Free Ratio Calculator - Simplify, Solve & Compare Ratios Online | Lembog
Our Ratio Calculator is a powerful, free online tool that helps you simplify ratios to their lowest terms, solve proportions using cross multiplication, convert ratios to fractions, decimals, and percentages, and compare two ratios to check if they are equivalent. Every calculation includes detailed step-by-step solutions.
What is a Ratio?
A ratio is a mathematical relationship between two or more quantities that shows how many times one value contains or is contained within another. Ratios are typically written in colon notation as A : B or as a fraction A/B. For example, a ratio of 3 : 4 means that for every 3 units of the first quantity, there are 4 units of the second.
Ratios are everywhere in daily life — from cooking recipes and map scales to financial analysis and screen dimensions. Understanding ratios is essential for proportional reasoning, which is one of the most important concepts in mathematics.
How to Simplify a Ratio
Simplifying a ratio means reducing it to its lowest terms, similar to simplifying a fraction. To simplify a ratio A : B, find the Greatest Common Divisor (GCD) of both numbers and divide each term by it.
Steps to Simplify
- Find the GCD of both numbers using the Euclidean algorithm
- Divide both terms of the ratio by the GCD
- Write the result in colon notation
Example: Simplify 24 : 36
GCD(24, 36) = 12
24 ÷ 12 = 2
36 ÷ 12 = 3
Simplified ratio: 2 : 3
The Euclidean algorithm for finding GCD works by repeatedly dividing the larger number by the smaller and replacing with the remainder:
gcd(a, b) = b === 0 ? Math.abs(a) : gcd(b, a % b)
This recursive process is fast and efficient, even for very large numbers.
How to Solve a Proportion
A proportion is an equation that states two ratios are equal: A : B = C : D. When you know three of the four values, you can solve for the unknown using cross multiplication.
Cross Multiplication Method
Cross multiplication works by multiplying diagonally across the equal sign:
A : B = C : D
A × D = B × C
To solve for any missing value, rearrange the cross multiplication equation.
Example: Solve 3 : 4 = 6 : D
3 × D = 4 × 6
3D = 24
D = 24 / 3
D = 8
Example: Solve A : 5 = 12 : 20
A × 20 = 5 × 12
20A = 60
A = 60 / 20
A = 3
Cross multiplication is the foundation of proportional reasoning and is used extensively in chemistry, physics, cooking, and finance.
Ratio to Fraction, Decimal, and Percent
Every ratio can be expressed in multiple forms. Converting between these forms helps you understand the relationship between quantities from different perspectives.
Ratio to Fraction
Write the ratio A : B as the fraction A / B, then simplify:
Ratio 3 : 4 = Fraction 3/4
Ratio to Decimal
Divide the first term by the second:
3 : 4 = 3 ÷ 4 = 0.75
Ratio to Percent
Convert to decimal first, then multiply by 100:
3 : 4 = 0.75 × 100 = 75%
Quick Reference Table
| Ratio | Fraction | Decimal | Percent |
|---|---|---|---|
| 1 : 1 | 1/1 | 1.0 | 100% |
| 1 : 2 | 1/2 | 0.5 | 50% |
| 1 : 3 | 1/3 | 0.333 | 33.33% |
| 1 : 4 | 1/4 | 0.25 | 25% |
| 2 : 3 | 2/3 | 0.667 | 66.67% |
| 3 : 4 | 3/4 | 0.75 | 75% |
| 3 : 5 | 3/5 | 0.6 | 60% |
| 5 : 8 | 5/8 | 0.625 | 62.5% |
| 9 : 16 | 9/16 | 0.5625 | 56.25% |
How to Compare Ratios
Two ratios are equivalent if their cross products are equal. For ratios A₁ : B₁ and A₂ : B₂, they are equivalent if:
A₁ × B₂ = A₂ × B₁
Alternatively, simplify both ratios — if they reduce to the same lowest terms, they are equivalent.
Example: Are 2 : 3 and 4 : 6 equivalent?
Cross multiply: 2 × 6 = 12 and 3 × 4 = 12
12 = 12 → Yes, they are equivalent
This works because both ratios simplify to 2 : 3.
Example: Are 3 : 5 and 4 : 7 equivalent?
Cross multiply: 3 × 7 = 21 and 5 × 4 = 20
21 ≠ 20 → No, they are NOT equivalent
Real-World Applications of Ratios
Cooking and Baking
Recipes use ratios extensively. A basic pancake batter uses a 1 : 1 : 1 ratio of flour, milk, and eggs by proportion. A vinaigrette dressing uses 3 : 1 oil to vinegar. Ratios let you scale recipes up or down while maintaining the same flavor and texture. If a recipe serves 4 people and you need to serve 12, you multiply every ingredient by 3 while keeping all ratios the same.
Maps and Scale Drawings
Map scales are ratios that relate distances on a map to real-world distances. A scale of 1 : 50,000 means that 1 cm on the map represents 50,000 cm (or 500 meters) in reality. Architects use similar scale ratios for blueprints. A house plan at 1 : 100 scale means every centimeter on paper represents 1 meter of actual construction.
Finance and Investment
Financial ratios are fundamental to investing. The price-to-earnings (P/E) ratio compares stock price to earnings per share. The debt-to-income ratio 25 : 100 (or 25%) tells lenders how much of your income goes toward debt. Return on investment (ROI) is expressed as a ratio of profit to cost. Currency exchange rates are ratios too — if 1 USD = 0.85 EUR, the ratio is 1 : 0.85.
Screen and Image Ratios
Aspect ratios define the shape of screens and images. Common aspect ratios include 16 : 9 (widescreen TV), 4 : 3 (classic TV), 21 : 9 (ultrawide monitors), and 1 : 1 (Instagram squares). The 16 : 9 ratio means the screen is 16 units wide for every 9 units tall. Understanding aspect ratios helps you choose the right image dimensions for different platforms without distortion.
Medicine and Chemistry
Drug dosages are often prescribed as ratios based on body weight. A dosage of 5 mg/kg means 5 milligrams per kilogram of body weight. Chemical solutions use molar ratios to determine reactant proportions. If the balanced equation shows a 2 : 1 ratio of hydrogen to oxygen, you need exactly twice as many hydrogen molecules.
Architecture and Design
The Golden Ratio (1 : 1.618) appears throughout architecture and design. The Parthenon, the Mona Lisa, and modern logos all incorporate this ratio. Designers use it to create visually pleasing proportions in layouts, typography, and product design. The rule of thirds in photography is based on dividing an image into a 1 : 1 : 1 grid.
Equivalent Ratios Table
An equivalent ratio is created by multiplying or dividing both terms by the same number. Here are ratios equivalent to 2 : 3:
| Multiply by | Ratio |
|---|---|
| × 1 | 2 : 3 |
| × 2 | 4 : 6 |
| × 3 | 6 : 9 |
| × 4 | 8 : 12 |
| × 5 | 10 : 15 |
| × 10 | 20 : 30 |
| × 100 | 200 : 300 |
All of these ratios simplify back to 2 : 3 when divided by their GCD.
How to Use Our Ratio Calculator
- Simplify Ratio — Enter two numbers A and B to reduce the ratio to its lowest terms using GCD
- Solve Proportion — Enter any three values of A : B = C : D and select which value to solve for
- Convert Ratio — Enter A : B to see it as a simplified fraction, decimal, and percentage
- Compare Ratios — Enter two ratios to check if they are equivalent using cross multiplication
Every mode shows complete step-by-step solutions so you can understand the math behind the result.
Ratio Properties and Formulas
Fundamental Proportion Formula
If A : B = C : D, then A × D = B × C
Simplification Formula
Simplified ratio = (A ÷ GCD(A,B)) : (B ÷ GCD(A,B))
Ratio to Percent
Percent = (A / B) × 100%
Combining Ratios
If A : B = p : q and B : C = r : s, then:
A : B : C = (p × r) : (q × r) : (q × s)
Dividing in a Ratio
To divide an amount X in the ratio A : B:
First share = X × A / (A + B)
Second share = X × B / (A + B)
For example, to divide 100 in ratio 2 : 3:
First = 100 × 2/5 = 40
Second = 100 × 3/5 = 60
Frequently Asked Questions
What is the difference between a ratio and a fraction?
A ratio compares two or more quantities (like 3 : 4), while a fraction represents one quantity divided by another (like 3/4). Every ratio can be written as a fraction, but not all fractions represent ratios of separate quantities. A ratio of 3 : 4 means "3 parts to 4 parts" while the fraction 3/4 means "3 out of 4 equal parts."
How do you simplify a ratio with decimals?
Multiply both terms by 10, 100, or 1000 (depending on the decimal places) to convert to whole numbers, then simplify normally. For example, 0.5 : 1.5 → multiply by 10 → 5 : 15 → divide by 5 → 1 : 3.
Can ratios have more than two terms?
Yes. Ratios can compare three or more quantities, such as 2 : 3 : 5 for a mixture of three ingredients. To simplify a multi-term ratio, divide all terms by their GCD. The ratio 12 : 18 : 30 simplifies to 2 : 3 : 5 (dividing each by 6).
What is cross multiplication?
Cross multiplication is a technique for solving proportions. Given A/B = C/D, you multiply the numerator of each fraction by the denominator of the other: A × D = B × C. This lets you find any unknown value when three of the four values are known.
How do you divide a quantity in a given ratio?
Add the ratio terms to get the total number of parts, then divide the quantity proportionally. For example, to divide 200 in the ratio 3 : 5, the total parts are 3 + 5 = 8. The first share is 200 × 3/8 = 75, and the second share is 200 × 5/8 = 125.
What is a proportion in math?
A proportion is a statement that two ratios are equal, written as A : B = C : D or A/B = C/D. Proportions are used to find unknown values when you know that two ratios are equivalent. They are essential for scaling, unit conversion, and percentage calculations.
How are ratios used in real life?
Ratios are used in cooking (recipe scaling), finance (debt-to-income, P/E ratios), maps (scale ratios), screen dimensions (aspect ratios), medicine (drug dosages), mixing paint or chemicals, gear ratios in machinery, probability (odds), and many other fields. Any time you compare two quantities or scale something proportionally, you are using ratios.
What does it mean for two ratios to be equivalent?
Two ratios are equivalent if they represent the same relationship between quantities, even if the actual numbers are different. 2 : 3 and 4 : 6 are equivalent because both simplify to 2 : 3. You can check equivalence using cross multiplication — if the cross products are equal, the ratios are equivalent.
How do you convert a ratio to a percentage?
Divide the first term by the second to get a decimal, then multiply by 100. For example, 3 : 4 → 3 ÷ 4 = 0.75 → 0.75 × 100 = 75%. This tells you that the first quantity is 75% of the second.
What is the Golden Ratio?
The Golden Ratio is approximately 1 : 1.618 (or 1.618 : 1). It appears naturally in spirals, flower petals, and seashells. Artists and architects use it to create aesthetically pleasing proportions. Mathematically, it occurs when the ratio of the larger quantity to the smaller equals the ratio of their sum to the larger.
Why Use Our Ratio Calculator
Our Ratio Calculator provides four powerful modes in one free tool. The Simplify Ratio mode reduces any ratio to lowest terms using the GCD algorithm. The Solve Proportion mode uses cross multiplication to find the missing value in any proportion. The Convert Ratio mode shows your ratio as a simplified fraction, decimal, and percentage simultaneously. The Compare Ratios mode checks equivalence using cross multiplication with full step-by-step explanations. Every calculation is instant and includes detailed solutions so you can learn the math, not just get the answer.