What is the Adding Fractions Calculator?
This calculator adds two fractions together and returns the answer as a fully simplified (reduced) fraction along with its decimal equivalent. It handles positive and negative numerators, automatically finds a common denominator, and reduces the result using the greatest common divisor (GCD), so you never have to simplify by hand.
How to use it
Enter the numerator and denominator of the first fraction, then the numerator and denominator of the second fraction. Press calculate. The tool shows the simplified sum, the unsimplified numerator and denominator (useful for checking your work), and the decimal value. Denominators cannot be zero.
The formula explained
To add fractions with different denominators, we put them over a common denominator. The simplest common denominator to use is the product of the two denominators, \(b\cdot d\). We rewrite \(a/b\) as \((a\cdot d)/(b\cdot d)\) and \(c/d\) as \((c\cdot b)/(b\cdot d)\), then add the numerators:
$$\frac{a}{b} + \frac{c}{d} = \frac{a\cdot d + c\cdot b}{b\cdot d}$$
Finally, we divide the top and bottom by their GCD to reduce the fraction to lowest terms.
Worked example
Add \(1/4 + 1/6\). Using the formula: numerator \(= 1\cdot 6 + 1\cdot 4 = 6 + 4 = 10\); denominator \(= 4\cdot 6 = 24\). So the unsimplified result is \(10/24\). The GCD of 10 and 24 is 2, so we divide both: \(10\div 2 = 5\) and \(24\div 2 = 12\). The simplified answer is $$\frac{5}{12} \approx 0.4167$$.
Key Terms Explained
- Numerator
- The top number of a fraction, indicating how many equal parts are taken. In \(\tfrac{3}{4}\), the numerator is 3.
- Denominator
- The bottom number of a fraction, indicating how many equal parts make up one whole. In \(\tfrac{3}{4}\), the denominator is 4. It can never be 0.
- Common denominator
- A shared denominator for two or more fractions, required before they can be added or subtracted. It can be any common multiple of the denominators; the smallest such value is the least common denominator (LCD), equal to the least common multiple (LCM) of the denominators.
- GCD (greatest common divisor)
- The largest whole number that divides two integers exactly, also called the greatest common factor (GCF). Dividing a fraction's numerator and denominator by their GCD reduces it. For example, \(\gcd(38,24)=2\).
- Simplified / lowest terms
- A fraction is in lowest terms when the numerator and denominator share no common factor other than 1 (their GCD is 1), so it cannot be reduced further — e.g. \(\tfrac{3}{5}\).
- Improper fraction
- A fraction whose numerator is greater than or equal to its denominator, representing a value of 1 or more — e.g. \(\tfrac{19}{12}\). It can be rewritten as a mixed number such as \(1\tfrac{7}{12}\).
FAQ
Do the denominators have to be the same? No. The calculator finds a common denominator automatically.
Can I add negative fractions? Yes — enter a negative numerator (e.g. -3 over 4). The sign is handled correctly and the denominator is kept positive.
Will the answer always be reduced? Yes. The result is divided by the greatest common divisor so it is always in lowest terms.
