The simplify fractions calculator above reduces any fraction to lowest terms and, more usefully, shows you how it got there: the greatest common divisor of the two numbers, the Euclidean algorithm steps used to find it, the prime factors shared by top and bottom, and the single division that produces the reduced fraction. Reducing 84/126 to 2/3 is easy to state and much harder to trust unless you can see that 42 divides both numbers exactly.
Arb Digital builds free tools that show their working, because a bare answer cannot be checked and therefore cannot be learned from. This page handles reduction on its own; if you need to reduce the result of an arithmetic operation, the general fraction calculator performs the operation and the reduction together.
What This Simplify Fractions Calculator Does
Enter a numerator and denominator and the tool returns the fraction in lowest terms, the greatest common divisor used, the mixed-number form, the decimal, and the percentage. If you supply a whole-number part, it simplifies the fractional portion of a mixed number and reports the improper form as well. Negative numerators are handled correctly, with the sign kept on the numerator rather than left floating on the denominator, which is the standard convention.
A denominator of zero returns an explanation instead of a number. So does a numerator of zero, though for a different reason: 0/7 simplifies to 0, since zero parts of any size is nothing. That is a legitimate answer rather than an error, and the tool says so.
How to Use It
- Enter the numerator — the number on top of the fraction.
- Enter the denominator — the number underneath. Any non-zero integer works, including negatives.
- Add a whole number if you have a mixed number. Entering 3 with 84/126 simplifies the fraction part and shows the improper form.
- Read the greatest common divisor in the results grid. That single number is what the reduction depends on.
- Follow the steps panel for the Euclidean algorithm and the shared prime factors, which show why that divisor is the largest one available.
The Method: Divide by the Greatest Common Divisor
Simplifying a fraction means finding an equivalent fraction with smaller numbers. Multiplying or dividing the numerator and denominator by the same non-zero value never changes a fraction's value, because you are effectively multiplying by a disguised 1: dividing 84/126 by 42/42 is dividing by 1. So the whole task reduces to finding the largest number that divides both, then applying it once.
That number is the greatest common divisor, also called the greatest common factor or highest common factor. For 84 and 126 it is 42, so 84 ÷ 42 = 2 and 126 ÷ 42 = 3, giving 2/3. You can get there in stages instead — divide both by 2 to get 42/63, then by 3 to get 14/21, then by 7 to get 2/3 — and the destination is identical. Using the greatest common divisor — the largest integer dividing both numbers exactly, in Wolfram MathWorld's phrasing — just does it in one move. If you need the divisor on its own for other work, our LCM and GCF calculator computes it directly.
The reliable way to find the greatest common divisor is the Euclidean algorithm, which is over two thousand years old and still what computers use. Divide the larger number by the smaller and keep the remainder; then divide the previous divisor by that remainder; repeat until the remainder is zero. The last non-zero remainder is the greatest common divisor. For 126 and 84: 126 ÷ 84 leaves 42; 84 ÷ 42 leaves 0; so the answer is 42. It never requires factoring anything, which is why it stays fast even on enormous numbers. Wolfram MathWorld describes the procedure and its properties under the Euclidean algorithm.
The Prime Factor View
There is a second route that is slower but far more revealing. Break both numbers into primes: 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. The greatest common divisor is the product of every prime they share, each taken to the lower of the two powers: 2¹ × 3¹ × 7¹ = 42. Cancelling those shared primes leaves 2 on top and 3 underneath.
This view explains why some fractions refuse to simplify at all. If the numerator and denominator share no prime, the greatest common divisor is 1 and the fraction is already in lowest terms — 15/28 cannot be reduced, since 15 is 3 × 5 and 28 is 2² × 7, with nothing in common. Numbers in that relationship are called coprime, and every fully simplified fraction has a coprime numerator and denominator by definition. Our prime factorization calculator produces these factor breakdowns if you want to see them for your own numbers.
Why Lowest Terms Matter Beyond Homework
The obvious reason is readability. 2/3 tells you instantly that the value is a bit over a half; 84/126 tells you nothing without effort. But there are practical reasons too. Comparing fractions is far easier in lowest terms, because two reduced fractions are equal if and only if they are written identically — a property that fails as soon as you allow unreduced forms, where 2/3 and 84/126 and 56/84 are all the same value wearing different clothes.
Reduced form also keeps numbers small in a chain of calculations. Multiply three unreduced fractions and the numbers grow multiplicatively; reduce at each step and they stay manageable. In programming, unreduced rational arithmetic is a classic source of integer overflow, which is why every rational-number library normalises after every operation. And in measurement, the reduced form is usually the one that matches a real tool: a socket set has a 3/4-inch fitting, never a 24/32-inch one.
Simplifying Mixed Numbers and Improper Fractions
A mixed number simplifies in its fractional part only. 3 84/126 becomes 3 2/3, and the whole number is untouched. If the fractional part is improper — say 3 7/4 — it needs converting first, because 7/4 is one and three quarters, so the whole thing is 4 3/4. This tool handles that case, and the mixed number calculator covers the conversion in both directions in more depth.
An improper fraction on its own can be simplified without becoming a mixed number. 22/8 reduces to 11/4, which is entirely acceptable as a final answer in most contexts and is often preferred in algebra, where 11/4 behaves better than 2 3/4 in further manipulation. Reducing and converting to a mixed number are two independent operations, and a question asking for one does not automatically require the other.
Fractions That Look Reducible But Are Not
A pattern that catches people out: sharing a digit is not sharing a factor. 21/31 looks like it should reduce because both contain a 1, but 31 is prime and does not divide 21, so the fraction is already in lowest terms. Similarly, both numbers being odd, or both being large, tells you nothing.
Quick divisibility checks save time. Both even means you can divide by 2. Both ending in 0 or 5 means you can divide by 5. If the digits of each number sum to a multiple of 3, both divide by 3 — 84 gives 8+4 = 12 and 126 gives 1+2+6 = 9, both multiples of 3, so 3 is a common factor. These tests are quick to apply mentally and will spot most reducible fractions immediately, though only the greatest common divisor guarantees you have finished.
One special case is worth naming: if the denominator is 1 after reducing, the fraction is a whole number. 45/9 reduces to 5/1, which is just 5, and writing it as a fraction afterwards is technically correct but rarely what anyone wants.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Stopping at a partial reduction. Dividing 84/126 by 2 gives 42/63, which still reduces. Check that the new numbers share no factor before you stop.
- Subtracting instead of dividing. Taking 40 off both 84 and 126 gives 44/86, a completely different value. Only multiplication and division preserve a fraction.
- Cancelling digits rather than factors. Crossing the 6 out of 16/64 to get 1/4 gives the right answer by pure coincidence and fails on almost every other fraction.
- Leaving the negative sign on the denominator. Write −2/3 rather than 2/−3; the sign belongs on the numerator or in front of the whole fraction.
- Assuming a large fraction must reduce. If the numerator and denominator are coprime, no reduction exists no matter how big the numbers are.
Related Free Tools From Arb Digital
Find the divisor on its own with the LCM and GCF calculator, see the prime breakdown with the prime factorization calculator, or generate equal-value fractions with the equivalent fractions calculator. For arithmetic, use the adding fractions calculator or the multiplying fractions calculator, and to go the other way, the decimal to fraction calculator. The free online tools hub lists the rest.
Frequently Asked Questions
Divide the numerator and the denominator by their greatest common divisor. For 84/126 the greatest common divisor is 42, so the fraction reduces to 2/3 in a single step.
It is the largest whole number that divides two numbers exactly. For 84 and 126 it is 42, because 42 goes into 84 twice and into 126 three times with nothing left over.
When the numerator and denominator share no common factor except 1. Such numbers are called coprime, and 15/28 is an example: 15 is 3 times 5 and 28 is 2 squared times 7, with no overlap.
No. Dividing the top and bottom by the same number is dividing by 1 in disguise, so the value stays identical while the numbers written down get smaller.
Yes, by reducing its fractional part while leaving the whole number alone, so 3 and 84/126 becomes 3 and 2/3. If the fractional part is improper, convert the extra whole out of it first.
The fraction equals zero, since no parts of any size still amounts to nothing. Zero over any non-zero denominator simplifies to 0.
A method for finding the greatest common divisor by repeatedly dividing and keeping the remainder until the remainder reaches zero. The last non-zero remainder is the divisor, and the method needs no factoring at all.
This tool is provided for educational and general reference use. Always check results against the method your course or specification requires.
