Fraction Calculator

What should I know about 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.

What should I know about 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.

What do I need to know about 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.

What's the key thing to understand about 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 to check it makes sense — this catches most arithmetic slips and is worth the extra 30 seconds in an exam.

What do I need to know about Converting Between Mixed Numbers and Improper Fractions?

A mixed number like 2¾ combines a whole number and a proper fraction, while an improper fraction like 11/4 expresses the same value with a numerator larger than its denominator. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the same denominator: 2¾ becomes (2×4+3)/4 = 11/4. To convert back, divide the numerator by the denominator — the whole-number quotient becomes the new whole number, and the remainder becomes the new numerator over the same denominator: 11/4 gives 11÷4 = 2 remainder 3, so 2¾. This conversion is a necessary first step before multiplying or dividing mixed numbers, since those operations require improper fractions to work correctly, and skipping straight to multiplying mixed numbers "as written" is one of the most common fraction mistakes.

How Do Common Fractions Convert to Decimals and Percentages?

Converting between fractions, decimals, and percentages is a frequent source of small errors, particularly with fractions that don't divide evenly (repeating decimals) or when a percentage needs converting back to a simplified fraction.

Worked example: comparing 5/8 and 0.6 to see which is larger. Converting 5/8 to a decimal (5÷8 = 0.625) shows it is larger than 0.6, even though at a glance a fraction with a smaller-looking numerator and denominator can be mistaken for a smaller value than a clean decimal. This conversion step — turning any fraction into its decimal form before comparing — removes the guesswork entirely and is more reliable than trying to find a common denominator when one value is already given as a decimal.

What's the Fastest Way to Compare Two Fractions Without a Calculator?

Cross-multiplication is the quickest manual method for comparing two fractions without finding a full common denominator: for a/b versus c/d, compare a×d against b×c — whichever product is larger corresponds to the larger fraction. This avoids the extra step of simplifying to a shared denominator when only a comparison (not an actual sum) is needed. For example, comparing 3/7 and 2/5: cross-multiply 3×5=15 and 7×2=14; since 15 is larger, 3/7 is the larger fraction. This method works for any two fractions with positive denominators and is especially useful for quick mental checks during exams, where finding a true common denominator would take unnecessary extra time compared to the two simple multiplications cross-multiplication requires.

What's a Common Mistake When Adding Fractions With Different Denominators?

The most frequent error is adding numerators and denominators straight across (e.g., treating 1/2 + 1/3 as 2/5) without first converting both fractions to a common denominator. The correct method finds the least common denominator — for 1/2 and 1/3, that's 6 — converts each fraction (1/2 becomes 3/6, 1/3 becomes 2/6), and only then adds the numerators: 3/6 + 2/6 = 5/6. The straight-across method gives 2/5 = 0.4, while the correct method gives 5/6 ≈ 0.833 — a large enough gap to immediately flag the shortcut as wrong on any rough sanity check, which is exactly why converting to decimals is a useful way to catch this specific mistake before submitting a worked answer.

Worked Example: Combining Three Recipe Measurements

A recipe calls for 3/4 cup of flour, 1/3 cup of sugar, and 1/8 cup of butter, and you want the total dry-and-fat volume before adding liquids. Finding a common denominator for 4, 3 and 8 gives 24: 3/4 becomes 18/24, 1/3 becomes 8/24, and 1/8 becomes 3/24. Adding the numerators gives 18 + 8 + 3 = 29, so the total is 29/24 cups — an improper fraction, since the numerator exceeds the denominator. Converting to a mixed number, 29/24 = 1 and 5/24 cups, or just over 1 1/5 cups in practical kitchen terms. This kind of three-way fraction addition with unrelated denominators (4, 3, 8) is exactly where manual arithmetic most often goes wrong, because it's tempting to add numerators and denominators directly (giving a meaningless 29/15) rather than converting every fraction to the same denominator first.