Example 1 — Simplify a fraction
Simplify \( \tfrac{18}{24} \). Compute \( g=\gcd(18,24)=6 \). Then \( \tfrac{18}{24}=\tfrac{18/6}{24/6}=\tfrac{3}{4} \).
Equivalent Fractions Calculator
Enter two fractions and click Calculate to check if they’re equivalent, see the cross-multiplication, and view simplified forms. Try the ready-made examples below.
Tip: Use whole numbers (integers). Negative values are allowed.
Must be non-zero.
Example shown forms an equivalent pair with the defaults.
Must be non-zero.
Result
Equation Preview
This shows the cross-multiplication check: A/B is equivalent to C/D if and only if
A × D = B × C.
Are the Fractions Equivalent?
If the cross-products match exactly, the two fractions represent the same rational number.
Simplified Forms
Each fraction is reduced to lowest terms using the greatest common divisor (GCD).
Helping Notes
- Denominators cannot be zero.
- Negative signs can appear in the numerator or denominator; results are normalized for clarity.
- Two fractions are equivalent if they simplify to the same lowest-terms fraction.
What is an Equivalent Fractions Calculator?
An Equivalent Fractions Calculator is a math tool that finds, verifies, and simplifies fractions that represent the same value. Two fractions are equivalent when they reduce to the same simplest form or, equivalently, when their cross-products match. This calculator lets you simplify a single fraction, compare two fractions for equivalence, or generate a list of equivalent fractions by multiplying numerator and denominator by a common nonzero factor. It also provides the algebraic steps, so students and educators can see exactly how each result is obtained. All formulas are written in LaTeX and will render responsively when you load MathJax or math.js on your page.
About the Equivalent Fractions Calculator
Fractions model parts of a whole and ratios. Many different-looking fractions can describe the same quantity. For example, \( \tfrac{1}{2} = \tfrac{2}{4} = \tfrac{50}{100} \). The calculator uses three core ideas: (1) scaling a fraction by any nonzero factor preserves its value, (2) greatest common divisor (GCD) reduction puts a fraction into lowest terms, and (3) cross-multiplication tests whether two fractions are equal without first simplifying. You can also request a specific denominator or numerator and the tool will compute the necessary scaling factor, if it exists, to reach that target exactly. It handles negative signs (placing them in the numerator by convention) and returns results in canonical form for clarity.
How to Use this Equivalent Fractions Calculator
- Enter a fraction \( \tfrac{a}{b} \) (with \(b \ne 0\)). Optionally enter a second fraction \( \tfrac{c}{d} \) to compare.
- Select an action: Simplify, Check Equivalence, Generate N Equivalents, or Match Target Numerator/Denominator.
- (Optional) Provide a target numerator or denominator to scale to (e.g., denominator \(100\)).
- Review the output: simplest form, proof via GCD or cross-multiplication, and any scaling factor used.
- Copy the steps for homework, lesson plans, or documentation.
Core Formulas (LaTeX for MathJax/math.js)
Equivalence by scaling: \[ \frac{a}{b} \sim \frac{ka}{kb} \quad \text{for any } k \in \mathbb{Z}\setminus\{0\},\ b \ne 0. \]
GCD simplification: \[ g=\gcd(|a|,|b|),\quad \frac{a}{b}=\frac{a/g}{\,b/g\,} \text{ in lowest terms}. \]
Equivalence test (cross-multiplication): \[ \frac{a}{b}=\frac{c}{d} \iff ad=bc,\quad b\ne0,\ d\ne0. \]
Scaling to a target denominator \(\hat b\): \[ k=\frac{\hat b}{b}\ \text{(integer required)},\quad \frac{a}{b}=\frac{k a}{\hat b}. \]
Scaling to a target numerator \(\hat a\): \[ k=\frac{\hat a}{a}\ \text{(integer required)},\quad \frac{a}{b}=\frac{\hat a}{k b}. \]
Examples (Illustrative)
Example 2 — Check if two fractions are equivalent
Are \( \tfrac{9}{12} \) and \( \tfrac{3}{4} \) equivalent? Test \(9\cdot4=36\) and \(12\cdot3=36\). Since \(ad=bc\), the fractions are equivalent.
Example 3 — Match a target denominator
Scale \( \tfrac{2}{5} \) to denominator \(40\). \(k=\tfrac{40}{5}=8\). New numerator \(=2\cdot8=16\). Result \( \tfrac{16}{40} \sim \tfrac{2}{5} \).
Example 4 — Zero numerator case
\( \tfrac{0}{7} \sim \tfrac{0}{100} \) because both equal \(0\). Denominator must never be zero.
FAQs
What are equivalent fractions?
Fractions that represent the same number, obtainable by multiplying or dividing numerator and denominator by the same nonzero integer.
How do I find equivalent fractions quickly?
Multiply numerator and denominator by any nonzero integer \(k\): \(\tfrac{a}{b} \to \tfrac{ka}{kb}\).
How do I check if two fractions are equivalent?
Use cross-multiplication: \(\tfrac{a}{b}=\tfrac{c}{d}\) iff \(ad=bc\) (with \(b,d\ne0\)).
What is the simplest form of a fraction?
When numerator and denominator share no common factor except 1; compute \(g=\gcd(|a|,|b|)\) and divide both by \(g\).
Can the calculator target a specific denominator or numerator?
Yes. It solves for an integer scaling factor \(k\) to reach your requested numerator or denominator exactly.
Do negative signs affect equivalence?
No. \(-\tfrac{a}{b}=\tfrac{-a}{b}=\tfrac{a}{-b}\). The tool returns a canonical sign placement.
Can I generate many equivalent fractions?
Yes—choose \(k=2,3,\dots\) to produce \( \tfrac{2a}{2b}, \tfrac{3a}{3b}, \dots \) infinitely many equivalents.
Does this work with mixed numbers or decimals?
Convert mixed numbers to improper fractions and decimals to rational form (if exact) before applying the same rules.
Is a zero denominator allowed?
No. Denominators must be nonzero. A zero numerator is allowed and represents the value \(0\).
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