Understanding Proportion

Proportion is a mathematical concept that is used to describe the relationship between two or more quantities, where their ratios are equal. It also represents the relationship between changing quantities in a consistent ratio. A proportion is written as a:b = c:d, where a, b, c, d are numbers.

Also, b and d must not be zero (b,d ≠ 0. Proportions are used in applications like scaling, cooking, map reading, and financial calculations. It is a fundamental concept in algebra and geometry.

The use of proportion dates back to ancient civilizations, where people used it in mathematics, architecture, and art. The Greeks, particularly Euclid, used proportions extensively in his work “Elements”, showing how proportions are used in ratios and geometric relationships.

The Golden Ratio, which is a special kind of proportion that is found in nature, was studied by the Greeks and later in the Renaissance by Leonardo da Vinci. Over time, proportions were used and became essential in fields like finance, physics, and engineering.

Proportion is an important concept for students as it helps them understand the relationship between quantities and develop problem-solving skills in mathematics and real-life situations. Proportions are important in topics and fields like ratios, scaling, probability, finance, cooking, map reading, and science. 

• Mastering proportion helps the students to make accurate comparisons and estimate values.
   
• Develop logical reasoning and critical thinking.
   
•  It is fundamental for advanced mathematical concepts and help students to improve academic performance and practical decision-making.

There are many properties used in proportions.

The most important properties are:

• Cross Multiplication Property
• Invertendo Property
• Alternendo Property
• Componendo and Dividendo Property
• Mean Proportional Property

Let us now see the meaning and some examples of the properties mentioned above:

Cross Multiplication Property:

Cross multiplication property is fundamental in proportions. This property states that in a proportion a/b = c/d, the product of the inner terms (means) is equal to the outer terms (extremes).

a x d = b x c 
Or, ad = bc

For example, 

If 4/6 = 6/9, then according to the cross multiplication property:

4 x 9 = 6 x 6
36 = 36

Invertendo Property:

The invertendo property states that if two ratios are equal, then their reciprocal is also equal. To show this as an equation:
 

For example, if 8/10 = 16/20, then according to the invertendo property:

10/8 = 20/16
 

Componendo and Dividendo Property:

The componendo property means that we have to add the numerator and denominator of each ratio, which then forms a new ratio. To show this property as a mathematical representation:
 

Then by applying componendo property:

The same goes to the dividendo property, which means subtracting the denominator from the numerator and forming another ratio. To show this property as a mathematical equation:

Then by applying the dividendo property:

For example, if 5/7 = 15/21, then by applying componendo property:

By applying the dividendo property:

Mean Proportional Property:

The mean proportional property states that, if there are three quantities a, b, and c and if they are in a continued proportion, then b is called the main proportional between a and c.
 
To show this as a mathematical equation:
 

For example, if 4, x, and 9 are in a continued proportion, then by applying the mean proportional property:

 

There are two main types of proportions. The two main types of proportions are mentioned below:

Direct Proportion:

Direct proportion means that two quantities are in direct proportion, which means when there is an increase in one quantity it causes a proportional increase in the other quantity.


To show this as a mathematical equation:

If x and y are directly proportional, then:

Where k is constant.

Inverse Proportion:

Inverse proportion means that when two quantities are inversely proportional, when one quantity is increased the other quantity decreases.

To show this as a mathematical representation:

If a and b are inversely proportional, then:  a x b = k

Where, k is a constant.

Understanding proportions helps students solve complicated mathematical problems and develop problem-solving skills. Learning some tips and tricks are essential for both academics and real-life situations.

 
Units Matter:

Students must ensure the units are consistent within each ratio. For example, if the students are comparing distances, then both the distances they are comparing must be in miles, or they must be in kilometers.

Simplify Ratios:

Students must be able to simplify the ratios before applying any properties. This will make solving the problems easier.

Practice:

Students must practice the concept of proportions. The more the students practice solving problems related to proportions, they will be able to solve problems with accuracy and speed.

We use proportions in many fields and tasks in daily life. Let us now see where the concept of proportions is being used:

Cooking and Baking:

We use the concept of proportions to scale the recipes of particular dishes being served, we also use proportions to maintain the ratios of the ingredients being used

Maps and Scale Models:

The concept of proportions is used to scale distances in maps, it is used in model building by architects and engineers to build cars, planes, and buildings.

Shopping and Finance:

We use proportions in unit pricing; to compare prices between different products, we use it in currency exchange; for converting money from one currency to another, we use it to calculate the discounts; to find the sale of a particular product after discount, and to calculate the interest; where simple interest calculations use proportions.

Construction and Engineering:

The concept of proportions is used in blueprint scaling; where architects use proportions to represent the dimensions of the buildings, it is used to mix the materials; where we use proportions to mix one quantity with another.