Fraction Simplifier

What lowest terms means

A fraction is in lowest terms, or simplest form, when the only number that divides both the numerator and the denominator is 1. The value never moves: 6/8, 9/12 and 3/4 all sit at exactly 0.75 on the number line. Reducing just swaps long words for short ones.

That swap matters more than it sounds. A recipe that says 4/8 of a stick of butter is harder to measure than one that says 1/2 a stick, and a drawing that calls for 10/16 of an inch is easy to misread on a tape. The University of Arkansas math blocks course makes the same point with a stick of butter: four of eight tablespoons is one half, and saying it the short way is what makes the fraction usable.

So the goal of simplifying is a fraction whose top and bottom share nothing. Every method on this page is a different route to the same finish line: find what the two numbers have in common, divide it out, and stop when nothing is left to divide.

The GCF shortcut: one division, done

The fastest route divides both numbers by the greatest common factor, the largest number that fits both. Take 84/126. The GCF of 84 and 126 is 42, and one division finishes the job: 84 ÷ 42 = 2 and 126 ÷ 42 = 3, so 84/126 = 2/3.

Finding that GCF needs no guessing. Run the Euclidean algorithm: divide the bigger number by the smaller and keep dividing the last divisor by the last remainder. Here 126 ÷ 84 leaves remainder 42, and then 84 ÷ 42 leaves remainder 0, so 42 is the GCF. The last nonzero remainder is always the answer.

Big ugly pairs are where this pays off. For 51/68, the algorithm gives 68 ÷ 51 remainder 17, then 51 ÷ 17 remainder 0, so the GCF is 17 and the fraction collapses to 3/4. Guessing small factors like 2 or 3 would never have found it, because both numbers are odd multiples of 17. The widget at the top of this page runs the same algorithm on whatever you type and shows the remainder steps.

The ladder method when the GCF is not obvious

When the big factor is hard to see, divide out small ones in stages and stop when nothing divides both numbers. This ladder takes more lines but each line is easy arithmetic, and the product of every number you divided by is the GCF.

Both ladders land on 2/3, which is the whole point: the ladder and the one-shot GCF division are the same math at different zoom levels. The BYU-Pathway simplifying lesson teaches the same idea as prime factorization: cancel every factor the top and bottom share. The ladder is also self-checking. If a run leaves an even top and bottom, another division is waiting; if no prime under 10 fits both, check the GCF before declaring the fraction finished, because a shared 11 or 13 can still be hiding.

Improper fractions become mixed numbers

An improper fraction has a numerator at least as big as its denominator, like 17/6. Reducing cannot shrink it below whole numbers, so the finishing touch is splitting it into a whole part plus a proper fraction: 17 ÷ 6 is 2 with remainder 5, so 17/6 = 2 5/6.

The whole part and the fraction part reduce independently. In 1 2/4 the fraction half reduces to 1 1/2; in 2 6/8 it reduces to 2 3/4. The widget handles this automatically: give it a whole number and a fraction, or an improper fraction on its own, and it reports both forms.

Watch the boundary cases. A fraction like 9/9 reduces to 1, not 0, because 9 ÷ 9 is 1. And when the denominator divides the numerator exactly, the mixed number has no fraction part left at all.

Decimals and percents never change

Reducing rewrites a fraction without moving its value, so the decimal and percent stay put. That makes them the fastest check on your work: 12/18 and 2/3 must both read 0.6667, and if your reduced fraction computes to a different decimal, a division went wrong somewhere.

The sixteenths are worth knowing on sight because tape measures, drill bits and wrench sizes all speak that language.

Notice the pattern in the reduced column: only the even sixteenths collapse to eighths, quarters and halves. Odd sixteenths like 5/16 or 7/16 are already in lowest terms because 16 is a power of 2 and shares no odd factor.

Cancel before you multiply

Reducing early saves the most work in multiplication, because common factors can be cancelled across the two fractions before anything is multiplied. Compute 4/9 x 3/8 the long way and you get 12/72, which still has to be reduced by 12 to reach 1/6. Cancel first and the numbers never get big: the 4 over the 8 leaves 1/2, the 3 over the 9 leaves 1/3, and 1/2 x 1/3 = 1/6 with nothing left to reduce.

The same trick on bigger numbers shows why it is worth the habit. 15/28 x 7/25 multiplies out to 105/700, but cancelling first makes it painless: the 7 over the 28 leaves 1/4, the 15 over the 25 leaves 3/5, and 3/5 x 1/4 = 3/20. Both routes land on 0.15, one with three-digit divisions and one without.

The rule behind the shortcut is that dividing a numerator and any denominator by the same number does not change the product, no matter which fraction each came from. What you must never do is cancel across a plus or minus sign; addition has no such shortcut, which is the next section's topic.

Moves that look like simplifying but are not

Three wrong moves show up again and again, and each one changes the value of the fraction.

Dividing across a plus sign. 2/3 + 1/3 = 3/3 = 1. Treating the sum like a fraction and reducing (2+1)/(3+3) to 3/6 gives 1/2, which is a different number entirely. Add first, reduce the sum afterwards.

Zeros in the wrong place. 0/7 simplifies to 0, and that is fine. But 7/0 is undefined, not 7, because nothing times 0 makes 7. Any fraction with a zero denominator has no value at all, and the widget flags it instead of pretending.

Losing a negative sign. Reduce -6/8 to -3/4 by dividing the 6 and the 8 by 2, keeping the minus on the numerator. Writing 6/-8 means the same number but reads badly; pick one convention and stay with it.

One more near-miss: decimals inside fractions. 0.5/2 is not in lowest terms because neither number is an integer; multiply top and bottom by 10 to get 5/20, then reduce to 1/4. Reducing only applies once both parts are whole numbers.

Where reduced fractions show up at home

Recipes lean on reduced fractions constantly. Halve a 3/4 cup of sugar and the direct answer is 3/8 of a cup, which is 6 tablespoons on a measuring spoon set. Say 6/8 of a cup out loud in a kitchen and nobody can picture it; say 3/8 and the measuring cups on the shelf already have the mark.

Tape measures run in sixteenths, so every reading arrives as a fraction begging to be reduced. A shelf cut at 10/16 of a foot is 5/8 of a foot, and the mark for 5/8 is the one the tape itself labels. Woodworkers reduce for a second reason: comparing 9/16 against 19/32 is easy once both sit as decimals, 0.5625 against 0.59375, and reduction is the first step toward that comparison.

Schoolwork closes the loop. A test score of 45/60 means nothing to a parent until it reads 3/4, and a scale drawing at 1/48 is unusable until it reads 1 inch = 4 feet. In every case the reduced form is the same quantity in words people use every day, which is the entire reason the skill exists.