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Simplifying Fractions Calculator — divide by the GCD, prove it back

One fraction in, its lowest-terms form out — with the greatest common divisor named, the division shown, and a multiply-back proof that the simplified fraction equals the original.

  • ✓100% free — no paywall, ever
  • ✓No sign-up, no app
  • ✓Every answer verified before you see it

In short

To simplify a fraction, find the greatest common divisor of its numerator and denominator and divide both by it: 12/18 has a GCD of 6, so it reduces to 2/3. Dividing the top and bottom by the same number never changes the fraction's value. VerifySolve shows the GCD, the division, and a multiply-back check that rebuilds the original fraction from the simplified one.

Simplifying, three cases

One method: divide top and bottom by their greatest common divisor. The cases differ only in what you find when you look for it.

The method

1218=12÷618÷6=23\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}

Already simplest

gcd⁡(7,11)=1  ⟹  711 cannot simplify\gcd(7, 11) = 1 \;\Longrightarrow\; \frac{7}{11} \text{ cannot simplify}

Improper result

64=32=112\frac{6}{4} = \frac{3}{2} = 1\tfrac{1}{2}

The multiply-back proof

23×66=1218  ✓\frac{2}{3} \times \frac{6}{6} = \frac{12}{18} \;\checkmark

Worked examples — reduced, proven, checked

Two reductions, one fraction that is already simplest, and one improper fraction that also becomes a mixed number.

12/18

  1. 1.GCD of 12 and 18 is 6
  2. 2.12 ÷ 6 = 2 and 18 ÷ 6 = 3

2/3

Verified by multiplying back: 2 × 6 = 12 and 3 × 6 = 18 ✓.

24/36

  1. 1.GCD of 24 and 36 is 12
  2. 2.24 ÷ 12 = 2 and 36 ÷ 12 = 3

2/3

Verified by multiplying back: 2 × 12 = 24 and 3 × 12 = 36 ✓. Same value as 12/18 — equivalent fractions reduce to the same form.

7/11

  1. 1.GCD of 7 and 11 is 1 — no common factor above 1
  2. 2.Nothing to divide by

7/11

Verified: 7 × 1 = 7 and 11 × 1 = 11 rebuild the original ✓. Already in lowest terms.

42/56

  1. 1.GCD of 42 and 56 is 14
  2. 2.42 ÷ 14 = 3 and 56 ÷ 14 = 4

3/4

Verified by multiplying back: 3 × 14 = 42 and 4 × 14 = 56 ✓.

Common mistakes — and how to dodge them

Stopping at a common factor that is not the greatest

12/18 ÷ 2 = 6/9 is a step, not the answer — 6/9 still simplifies (÷ 3) to 2/3. Dividing by any common factor repeatedly does reach lowest terms eventually, but stopping early leaves the job half done. Going straight for the GCD finishes in one division.

Dividing only the numerator

12/18 does not become 2/18. Both parts must be divided by the same number, or the value changes: 2/18 is a much smaller number than 12/18. If your simplified fraction looks wildly different in size, one side was divided and the other was not.

“Simplifying” by subtracting the same number

12/18 minus 1 top and bottom is 11/17 — a different number entirely. Subtraction does not preserve fraction value; division by a shared factor does. The multiply-back proof is the habit that catches this: 11 × 1 ≠ 12, so 11/17 cannot equal 12/18.

Simplifying Fractions Calculator — questions

How do you simplify a fraction?

Find the greatest common divisor (GCD) of the numerator and denominator, then divide both by it. For 12/18 the GCD is 6, so 12 ÷ 6 = 2 and 18 ÷ 6 = 3: the fraction simplifies to 2/3. Dividing top and bottom by the same number never changes the value.

What if the fraction is already in lowest terms?

The calculator says so plainly. 7/11 has a GCD of 1 — no number bigger than 1 divides both — so it cannot be simplified further. “Already simplest” is a real answer, and the check still runs: 7 × 1 = 7 and 11 × 1 = 11 rebuilds the original.

Does simplifying change the value of the fraction?

No. 12/18 and 2/3 are the same number written two ways — both equal 0.666… Dividing the numerator and denominator by the same number is multiplying by 1 in disguise, which is why the multiply-back proof works: 2 × 6 = 12 and 3 × 6 = 18.

Can it simplify improper fractions?

Yes. 6/4 simplifies to 3/2, and because 3/2 is top-heavy the calculator also writes it as the mixed number 1 1/2. The GCD step and the mixed-number conversion are separate visible steps.

How is a simplification verified?

By multiplying back: the simplified numerator and denominator, each times the GCD, must rebuild the original fraction exactly. 2 × 6 = 12 and 3 × 6 = 18 proves that 12/18 = 2/3. The check is computed and printed on the page.

Is the simplifying fractions calculator free?

Yes — free, no sign-up, no app, full steps and verification included.

Keep solving

Fraction Calculator →

The hub: all four operations in one box, plus simplifying.

Adding Fractions Calculator →

Sums end with this same simplification step, after the common denominator.

Subtracting Fractions Calculator →

Differences end with this same simplification step.

Multiplying Fractions Calculator →

Cross-cancelling is simplifying before multiplying instead of after.

Dividing Fractions Calculator →

Quotients are simplified after the flip-and-multiply.

Mixed Number Calculator →

Top-heavy results become mixed numbers — the step after simplifying.

A note on honest use: VerifySolve is built for learning. Use the steps to understand the method, then close the tab and try one on your own — that is where the grade actually comes from. If your teacher or school has a policy on calculators, follow it.