Ratio Calculator
Use this ratio calculator to simplify ratios, find equivalent ratios, solve missing values in proportions, divide a total amount by a ratio, and convert ratios into fractions, decimals, and percentages. A ratio compares two or more quantities. For a two-part ratio, the basic form is \[a:b\] and the simplified form is found by dividing both parts by their greatest common factor: \[\frac{a}{g}:\frac{b}{g},\qquad g=\gcd(a,b)\] Enter your values below and the calculator will show the result with step-by-step working.
Calculate Ratios
Choose the type of ratio calculation. You can simplify a ratio, scale it, solve a proportion, or divide a total amount in a given ratio.
Result
For whole-number ratios, the simplified ratio is found by dividing all parts by their greatest common factor.
Ratio Formula
A ratio compares quantities by division. The ratio \(a:b\) can also be written as the fraction \(\frac{a}{b}\), as long as \(b\ne0\).
To simplify a whole-number ratio, divide each part by the greatest common factor:
Two ratios are equivalent if they have the same value as fractions:
For a proportion, cross multiplication gives:
Part-to-part ratio
A part-to-part ratio compares one part directly with another part, such as boys to girls, red paint to blue paint, or wins to losses.
Part-to-whole ratio
A part-to-whole ratio compares one part with the total, such as boys to all students or red paint to total paint mixture.
How to Use the Ratio Calculator
- Select the ratio calculation type from the dropdown menu.
- Enter ratio part \(a\) and ratio part \(b\).
- For equivalent ratios, enter the scale factor in the extra field.
- For proportions, enter the known values and let the calculator solve the missing value.
- For dividing a total by a ratio, enter the total amount and the ratio parts.
- Choose the number of decimal places for decimal results.
- Click Calculate Ratio to see the simplified result and formula steps.
Ratio Formulas and Rules
| Concept | Formula | Meaning |
|---|---|---|
| Ratio as fraction | \[a:b=\frac{a}{b}\] | A two-part ratio can be viewed as a fraction comparing the first part to the second part. |
| Simplified ratio | \[a:b=\frac{a}{g}:\frac{b}{g}\] | Divide both parts by \(g=\gcd(a,b)\). |
| Equivalent ratio | \[a:b=ka:kb\] | Multiplying both parts by the same nonzero number gives an equivalent ratio. |
| Proportion | \[\frac{a}{b}=\frac{c}{d}\] | Two ratios are equal. |
| Cross multiplication | \[ad=bc\] | Used to solve missing values in equivalent ratios. |
| Missing value | \[\frac{a}{b}=\frac{c}{x}\Rightarrow x=\frac{bc}{a}\] | Solves for the fourth value in a proportion. |
| Divide total by ratio | \[\text{Unit Value}=\frac{T}{a+b}\] | Find the value of one ratio unit when a total is shared in ratio \(a:b\). |
| First share | \[\text{First Share}=\frac{a}{a+b}T\] | The first part of total \(T\). |
| Second share | \[\text{Second Share}=\frac{b}{a+b}T\] | The second part of total \(T\). |
Worked Examples
Example 1: Simplify a ratio
Simplify the ratio \(12:18\).
The simplified ratio is \(2:3\).
Example 2: Find an equivalent ratio
Find an equivalent ratio to \(4:7\) using a scale factor of \(5\).
The equivalent ratio is \(20:35\). It simplifies back to \(4:7\).
Example 3: Solve a missing ratio value
Solve \(2:3=10:x\).
The missing value is \(15\), so \(2:3=10:15\).
Example 4: Divide a total by a ratio
Divide \(100\) in the ratio \(2:3\).
The two shares are \(40\) and \(60\).
Example 5: Convert a ratio to percentages
Convert the ratio \(3:2\) into part-to-whole percentages.
The first part is \(60\%\) of the whole, and the second part is \(40\%\) of the whole.
Complete Guide to Ratios
A ratio is a comparison between two or more quantities. It tells how much of one quantity there is compared with another quantity. Ratios are used in mathematics, science, cooking, maps, finance, business, design, construction, chemistry, statistics, probability, sports, and everyday decision-making. Whenever two quantities are compared by relative size rather than simple difference, a ratio is involved.
The simplest ratio has two parts, such as \(2:3\). This means that for every \(2\) units of the first quantity, there are \(3\) units of the second quantity. The numbers in a ratio do not always give the exact actual quantities. They often give the relationship between the quantities. For example, if orange juice and water are mixed in the ratio \(1:4\), then every \(1\) part of juice goes with \(4\) parts of water. The actual amount could be \(1\) cup juice and \(4\) cups water, or \(2\) cups juice and \(8\) cups water.
Ratio notation
Ratios can be written in several ways. The ratio of \(a\) to \(b\) can be written as \(a:b\), \(a\text{ to }b\), or \(\frac{a}{b}\). These forms are related, but the context matters. The colon form is common when comparing parts. The fraction form is useful when simplifying or solving equations. The word form is common in explanations.
For example, the ratio \(8:12\) can be written as \(\frac{8}{12}\). Simplifying the fraction gives \(\frac{2}{3}\), so the simplified ratio is \(2:3\). This means \(8:12\) and \(2:3\) describe the same relationship.
Simplifying ratios
To simplify a ratio, divide every part of the ratio by the greatest common factor. The greatest common factor is the largest number that divides all parts evenly. For a two-part ratio \(a:b\), if \(g=\gcd(a,b)\), then the simplified ratio is:
For example, \(24:36\) has greatest common factor \(12\). Dividing both parts by \(12\) gives \(2:3\). Simplifying ratios makes them easier to understand and compare.
Equivalent ratios
Equivalent ratios describe the same relationship using different numbers. You can create an equivalent ratio by multiplying or dividing every part by the same nonzero number:
For example, \(2:3\), \(4:6\), \(6:9\), and \(20:30\) are equivalent ratios. Each one simplifies to \(2:3\). Equivalent ratios are useful when scaling recipes, maps, models, drawings, and proportions.
Ratios and fractions
A two-part ratio can be connected to a fraction. The ratio \(a:b\) can be written as \(\frac{a}{b}\). This fraction compares the first quantity to the second. If \(a:b=3:4\), then the first quantity is \(\frac{3}{4}\) of the second quantity. However, this does not mean the first quantity is \(\frac{3}{4}\) of the total. The total is \(3+4=7\) parts, so the first quantity is \(\frac{3}{7}\) of the whole.
This distinction is very important. In a ratio \(3:4\), the part-to-part comparison is \(\frac{3}{4}\). The first part as a fraction of the whole is \(\frac{3}{3+4}=\frac{3}{7}\). Many ratio mistakes happen because students mix up part-to-part and part-to-whole thinking.
Part-to-part ratios
A part-to-part ratio compares one part directly with another part. For example, if a class has \(12\) boys and \(18\) girls, the ratio of boys to girls is \(12:18\), which simplifies to \(2:3\). This means that for every \(2\) boys, there are \(3\) girls.
Part-to-part ratios are useful when comparing categories inside a group. Examples include red balls to blue balls, teachers to students, sugar to flour, wins to losses, or adults to children. The total is not directly shown in the ratio unless you add the parts.
Part-to-whole ratios
A part-to-whole ratio compares one part with the total. Using the same class example, if there are \(12\) boys and \(18\) girls, the total is \(30\) students. The ratio of boys to total students is \(12:30\), which simplifies to \(2:5\). The ratio of girls to total students is \(18:30\), which simplifies to \(3:5\).
Part-to-whole ratios are closely related to percentages. If boys are \(\frac{12}{30}\) of the class, then boys are \(40\%\) of the class. If girls are \(\frac{18}{30}\), then girls are \(60\%\) of the class.
Solving proportions
A proportion is an equation stating that two ratios are equal. For example:
Proportions are often used to solve missing values. If \(\frac{2}{3}=\frac{10}{x}\), cross multiply:
This works because equivalent ratios have equal cross products. If \(\frac{a}{b}=\frac{c}{d}\), then \(ad=bc\).
Dividing a quantity in a ratio
One of the most common ratio applications is dividing a total into parts. Suppose \(100\) is divided in the ratio \(2:3\). The total number of ratio units is \(2+3=5\). Each unit is worth \(\frac{100}{5}=20\). Therefore, the first share is \(2(20)=40\), and the second share is \(3(20)=60\).
The general formulas are:
These formulas work for sharing money, ingredients, quantities, marks, materials, and many real-world totals.
Ratios with three or more parts
Ratios can have more than two parts, such as \(2:3:5\). This might describe a paint mixture with \(2\) parts red, \(3\) parts blue, and \(5\) parts white. The total number of parts is \(2+3+5=10\). If the total mixture is \(100\) liters, then each part is \(10\) liters. The quantities are \(20\), \(30\), and \(50\) liters.
Although this calculator focuses mainly on two-part ratios, the same principles extend to multiple parts. Add all ratio parts, divide the total by that sum, then multiply each ratio part by the unit value.
Ratios and rates
A rate is a special type of ratio that compares quantities with different units. For example, speed is a ratio of distance to time, such as \(60\text{ km}:1\text{ hour}\). Price per item is also a rate, such as \(\$12:3\text{ items}\), which simplifies to \(\$4:1\text{ item}\). Unit rates are ratios where the second quantity is \(1\).
Rates are important in real life because they help compare efficiency, cost, speed, density, concentration, and productivity. If one car travels \(300\) kilometers in \(5\) hours, its speed is \(\frac{300}{5}=60\) kilometers per hour.
Ratios in maps and scale drawings
Maps and scale drawings use ratios to represent large objects on smaller drawings. A map scale might say \(1:100000\), meaning \(1\) unit on the map represents \(100000\) of the same units in real life. If \(1\) centimeter on the map represents \(100000\) centimeters in real life, then it represents \(1\) kilometer.
Scale drawings use equivalent ratios. If a model car is built at scale \(1:24\), then every \(1\) unit on the model represents \(24\) units on the real car. Ratios make it possible to shrink or enlarge objects while preserving proportions.
Ratios in recipes and mixtures
Recipes often use ratios to keep flavors and textures consistent. If a recipe uses flour and sugar in the ratio \(3:1\), then for every \(3\) parts flour, there is \(1\) part sugar. Doubling or tripling the recipe means multiplying both parts by the same scale factor. The ratio remains the same even though the total quantity changes.
Mixtures also use ratios in chemistry, cleaning solutions, paint, fuel, and medicine. A ratio ensures that the relationship between ingredients stays constant. If only one ingredient is increased, the ratio changes, and the final mixture may not behave the same way.
Ratios and percentages
Ratios can be converted to percentages when you compare each part with the total. For a two-part ratio \(a:b\), the total number of parts is \(a+b\). The first part as a percentage is:
The second part as a percentage is:
For example, in the ratio \(2:3\), the total is \(5\) parts. The first part is \(\frac{2}{5}=40\%\), and the second part is \(\frac{3}{5}=60\%\).
Ratios with decimals
Ratios can include decimals, but they are usually easier to interpret as whole numbers. To simplify a decimal ratio, multiply all parts by a power of \(10\) to remove decimals, then simplify. For example, \(1.5:2.5\) can be multiplied by \(10\) to get \(15:25\), which simplifies to \(3:5\).
Decimal ratios are common in measurement and science, but whole-number ratios are often easier for communication. If the exact decimal ratio matters, keep enough precision before rounding.
Ratios with units
When ratios compare quantities with the same unit, the units cancel. For example, \(10\text{ cm}:15\text{ cm}\) simplifies to \(2:3\). When ratios compare quantities with different units, the result is a rate, and the units should be kept. For example, \(120\text{ miles}:2\text{ hours}\) becomes \(60\text{ miles per hour}\).
Always check whether the units are the same before simplifying. If one length is in meters and another is in centimeters, convert them to the same unit before writing the ratio.
Common mistakes with ratios
A common mistake is reversing the order of a ratio. The ratio \(2:3\) is not the same as \(3:2\). Order matters because the first number refers to the first quantity and the second number refers to the second quantity. If the ratio of boys to girls is \(2:3\), then the ratio of girls to boys is \(3:2\).
Another common mistake is confusing part-to-part and part-to-whole ratios. In the ratio \(2:3\), the first part compared with the second is \(\frac{2}{3}\), but the first part compared with the whole is \(\frac{2}{5}\). A third mistake is simplifying by subtracting instead of dividing. Ratios are simplified by dividing all parts by a common factor, not by subtracting the same number.
How to check a ratio answer
To check a simplified ratio, make sure the simplified ratio can be scaled back to the original ratio. For example, \(12:18\) simplifies to \(2:3\). Multiplying \(2:3\) by \(6\) gives \(12:18\), so the simplification is correct. You can also check by comparing fractions: \(\frac{12}{18}=\frac{2}{3}\).
To check a proportion, cross multiply. If \(a:b=c:d\), then \(ad=bc\). For example, \(2:3=10:15\) because \(2(15)=3(10)=30\).
Summary
A ratio compares quantities. Ratios can be written with a colon, words, or fractions. To simplify a ratio, divide all parts by the greatest common factor. To create an equivalent ratio, multiply all parts by the same nonzero scale factor. To solve proportions, use cross multiplication. To divide a total by a ratio, add the ratio parts to find total units, divide the total by the number of units, then multiply each ratio part by the unit value. Understanding ratios is essential for proportions, percentages, rates, scale drawings, recipes, mixtures, probability, finance, and many real-world applications.
Common Mistakes with Ratios
Reversing the order
The ratio \(a:b\) is not the same as \(b:a\). Always match the order to the quantities in the question.
Confusing part-to-part and part-to-whole
In \(2:3\), the first part is \(\frac{2}{3}\) of the second part, but \(\frac{2}{5}\) of the whole.
Subtracting instead of dividing
Ratios are simplified by dividing all parts by a common factor, not by subtracting the same number from each part.
Ignoring units
If two quantities have the same kind of unit, convert them to the same unit before simplifying the ratio.
Ratio Calculator FAQs
What is a ratio?
A ratio is a comparison between two or more quantities. For example, \(2:3\) means two parts of the first quantity for every three parts of the second quantity.
How do you simplify a ratio?
Divide every part of the ratio by the greatest common factor. For example, \(12:18\) simplifies to \(2:3\).
What is an equivalent ratio?
An equivalent ratio is a ratio with the same value. You can create one by multiplying or dividing every part by the same nonzero number.
How do you solve a proportion?
Use cross multiplication. If \(\frac{a}{b}=\frac{c}{d}\), then \(ad=bc\).
How do you divide a total by a ratio?
Add the ratio parts, divide the total by that sum, then multiply each ratio part by the unit value.
What is the difference between part-to-part and part-to-whole ratio?
A part-to-part ratio compares one part with another part. A part-to-whole ratio compares one part with the total.
Can a ratio be written as a fraction?
Yes. The ratio \(a:b\) can be written as \(\frac{a}{b}\) when \(b\ne0\).
Can ratios have decimals?
Yes, but decimal ratios are often converted to whole-number ratios by multiplying every part by a power of \(10\) and then simplifying.
Does ratio order matter?
Yes. The ratio \(2:3\) is different from \(3:2\) because the first and second quantities are reversed.
How do you convert a ratio to a percentage?
For a ratio \(a:b\), the first part as a percentage of the whole is \(\frac{a}{a+b}\times100\%\), and the second part is \(\frac{b}{a+b}\times100\%\).
