Fraction Calculator

You can only add quantities that are the same size, and the denominator is what states the size. Thirds and halves are different units, so 1/2 + 1/3 cannot be combined until both are rewritten in one shared unit — exactly as you could not add 3 feet to 4 inches without converting first. The reliable method is to cross-multiply: a/b + c/d = (a·d + c·b) / (b·d). For 1/2 + 1/3 that gives (3 + 2)/6 = 5/6. Multiplying the denominators always produces a common denominator, though not always the smallest one. Using the least common multiple keeps the numbers down. For 1/4 + 1/6, cross-multiplying gives 10/24, which reduces to 5/12; going through the LCD of 12 gives 3/12 + 2/12 = 5/12 directly. Same answer, smaller arithmetic. And the mistake to name explicitly: do not add the denominators. 1/2 + 1/3 is not 2/5. You can see it fails immediately — 2/5 is smaller than 1/2, and adding a positive quantity cannot make the total shrink. That sanity check catches the error every time.

Multiplying is the easy one: numerator times numerator, denominator times denominator, a/b × c/d = ac/bd. No common denominator needed. And "of" means multiply — two thirds of three quarters is 2/3 × 3/4 = 6/12 = 1/2. Dividing means multiplying by the reciprocal: a/b ÷ c/d = a/b × d/c. Everyone learns the rule; far fewer are told why it is true, and the why is not complicated. Division asks how many of these fit into that. How many halves fit into 3? Six of them — so 3 ÷ 1/2 = 6, and dividing by one half is the same as multiplying by two. Once you read division that way, the surprising part disappears: dividing by a number smaller than 1 gives a bigger answer, because small pieces fit in more times. There is also an algebraic route. Multiplying a fraction by its reciprocal always gives 1, since c/d × d/c = cd/dc = 1. So multiplying both parts of the division by d/c clears the divisor completely and leaves a/b × d/c. The reciprocal is simply the number that cancels the divisor out.

Multiplying a fraction by the same number top and bottom does not change its value — 1/2, 2/4, 3/6 and 50/100 are all the same quantity wearing different clothes, because each is the previous one multiplied by a disguised 1. Simplifying runs that in reverse. Divide the numerator and denominator by their greatest common divisor and you reach lowest terms, the unique form in which the two numbers share no factor. 6/8 has a GCD of 2, so it becomes 3/4. 24/36 has a GCD of 12 and becomes 2/3. The fastest way to find that GCD on awkward numbers is the Euclidean algorithm: repeatedly replace the larger number with its remainder against the smaller until the remainder hits zero. For 24 and 36, 36 mod 24 = 12, then 24 mod 12 = 0, so the GCD is 12 — two steps, no factoring. A denominator of 1 means the result is a whole number, and a numerator larger than the denominator gives an improper fraction such as 11/4, which some contexts prefer to write as the mixed number 2 3/4. Neither form is more correct; improper fractions are simply easier to compute with, which is why they are standard in algebra.

A fraction bar is a division sign. 5/6 means 5 divided by 6, which is why converting to a decimal needs nothing more than long division: 0.8333…, repeating forever. Which fractions terminate and which repeat is entirely predictable. Reduce the fraction to lowest terms and look at the denominator: if its only prime factors are 2 and 5 — the primes that divide 10 — the decimal terminates. Everything else repeats. So 1/8 = 0.125 stops, 1/20 = 0.05 stops, but 1/3, 1/6 and 1/7 cannot. The repeating ones can run surprisingly long. 1/7 = 0.142857142857… cycles through six digits before starting over, and the same six digits in the same cyclic order appear for 2/7, 3/7 and the rest. That is the real argument for keeping fractions as fractions. 1/3 is exact; 0.333 is not, and the error compounds through a calculation. Three thirds add to exactly 1, while three copies of 0.333 give 0.999. Convert to a decimal at the end, for display, and not before.

Which is bigger, 3/7 or 4/9? Converting both to decimals works, but cross-multiplying is faster and exact: compare 3 × 9 = 27 against 4 × 7 = 28. The larger product sits above the larger fraction, so 4/9 wins. Two shortcuts cover most cases without any arithmetic. With the same denominator, the bigger numerator wins — 5/8 beats 3/8. With the same numerator, the bigger denominator loses, because the whole is being cut into more and smaller pieces: 3/5 beats 3/8. That second one runs against instinct and is worth rehearsing. Benchmarking against 1/2 handles a lot of everyday comparisons: a fraction exceeds one half whenever the numerator is more than half the denominator. 7/13 clears it; 6/13 does not. Before accepting any result from this calculator, run the estimate. Adding two fractions each near one half should land near 1. Multiplying two proper fractions must give something smaller than both of them. Dividing by a fraction below 1 must give something larger than what you started with. Answers that violate those bounds are wrong regardless of how tidy they look.