Proportion

Proportions describe the relationship between two or more quantities whose ratios are equal. It is written as a:b = c:d, where a, b, c, and d are numbers and b, d ≠ 0. Proportions represent the relationship between changing quantities in a consistent ratio. It is used in algebra, geometry, cooking, map reading, scaling, and financial calculations. 

The concept of proportions dates back to ancient civilizations, particularly among the Greeks, where Euclid used them in mathematics, architecture, and art. In Euclid’s Elements, he explains how proportions are used in ratios and geometric relationships. A special proportion found in nature, the Golden Ratio was studied by the Greeks and later by Leonardo da Vinci. Over time, proportions became an essential concept in finance, physics, and engineering.

Ratio and proportion are used to represent relationships between numbers and to compare them. In this section, we will learn the difference between ratio and proportion. 

Proportions follow specific properties that make it easier to solve ratio-related problems. The properties of proportions are: 

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

Cross Multiplication Property:

In a proportion a/b = c/d, the product of the means equals the product of the extremes: \(a × d = b × c\). 

For example, If \({4 \over 6} = {6 \over 9} \), then \(4 \times 9 = 6 \times 6 = 36\)

Invertendo Property:

Invertendo property states that if two ratios are equal, then their reciprocal is also equal. If  \({{a \over b} = {c \over d}} \implies {{b \over a} = {d \over c}} ​\)

For example, \({8 \over 10} = {16 \over 20} \implies {10\over 8} = {20\over 16}\)

Componendo and Dividendo Property:

According to the componendo property, when we add the numerator and denominator of each ratio to forms a new ratio. If \({a \over b} = {c \over d}\) then \({{a + b} \over b} = {{c + d} \over d}\). 

​

 For example, if \({5 \over 7} = {15\over 21}\), then \({{5 + 7} \over 7} = {{15 + 21} \over 21} \implies {12 \over 7} = {36 \over 21}\)

Dividendo property states that, subtract the denominator from the numerator of each ratio to form a new ratio. If \({a \over b} = {c \over d} \implies {{a - b} \over b} = {{c - d} \over d}\)

For example, if  \({5 \over 7} = {15\over 21}\), then \({{5 - 7} \over 7} = {{15 - 21} \over 21} \implies {-2 \over 7} = {-6 \over 21}\)

Mean Proportional Property:

The mean proportional property states that, if there are three quantities a, b, and c are in continued proportion, then b is the mean proportional between a and c. 

If \({a \over b} = {b \over c} \implies b^2 = a \times c\)
 

For example, if 4, \(x\), and 9 are in continued proportion, then:
 

\({4 \over x} = {x \over 9}\)

\(x^2 = 4 \times 9 = 36\)

\(x\) \(=\) \(6\)
 

Based on the relationship between the quantities, there are two main types of proportions. They are direct proportion and inverse proportion. 

Direct Proportion:

Two quantities are in direct proportion if an increase in one quantity causes a proportional increase in the other, and a decrease in one quantity causes a proportional decrease in the other. 


If x and y are directly proportional, then:

\( x \propto y \quad \text{or} \quad \frac{x}{y} = k \)

Where k is constant.

Inverse Proportion:

Two quantities are in inverse proportion if an increase in one quantity causes a proportional decrease in the other, whereas when one quantity increases, the other decreases. 

It can be represented as: \(a \propto \frac{1}{b} \quad \text{or} \quad a \times b = k \)Where, k is a constant.

Understanding proportions helps students to solve complex math problems and develop strong problem-solving skills. In this section, we will learn a few tips and tricks to master proportion

• Always make sure the units are consistent within each ratio. For example, when comparing distances, both should be in miles or kilometers, not a mix.
   

• Students should verify their answers using the cross multiplication property to avoid errors. 
   

• Parents should encourage their children to practice it through daily examples, such as dividing snacks, comparing distances, or planning allowances. 
   

• Always reduce ratios to their lowest terms before applying proportion properties. 
   

• Teachers can use interactive methods like quizzes, worksheets, and visual aids.

We use proportions in many fields and tasks in our 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 a particular dish. 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 also used by architects and engineers to build model 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. Furthermore, 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 also used to mix the materials; where we use proportions to mix one quantity with another.