# Ratio and Proportion Calculator Simplify ratios, solve proportions, and divide quantities by ratio instantly with this free step-by-step ratio calculator. --- - **Canonical URL:** https://dothecalculation.com/calculators/ratio-proportion-calculator - **Category:** Math calculators - **Publisher:** Do The Calculation (https://dothecalculation.com) - **Cost:** Free, no account or sign-up required - **Privacy:** Runs entirely in the browser; inputs are never sent to a server - **Methodology:** https://dothecalculation.com/methodology --- ## Ratio & Proportion solver Simplify multi-part ratios, solve proportions for missing variables, and partition values into exact share distributions. - Simplifies 2-part and 3-part integer ratios - Proportion solver (A:B = C:D) - Total value partition divider with SVG bar graphs ## Introduction to Ratios and Proportions A ratio is a mathematical comparison of two or more numbers, representing their relative sizes. It is commonly written with a colon, such as \(A:B\), or as a fraction \(A/B\). Ratios are fundamental in everyday life: they describe recipe ingredients (e.g., 2 cups of flour to 1 cup of sugar), map scales (e.g., 1 inch to 10 miles), business profit shares, and financial asset allocations. A proportion is an equation stating that two ratios are equal, written as \(A:B = C:D\) or \(\frac{A}{B} = \frac{C}{D}\). Solving a proportion involves finding a missing value when the other three are known. This is most commonly solved using cross-multiplication, where the product of the extremes equals the product of the means (\(A \cdot D = B \cdot C\)). This calculator simplifies and solves ratios and proportions. By selecting between ratio simplification, proportion variable solving, or total partition dividing, the user can solve algebraic equations and view proportional distributions on a stacked bar chart. ## Simplifying Ratios and Solving Proportions To simplify a ratio (like \(24:36\)), we find the Greatest Common Divisor (GCD) of all parts and divide each part by the GCD: \[\gcd(24, 36) = 12 \implies \frac{24}{12} : \frac{36}{12} = 2:3\] To solve a proportion equation \(\frac{A}{B} = \frac{C}{D}\) for a missing variable \(x\), we cross-multiply: \[A \cdot D = B \cdot C \implies x = \frac{B \cdot C}{D} \quad (\text{if } A \text{ is } x)\] This calculator handles proportions with a variable in any of the four positions (\(A\), \(B\), \(C\), or \(D\)). ## Partitioning Totals into Proportional Shares A common real-world application of ratios is dividing a total quantity (such as money or weight) into shares defined by a ratio \(x:y:z\). The steps to partition a total value \(T\) are: 1. Sum all parts of the ratio: \(S = x + y + z\). 2. Calculate the value of a single share: \(U = T / S\). 3. Multiply each ratio part by \(U\) to find the individual shares: \[\text{Share } 1 = x \cdot U, \quad \text{Share } 2 = y \cdot U, \dots\] ## How to Use This Calculator Choose Simplify Ratio, Solve Proportion, or Divide Total Amount. For simplifying, enter a colon-separated ratio; for solving, mark the unknown part as "x"; for dividing, enter the total and the split ratio. ## Worked Example: Simplify 12:18:24 $$\gcd(12, 18, 24) = 6$$ $$\frac{12}{6} : \frac{18}{6} : \frac{24}{6} = 2:3:4$$ ## Real-World Uses for Ratios and Proportions Recipe scaling is a proportion problem: if 2 cups of flour serves 4 people, how much flour serves 10? Map and blueprint scales, paint and concrete mixing ratios, medication dosing by body weight, aspect ratios for photos and video, and splitting a bill, tip pool, or business profit by contribution all reduce to the same three operations this calculator handles — simplify, solve for the missing term, or partition a total. Partitioning is the one people reach for most often without realizing it's a ratio problem: splitting a $600 profit 2:3:5 between three partners, or dividing a fertilizer mix 3:1 between nitrogen and phosphate, is exactly the total-partition mode above. ## Related Calculators For percent-based comparisons instead of ratios, use the [percentage calculator](/calculators/percentage-calculator). To reduce a single fraction the same way this tool reduces ratios, see the [fraction calculator](/calculators/fraction-calculator). ## Frequently asked questions ### What is a ratio? A ratio is a comparison of two or more quantities, showing how many times one value contains another. ### What is a proportion? A proportion is a statement that two ratios are equal, written as an equation: a/b = c/d. ### How do you simplify a ratio? Divide all terms of the ratio by their Greatest Common Divisor (GCD). ### What is cross-multiplication? Cross-multiplication is a method to solve proportions: if a/b = c/d, then ad = bc. ### How do you divide a total into a ratio? Add the ratio terms together, divide the total by this sum to get the unit share, and multiply each ratio term by the unit share. ### Can ratios contain more than two parts? Yes, ratios can compare three or more parts, such as 2:3:5, which is common in mixture allocations. ### What is a direct proportion? A relationship where two variables increase or decrease together at a constant rate: y = kx. ### What is an inverse proportion? A relationship where one variable increases as the other decreases: y = k/x. ### How do you convert a ratio to a percentage? For ratio A:B, the percentage of part A is A / (A+B) * 100%. ### What is the golden ratio? A special mathematical ratio approximately equal to 1.618:1, believed to be aesthetically pleasing. ### Can a ratio contain decimals or fractions? Yes, but standard form requires ratios to be expressed using the smallest possible whole numbers. Decimals/fractions are multiplied by a common factor to convert them to integers. ### What is a unit rate? A unit rate is a ratio simplified so that its second term is exactly 1, such as miles per hour (60:1). ## Related concepts - **Cross-Multiplication** — The method used to solve equivalent fraction proportions. - **Unit Rate** — A ratio comparing a quantity to a single unit of another. - **Greatest Common Divisor (GCD)** — The largest integer that divides all parts of a ratio. ## Related guides - [Ratio and Proportion Guide: Simplify, Solve, Split Totals](https://dothecalculation.com/blog/math/ratio-and-proportion) — Learn how to simplify ratios, solve proportions, and split totals with worked examples tied to the live DTC ratio calculator. ## Related calculators - [Cross Multiplication Calculator](https://dothecalculation.com/calculators/cross-multiplication-calculator) — Use our free cross multiplication calculator to solve equations with fractions. Shows step-by-step work for proportions and fraction comparisons. - [System of Linear & Nonlinear Equations Solver](https://dothecalculation.com/calculators/system-of-equations-calculator) — Solve a system of two linear equations via Cramer’s Rule, or a line-and-parabola nonlinear system via substitution, with clear steps and an intersection graph. - [Linear Programming Solver (Graphical Method)](https://dothecalculation.com/calculators/linear-programming-calculator) — Optimize a 2-variable objective function subject to linear inequality constraints using the graphical corner-point method with this free solver. - [Arithmetic & Geometric Sequence Solver](https://dothecalculation.com/calculators/number-sequence-calculator) — Generate terms, calculate general formulas, and sum series for arithmetic and geometric progressions with this free solver. - [Fibonacci & Golden Ratio Calculator](https://dothecalculation.com/calculators/fibonacci-golden-ratio-calculator) — Generate Fibonacci numbers, check sequence membership, and solve Golden Ratio proportions instantly with this free math calculator. - [Exponent Calculator](https://dothecalculation.com/calculators/exponent-calculator) — Solve exponent equations, simplify fractional powers into radicals, and compute large scientific notations with this free calculator. --- _This calculator is designed for academic, scientific, and planning purposes. Numerical limits, rounding parameters, or input precision rules should be verified manually before deploying calculations in engineering layouts or peer-reviewed papers._ --- _Source: [Do The Calculation](https://dothecalculation.com/calculators/ratio-proportion-calculator). Quote freely with attribution and a link to this page._