
Yes. Enter a negative numerator (for example, -1/2) and the engine treats it as a signed rational. -1/2 + 1/4 = -1/4. The negative sign always stays on the numerator, and denominators are kept positive.
What if one denominator is zero?
Division by zero is undefined, so the calculator rejects any input whose denominator is zero and shows an error instead of returning a nonsense value. Check both denominator fields before trusting the result.
Do I need to simplify the answer myself?
No. The engine divides the result by the greatest common divisor of the numerator and denominator, so the answer is always in lowest terms. The "Fully Reduced" flag is always true on this calculator.
How do I add mixed numbers like 2 1/2 and 3 3/4?
Convert each mixed number to an improper fraction first: 2 1/2 = 5/2 and 3 3/4 = 15/4. Then add them as fractions. On this page: 5/2 + 15/4 = 10/4 + 15/4 = 25/4 = 6 1/4. The result card prints the mixed-number form for you.
Why is my decimal result repeating?
Many fractions do not terminate in decimal form. 1/3 = 0.3333…, and 5/12 = 0.41666… The fraction form on this page is exact; the decimal is shown for convenience. If you need an exact answer, trust the reduced fraction, not the decimal.
Disclaimer
This fraction addition calculator is provided for informational purposes only and does not constitute mathematical, educational, or professional advice. Results are computed with exact rational arithmetic (Python's fractions.Fraction), but you are responsible for verifying any calculation that affects money, construction, cooking, grading, or other real-world decisions. Always double-check critical arithmetic by hand or with a second tool.
For our editorial standards see the About page, and for data handling see the Privacy Policy; we do not store the fractions you enter on this page.