Fraction Calculator

Adding and Subtracting

To add or subtract fractions, they must share a common denominator — you can only combine parts of the same size. The reliable method is to find the lowest common multiple (LCM) of the denominators and convert each fraction to that denominator before adding the numerators. Worked example: 3/4 + 1/6. The LCM of 4 and 6 is 12. Convert 3/4 to 9/12 (multiply top and bottom by 3) and 1/6 to 2/12 (multiply by 2). Now add the numerators: 9/12 + 2/12 = 11/12. Subtraction works identically — 9/12 − 2/12 = 7/12 — just subtract the numerators once the denominators match. A common shortcut is to multiply the denominators together to get a common denominator (4 × 6 = 24), which always works but can give larger numbers you'll then need to simplify. For mixed numbers (like 2 and 1/3), either convert to improper fractions first (2 and 1/3 = 7/3) or add the whole parts and fractional parts separately, carrying where the fractions sum to more than one whole. The most frequent mistakes are adding the denominators (3/4 + 1/6 is not 4/10) and forgetting to convert both fractions to the common denominator. Always simplify the final answer.

Multiplying and Dividing

Multiplying and dividing fractions is actually simpler than adding them, because you don't need a common denominator. To multiply, multiply the numerators together and the denominators together: 3/4 × 2/5 = (3×2)/(4×5) = 6/20, which simplifies to 3/10. A useful trick is to 'cancel' common factors before multiplying — in 3/4 × 8/9, the 8 and 4 share a factor of 4, and the 3 and 9 share a factor of 3, so cancelling first gives 1/1 × 2/3 = 2/3 with much smaller numbers. To divide, multiply by the reciprocal (the flipped version) of the second fraction: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 and 7/8. The phrase 'keep, change, flip' (keep the first fraction, change ÷ to ×, flip the second) is a popular memory aid. Dividing by a fraction less than 1 makes the result larger, which surprises people but makes sense — how many quarters fit into 3? (Twelve.) For mixed numbers, convert to improper fractions before multiplying or dividing. The most common error is flipping the wrong fraction when dividing, or flipping during multiplication; only the divisor (the fraction after the ÷ sign) gets flipped, and only when dividing.

Simplifying

Simplifying (or 'reducing') a fraction means writing it in its lowest terms, where the numerator and denominator share no common factor other than 1. You do this by dividing both top and bottom by their greatest common divisor (GCD). Worked example: 15/20. The GCD of 15 and 20 is 5, so dividing both by 5 gives 3/4 — the simplest form. If you don't spot the GCD immediately, you can simplify in steps: 24/36 → divide by 2 → 12/18 → divide by 2 → 6/9 → divide by 3 → 2/3. You'll arrive at the same answer either way; using the GCD just does it in one step. A fraction is fully simplified when the GCD of numerator and denominator is 1 (they are 'coprime'). Simplifying matters because it makes fractions easier to compare, work with, and understand — 3/4 is instantly clearer than 75/100, though both are equal. It's standard practice to give final answers in simplest form unless a question asks otherwise. Converting between forms is related: 3/4 = 0.75 (divide numerator by denominator) = 75% (multiply the decimal by 100). To go from a decimal to a fraction, write it over the appropriate power of ten (0.75 = 75/100) and simplify. Recognising common equivalents (1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 1/3 ≈ 0.333) speeds up mental work.

Exam Tips and Common Errors

Common mathematical errors to avoid: sign errors when moving terms across an equation (changing sign), order of operations (BIDMAS/BODMAS — brackets, indices, division and multiplication, addition and subtraction), not checking whether answers are reasonable (a negative length or probability outside 0-1 indicates an error), and rounding too early in multi-step calculations (carry extra decimal places until the final step). Always substitute your answer back into the original equation or problem