Mathematics · Algebra

Ratio Calculator and Simplifier

Simplify a two-number whole-number ratio and calculate its decimal relationship.

Runs locally
Your numbers

Inputs and results stay in this browser. Change one value at a time to explore the relationship.

Your inputCalculatedPassed forward in chains
First simplified term3
Second simplified term4
a ÷ b0.75

Calculation steps

  1. Greatest common divisor of 18 and 24 is 6.
  2. Divide both terms by 6.
  3. Simplified ratio = 3:4.

Understand Ratio simplifier

One idea, three depths

Choose how deeply to explain Ratio simplifier

Ratio simplifier: Simplify a two-number whole-number ratio and calculate its decimal relationship.

Age 5Explain it to a 5-year-oldStart with a picture

Imagine using Ratio simplifier to answer this question: simplify a two-number whole-number ratio and calculate its decimal relationship? Enter First term and Second term; the calculator shows First simplified term. For example: The ratio 18:24 simplifies to 3:4 because the greatest common divisor is 6. The answer tells you First simplified term.

Age 15Explain it to a 15-year-oldConnect it to the formula

Dividing both terms by their greatest common divisor preserves the relationship while producing the smallest equivalent whole-number ratio. The rule is Simplified ratio = a ÷ gcd(a,b) : b ÷ gcd(a,b). Its input values are First term, Second term, and the main result is First simplified term. For example: The ratio 18:24 simplifies to 3:4 because the greatest common divisor is 6.

CollegeExplain it at college levelState the model precisely

This calculator evaluates the stated ratio simplifier relation over the valid integer domain stated below. The implemented relation is Simplified ratio = a ÷ gcd(a,b) : b ÷ gcd(a,b), evaluated from First term, Second term to produce First simplified term. Dividing both terms by their greatest common divisor preserves the relationship while producing the smallest equivalent whole-number ratio. Both ratio terms must be divided by the same nonzero factor.

Inputs and valid domain

  • First term must be an integer.
  • Second term must be an integer.

Important boundary: Both ratio terms must be divided by the same nonzero factor.

The formula

Simplified ratio = a ÷ gcd(a,b) : b ÷ gcd(a,b)

How the calculator works through it

It substitutes First term, Second term into the formula and exposes every numerical step above. The main output is First simplified term, accompanied by Second simplified term, a ÷ b.

Read the result correctly

The First simplified term is the direct answer to “simplify a two-number whole-number ratio and calculate its decimal relationship.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.

A worked check

The ratio 18:24 simplifies to 3:4 because the greatest common divisor is 6.

Where this model stops being reliable

Both ratio terms must be divided by the same nonzero factor.

Learn it by changing one value

Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.

Dictionary terms behind this calculator

Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.

These foundations help you understand why Ratio simplifier works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.

Hard requirements

  • Reading formulas and substituting values

    Ratio simplifier uses Simplified ratio = a ÷ gcd(a,b) : b ÷ gcd(a,b). You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.

    Review this foundation about 4 min

Strong support

Optional enrichment

  • Powers and exponents

    Powers are not required for every Ratio simplifier calculation, but they make related algebraic forms and code easier to read.

    Review this foundation about 4 min
Learn the missing foundationsI already know these — show the code

Mathematics → algorithm → program

Implement this calculation in code

These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.

Algorithm

  1. Compute gcd(|a|, |b|).
  2. Divide both ratio terms by that common divisor.
  3. Return the first simplified term.
Python
            from math import gcd

def first_ratio_term(a: int, b: int) -> int:
    return a // gcd(a, b)

assert first_ratio_term(18, 24) == 3
          
Current calculator valuesUpdates when you change an input above.
              
            
C
            #include <assert.h>
#include <stdint.h>
int64_t gcd64(int64_t a,int64_t b){if(a<0)a=-a;if(b<0)b=-b;while(b){int64_t r=a%b;a=b;b=r;}return a;}
int64_t first_ratio_term(int64_t a,int64_t b){return a/gcd64(a,b);}
int main(void){assert(first_ratio_term(18,24)==3);}
          
Current calculator valuesUpdates when you change an input above.
              
            
C++
            #include <cassert>
#include <cstdint>
#include <numeric>
std::int64_t first_ratio_term(std::int64_t a,std::int64_t b){return a/std::gcd(a,b);}
int main(){assert(first_ratio_term(18,24)==3);}
          
Current calculator valuesUpdates when you change an input above.
              
            
Linux x86-64 assembly

x86-64 NASM · System V ABI · Linux · integer arguments in rdi, rsi and rdx

            ; int64_t first_ratio_term(int64_t a, int64_t b)
global first_ratio_term
section .text
gcd64:
    mov rax, rdi
    test rax, rax
    jns .a_ok
    neg rax
.a_ok:
    mov rcx, rsi
    test rcx, rcx
    jns .b_ok
    neg rcx
.b_ok:
    test rcx, rcx
    jz .done
.loop:
    xor rdx, rdx
    div rcx
    mov rax, rcx
    mov rcx, rdx
    test rcx, rcx
    jnz .loop
.done:
    ret

first_ratio_term:
    push rbx
    mov rbx, rdi
    call gcd64
    mov rcx, rax
    mov rax, rbx
    cqo
    idiv rcx
    pop rbx
    ret
          
Current calculator valuesUpdates when you change an input above.
              
            
MATLAB
            function result = first_ratio_term(a, b)
    divisor = gcd(abs(round(a)), abs(round(b)));
    result = round(a) / divisor;
end
          
Current calculator valuesUpdates when you change an input above.
              
            
Wolfram Language
            ClearAll[mwCalculate];
mwCalculate[a_Integer, b_Integer] := First[{a, b}/Apply[GCD, {a, b}]];
          
Current calculator valuesUpdates when you change an input above.
              
            

Continue in mathematical software

The downloaded file includes your current inputs and first calculated result. It is created locally.

Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.

Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations

Standards, reading and academic references

Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.

Algebra and Trigonometry 2e

Read the related free OpenStax mathematics chapters
Cite this book
APA 7
Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
MLA 9
Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
Chicago author-date
Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.

OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.

Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS

These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.

APA 7

MW SysArc. (2026, July 21). Ratio Calculator and Simplifier. MW SysArc Tools. https://math.mwsysarc.com/algebra/ratio-calculator

MLA 9

MW SysArc. “Ratio Calculator and Simplifier.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/algebra/ratio-calculator. Accessed 31 Aug. 2026.

Chicago 17

MW SysArc. “Ratio Calculator and Simplifier.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/algebra/ratio-calculator.

Harvard

MW SysArc (2026) ‘Ratio Calculator and Simplifier’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/algebra/ratio-calculator (Accessed: 31 August 2026).

BibTeX and RIS records

BibTeX

@misc{mwsysarc_ratio_2026,
  author = {{MW SysArc}},
  title = {Ratio Calculator and Simplifier},
  howpublished = {MW SysArc Tools},
  year = {2026},
  url = {https://math.mwsysarc.com/algebra/ratio-calculator},
  note = {Published July 21, 2026; accessed August 31, 2026}
}

RIS

TY  - ELEC
AU  - MW SysArc
TI  - Ratio Calculator and Simplifier
T2  - MW SysArc Tools
PY  - 2026
DA  - 2026-07-21
Y2  - 2026-08-31
UR  - https://math.mwsysarc.com/algebra/ratio-calculator
N1  - Published July 21, 2026
ER  -

Clear answers

Frequently asked questions

What does the Ratio simplifier do?

Simplify a two-number whole-number ratio and calculate its decimal relationship.

How does the Ratio simplifier work?

The calculator applies Simplified ratio = a ÷ gcd(a,b) : b ÷ gcd(a,b). Dividing both terms by their greatest common divisor preserves the relationship while producing the smallest equivalent whole-number ratio.

What can I learn from the Ratio simplifier?

It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.

Does MW SysArc receive or store what I enter?

No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.

How should I use the result?

Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.

Last reviewed . Calculations tested .

MW SysArc Certified