About the Fraction Calculator
Fractions are the first piece of mathematics where the rules stop feeling obvious. Whole numbers behave the way counting behaves. Fractions do not: adding them is fiddly, multiplying them is easy, and dividing them involves turning one of them upside down for reasons nobody explains at the time.
This calculator performs all four operations on two fractions and shows the working, including the lowest common denominator, the reduction step, and the answer expressed as a mixed number and a decimal.
More usefully, the working is laid out to make the why visible rather than just the answer. If you are checking homework, the intermediate steps are the part worth reading — the answer on its own teaches nothing.
How to Use the Fraction Calculator
Choose an operation: add, subtract, multiply or divide.
Enter the two fractions as four whole numbers — a numerator and a denominator for each. The numerator is the top number, the denominator the bottom one.
Negative numbers are accepted. If you enter a negative denominator the result is tidied so the sign sits on the numerator instead, because 1/−2 and −1/2 are the same value and keeping the sign in one place makes fractions easier to compare.
Press Calculate. You get the answer in lowest terms, the same value as a mixed number if it is improper, and the decimal equivalent.
A denominator of zero will be refused. So will dividing by a fraction whose numerator is zero, since that fraction is worth nothing and nothing divides into anything.
How Fraction Arithmetic Is Calculated
The denominator is the key to all of it. It does not count anything — it names the size of the pieces being counted. In 3/8, the 8 says the whole has been cut into eighths and the 3 says you have three of them.
That single idea explains why each operation behaves the way it does.
Adding and subtracting need the pieces to be the same size first. You cannot add thirds to quarters any more than you can add apples to oranges and get seven of something. Rewrite both fractions over a shared denominator, and the numerators become countable — at which point you simply add them.
Multiplying needs no such thing, because multiplying fractions means taking a fraction of a fraction. Half of a third is a sixth. You get there by multiplying the tops and multiplying the bottoms, and no common denominator is involved. This is why multiplication, which people expect to be the harder operation, is the easy one.
Dividing is multiplication in disguise. Dividing by a number is the same as multiplying by its reciprocal — dividing by 2 and multiplying by 1/2 give the same answer. Flipping the second fraction converts the division into a multiplication, and you are back to the easy case.
Fraction Formula
Addition: a/b + c/d = (a x d + c x b) / (b x d)
Subtraction: a/b - c/d = (a x d - c x b) / (b x d)
Multiplication: a/b x c/d = (a x c) / (b x d)
Division: a/b / c/d = (a x d) / (b x c)For addition and subtraction the calculator uses the product of the two denominators, b x d, which is always a valid common denominator even if it is not the smallest one. The reduction step at the end brings it down to lowest terms regardless, so the final answer is identical. The lowest common denominator is shown separately, because that is the figure a textbook exercise usually asks for.
Step-by-Step Example
Add 1/2 and 1/3.
Step 1 — find a common denominator. The lowest common multiple of 2 and 3 is
- Both halves and thirds can be expressed in sixths.
Step 2 — rewrite both fractions. 1/2 becomes 3/6 and 1/3 becomes 2/6. The value of each is unchanged; only the size of the pieces has.
Step 3 — add the numerators. 3 + 2 = 5, over the shared denominator of 6. The answer is 5/6.
Step 4 — reduce. 5 and 6 share no factor other than 1, so 5/6 is already in lowest terms.
Now divide 1/2 by 1/3, which catches people out because the answer is bigger than what you started with.
1/2 / 1/3 = 1/2 x 3/1 = 3/2 = 1 1/2That is correct, and the intuition behind it is worth having: the question "how many thirds fit into a half?" has the answer "one and a half of them". Dividing by a number smaller than one always produces something larger, for fractions exactly as for whole numbers.
Understanding Your Result
The headline is the answer in lowest terms, which is the conventional way to write a fraction. It is not more correct than an unreduced form — 6/8 and 3/4 are the same number — but reducing gives every value a single canonical form, so two answers can be compared at a glance.
As a mixed number appears when the answer is improper, meaning the numerator is larger than the denominator. 7/3 and 2 1/3 are the same quantity written two ways. Mixed numbers are easier to picture; improper fractions are easier to calculate with, which is why the calculator gives you both.
As a decimal is the same value again in a third notation. Some fractions convert exactly and some repeat forever — 1/4 is 0.25, 1/3 is 0.333… — which is a property of the denominator rather than a rounding artefact.
The working shows what the answer was reduced by. If it says "reduce by 4", the unreduced answer had a common factor of 4 in the top and bottom.
When Should You Use This Calculator?
Checking homework. Fraction arithmetic is where most early algebra errors originate, and the step-by-step working shows where a wrong answer went wrong rather than merely that it did.
Cooking and scaling recipes. Two thirds of a recipe calling for 3/4 cup is a multiplication, and the answer is 1/2 cup — the sort of thing worth getting right before the flour goes in.
Measurements in imperial units. Timber, drill bits and spanners come in sixteenths and thirty-seconds. Adding 3/8 and 5/16 is a fraction problem whether or not it feels like one.
Splitting quantities. Working out how many 3/4-cup portions a 6-cup batch makes is a division: 6 / (3/4) = 8.
Probability and ratios, where fractions are the natural notation and exact answers matter more than decimal approximations.
Common Mistakes
Adding the denominators. 1/2 + 1/3 is not 2/5. This is the most common fraction error there is, and it comes from treating the denominator as a count rather than a size. Adding the bottoms would mean the pieces changed size just because you counted more of them.
Finding a common denominator for multiplication. It does no harm, but it is unnecessary work, and it suggests the operation has been confused with addition.
Forgetting to flip when dividing. Multiplying straight across during a division gives the reciprocal of the right answer, which is a plausible-looking number and therefore easy to miss.
Flipping the wrong fraction. It is the second one — the divisor — that gets inverted. Flipping the first gives a different answer entirely.
Leaving the answer unreduced. Not wrong mathematically, but it is the convention in almost every context, and in an exam it usually costs a mark.
Mishandling negative mixed numbers. −2 1/2 means −(2 + 1/2), which is −5/2. Reading it as −2 + 1/2 gives −3/2 and is a full unit out.
Assuming a bigger denominator means a bigger fraction. 1/8 is smaller than 1/4, because dividing something into more pieces makes each piece smaller. The denominator works in the opposite direction from intuition.