Ratio Calculator

Simplify Ratio for
Simplified Ratio  =  2 : 1
GENERATE WORK

Ratio - work with steps

Input Data :
Given Ratio = 6:3

Objective :
Express the ratio in simplest form

Formula :
Simplified Ratio = frac{A}{GCD(A & B)} : frac{B}{GCD(A & B)}
Simplified Ratio = frac{A}{GCD(A, B & C)} : frac{B}{GCD(A, B & C)} : frac{C}{GCD(A, B & C)}

Solution :
gcd(6, 3) = 3
Simplified Ratio = 6/3 : 3/3
Simplified Ratio = 2 : 1

Ratio Calculator, formula, work with steps and practice problems to express the ratio of two or three numbers to its simplest form. It uses two or three whole numbers, integers, decimal numbers, fractions or mixed numbers $A, B$ and $C$ (optional) and simplifies ratios of the form $A : B: C$.
It is necessary to follow the next steps:

1. Enter two or three numbers in the box. These numbers must be three whole numbers, integers, decimal numbers, fractions or mixed numbers;
2. Press the "GENERATE WORK" button to make the computation;
3. Ratio calculator will reduce the ratio of two or three numbers to its simplest form.
Input : Two or three whole numbers, integers, decimal numbers, fractions or mixed numbers;
Output: 2 numbers ratio "$A:B$" or 3 numbers ratio "$A:B:C$".

Ratio Formulas
Simplified 2 Numbers Ratio Formula : The ratio of two whole numbers $A:B$ is determined by the formula
$$A:B=\frac{A}{GCF(A, B)}:\frac{B}{GCF(A, B)}$$

Simplified 3 Numbers Ratio Formula : The ratio of three whole numbers $A:B$ is determined by the formula
$$A:B:C=\frac{A}{GCF(A, B, C)}:\frac{B}{GCF(A, B, C)}:\frac{C}{GCF(A, B, C)}$$

What is Ratio?

A 2 numbers ratio is a relationship between two real or complex numbers indicating how many times the first number contains the second. The numbers in a ratio may be any magnitudes, such as measurements of lengths, weights, time, etc.
A ratio may be either specified by giving both constituting numbers, written as

"$A$ to $B$" or "$A:B$", or $\frac{A}{B}$
Since a ratio is a quotient, its denominator cannot be zero, i.e. $B\ne0$. The numbers $A$ and $B$ are called terms of the ratio, $A$ is the antecedent and $B$ is the consequent.
Ratios should be expressed in simplified form. For example, the ratio $35:42$ is equivalent to $5:6$. It is clear that 5 and 6 are relatively prime unlike 35 and 42. To simplify calculations, it is better to consider ratios with relatively prime antecedents and consequents.
If two ratios are equal, $A:B=C:D$, they are called the equivalent ratios. Equivalent ratios can be found by multiplying the numerator and the denominator by the same non-zero number.
In many cases, ratios consist of three or more terms. A 3 numbers ratio is an extended ratios $A:B:C$ compare three real or complex numbers $A,B$ and $C$. The ratio $A:B:C$ means that the ratio of the first two numbers is $A:B$, the ratio of the last two numbers is $B:C$, and the ratio of the first and last numbers is $A:C$.
An equation which express the equality of two ratios $A:B$ and $C:D$ is called a proportion, or mathematically
$$A:B=C:D\quad\mbox{or}\quad A:B::C:D$$
The numbers $A$, $B$, $C$ and $D$ are terms of the proportion. Usually, we say ($A$ is to $B$) as ($C$ is to $D$).

How to Find the Simplified Ratio?

To find the simplified ratio we need to convert all values to whole numbers and to reduce the whole numbers to lowest terms using the greatest common factor (GCF). We distinguish two cases:

• If $A,B$ and $S$ are whole numbers;
1. List the factors of the first number $A$;
2. List the factors of the second number $B$
3. List the factors of the third number $C$;
4. Find the GFC of $A,B$ and $C$, $GCF(A, B, C)$. If the $GCF(A, B, C)= 1$, then the ratio is in simplest form;
5. Divide $A,B$ and $C$ by the GCF, to get
$$A:B:C=\frac{A}{GCF(A, B, C)}:\frac{B}{GCF(A, B, C)}:\frac{C}{GCF(A, B, C)}$$
• If $A,B$ and $C$ are not whole numbers.
• Mixed number to ratio: If some of the numbers A, $B$ or C are mixed numbers, then convert mixed numbers to improper fractions. If numbers $A,B$ and C are like fractions, multiply fractions by the denominator to eliminate it. If numbers A and B are unlike fractions, find the $LCD(A, B, C)$ and rewrite the fractions with the common LCD as the denominator. Then multiply fractions by the common denominator to eliminate it.
• Decimal numbers to ratio: If some of the numbers are decimal numbers, multiply all values by the same factor of $10$ to eliminate all decimal places.
Let us consider the following example to find out corresponding profit shares based on the investment proportion.

Example Question: Three persons $A,$ $B$ and $C$ started a business by investing $\$5000$,$\$7000$ and $\$6000$, respectively. At the end of the same year, they earned a profit of$\$18000$. What are their profits share based on their investment proportion?

Solution: Because $A = 5000$, $B = 7000$, and $C = 6000$, the investment ratio is $5 : 7 : 6$. Since the investment ratio is $5 : 7 : 6$, the profit for persons $A,$ $B$ and $C$ are $5x$, $7x$ and $6x$, respectively. To find the profit sharing for each person find the sum of all ratio $$5x + 7x + 6x = 18000$$ By solving this equation for $x$, we obtain $$x=\frac{18000}{18}=\1000$$ The profit for the person $A$ is $5\cdot 1000=\$5000$, the profit for the person$B$is$7\cdot 1000=\$7000$ and the profit for the person $C$ is $6\cdot 1000=\$6000$. To verify the resulting sum up all the profit share should bring a total profit of$\$18000$ or use this ratio calculator.
The Ratio Calculator work with steps shows the complete step-by-step calculation to simplify a ratio $6:3$ using the simplified ratio formula. For any other numbers, just supply two or three numbers and click on the "Generate Work" button. The grade school students may use this Ratio Calculator to generate the work, verify the results of dividing fractions, comparing ratios or proportions, derived by hand or do their homework problems efficiently.

Applications or Use of Ratio

The ratio can be applied to all physical quantities, finance and numbers such as the amount of sharing, investment, profit, loss, time taken, time, work, distance, speed etc. Two magnitudes are in the golden ratio if their ratio is equal to the ratio of the sum of the magnitudes to the larger of the magnitudes. This means, for magnitudes $a>b>0$, the golden ratio is

$$\frac{a+b}{a}=\frac{a}{b}=\frac{1+\sqrt{5}}{2}$$
The golden ratio is also called the golden mean or golden section. The division of a segment into the golden ratio is important in the geometry, especially in constructing regular pentagrams and pentagons.
The ratio calculator can help to find equal ratios if we have three of the four parts of the two ratios. It is also useful in solving similar triangles. For instance, if the sides of one triangle are in proportion with the sides of another triangle, then these triangles are similar. It can also be used when resizing photos or video, etc.

Practice Problems for Simplifying Ratio

Practice Problem 1:
The perimeter of a rectangle is $100$ centimeters. The ratio of its sides is $3:7$. Find the area of this rectangle.

Practice Problem 2:
he measures of the angles of a triangle are in the ratio $1:2:3$. Find the measures of the angles of the triangle.

Practice Problem 3:
The ratios of the side lengths of the triangle $\Delta ABC$ to the corresponding side lengths of $\Delta A'B'C'$ are in ratio $2:1$. Find perimeters of these triangles.

The ratio calculator, formula, work with steps, applications and practice problems would be very useful for grade school students of K-12 education to learn how to find the ratio of two or three numbers and where it is applicable. It is helpful in many situations in mathematics, other fields of science and real life.