Why must a denominator be non-zero?

A fraction represents division by its denominator. Division by zero has no defined finite value.

Why do I invert the second fraction when dividing?

Multiplying by d/c reverses multiplication by c/d, so (a/b) ÷ (c/d) equals (a/b) × (d/c).

Is the decimal answer always exact?

Only fractions whose reduced denominator contains no prime factors other than 2 and 5 terminate in base ten. Others repeat and must be rounded for display.

How is a negative sign normalized?

The final denominator is kept positive, so any negative sign is placed on the numerator.

Why is the result -21/10 instead of -2 1/10?

The tool normalizes output as a reduced improper fraction and supplies a decimal; it does not format mixed numbers.

What question does Fraction calculator answer?

Use the Fraction calculator for fast, clear results with formulas and practical examples. The implemented relationship is R=reduce_gcd({(x₁x₅+x₄x₂)/(x₂x₅); (x₁x₅−x₄x₂)/(x₂x₅); x₁x₄/(x₂x₅); x₁x₅/(x₂x₄)}); selector=x₃; x₁=First numerator; x₂=First denominator; x₃=Operation; x₄=Second numerator; x₅=Second denominator.

Which inputs does Fraction calculator use?

It uses the visible fields First numerator, First denominator, Operation, Second numerator, Second denominator. No hidden value is substituted for an omitted required input.

What happens when Fraction calculator receives an invalid or extreme value?

The page rejects missing, ambiguous, out-of-range, or non-finite values and refuses to display a non-finite result.

Why can Fraction calculator differ from another result?

Different unit conventions, endpoint policies, fee or rate assumptions, formula domains, and premature rounding can produce a different answer.

Can I use Fraction calculator as professional advice?

No. It answers the displayed mathematical question and cannot account for omitted real-world facts.

How can I verify the Fraction calculator result?

Recalculate the two evaluated examples without rounding intermediate values and compare the declared formula or convention with Elementary rational arithmetic.

Fraction calculator

Result5/6

Decimal0.8333333333

Formula

Fraction calculator evaluates the registered expression “R=reduce_gcd({(x₁x₅+x₄x₂)/(x₂x₅); (x₁x₅−x₄x₂)/(x₂x₅); x₁x₄/(x₂x₅); x₁x₅/(x₂x₄)}); selector=x₃; x₁=First numerator; x₂=First denominator; x₃=Operation; x₄=Second numerator; x₅=Second denominator” with the validated inputs. Full calculation precision is retained until the result is formatted for display.

R=reduce_gcd({(x₁x₅+x₄x₂)/(x₂x₅); (x₁x₅−x₄x₂)/(x₂x₅); x₁x₄/(x₂x₅); x₁x₅/(x₂x₄)}); selector=x₃; x₁=First numerator; x₂=First denominator; x₃=Operation; x₄=Second numerator; x₅=Second denominator

Elementary rational arithmetic

What this calculator answers

Use this calculator when both operands are ratios of integers. The exact reduced fraction is the primary answer; the decimal is supplementary and may repeat indefinitely.

Input definitions

First numerator and denominator: The integer a and non-zero integer b in the first fraction a/b.
Operation: Addition, subtraction, multiplication, or division.
Second numerator and denominator: The integer c and non-zero integer d in the second fraction c/d. Division also requires c ≠ 0.

Formulas and variables

Addition and subtraction

a/b ± c/d = (ad ± bc) / bd

b ≠ 0
d ≠ 0

The resulting numerator and denominator are divided by their greatest common divisor.

Multiplication

a/b × c/d = ac / bd

b ≠ 0
d ≠ 0

Cross-cancellation before multiplication is equivalent and can keep integers smaller.

Division

a/b ÷ c/d = ad / bc

b ≠ 0
c ≠ 0
d ≠ 0

Dividing by a fraction means multiplying by its reciprocal.

Assumptions and rounding

All four entered parts are integers within JavaScript’s safe integer range.
Denominators are normalized so the final denominator is positive.
The fraction is reduced exactly by the greatest common divisor before a decimal is shown.
Decimal display may round a repeating expansion, while the fraction remains exact.

Worked examples

Add one half and one third

Compute 1/2 + 1/3.

Calculation

Use common denominator 2 × 3 = 6.
Cross-products: 1 × 3 + 1 × 2 = 3 + 2 = 5.
The result is 5/6; gcd(5, 6) = 1, so it is already reduced.

Result: 5/6, approximately 0.8333333333.

Divide three quarters by two fifths

Compute 3/4 ÷ 2/5.

Calculation

Invert the second fraction: 2/5 becomes 5/2.
Multiply: 3 × 5 / (4 × 2) = 15/8.
gcd(15, 8) = 1; as a mixed number, 15/8 = 1 7/8.

Result: 15/8, or 1.875.

Questions people ask

Why must a denominator be non-zero?: A fraction represents division by its denominator. Division by zero has no defined finite value.
Why do I invert the second fraction when dividing?: Multiplying by d/c reverses multiplication by c/d, so (a/b) ÷ (c/d) equals (a/b) × (d/c).
Is the decimal answer always exact?: Only fractions whose reduced denominator contains no prime factors other than 2 and 5 terminate in base ten. Others repeat and must be rounded for display.
How is a negative sign normalized?: The final denominator is kept positive, so any negative sign is placed on the numerator.

Boundaries and common mistakes

Do not add denominators when adding fractions; use a common denominator and cross-products.
Division by a fraction with numerator zero is also division by zero.
A rounded decimal should not replace the reduced fraction when an exact answer is required.

Sources and scope

OpenStax Prealgebra 2e — Add and Subtract FractionsApplies to: Common denominators and reduction for fraction addition and subtraction.
OpenStax Prealgebra 2e — Multiply and Divide FractionsApplies to: Numerator-and-denominator multiplication, reciprocal division, and reduction.

Variables, defaults, and limits

First numerator: First numerator is a number input. The displayed starter value is 1; it is an example, not a hidden assumption.; Default: 1; Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Numeric entries must be finite and unambiguous.
First denominator: First denominator is a number input. The displayed starter value is 2; it is an example, not a hidden assumption.; Default: 2; Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Numeric entries must be finite and unambiguous.
Operation: Operation is a choice input. The displayed starter value is Add; it is an example, not a hidden assumption.; Default: Add; Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Available choices: Add, Subtract, Multiply, Divide.
Second numerator: Second numerator is a number input. The displayed starter value is 1; it is an example, not a hidden assumption.; Default: 1; Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Numeric entries must be finite and unambiguous.
Second denominator: Second denominator is a number input. The displayed starter value is 3; it is an example, not a hidden assumption.; Default: 3; Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Numeric entries must be finite and unambiguous.

Worked examples