Final Report

INDRODUCTION
Ratio and Proportion are used for comparison in mathematics. Ratio is a comparison of two quantities
while Proportion is a comparison of two ratios.

Ratio and proportion are mostly about fractions. When you see a fraction written as a: b, that’s a ratio. A
proportion is when two ratios are the same. In both cases, a and b are just any two whole numbers.
Understanding ratios and proportions is really important. It helps you grasp many ideas in math and
science. Let’s learn about Ratios and Proportions in detail, including their types, formulas, and examples.

Table of Content
• Ratio and Proportion Definition
• Ratio Properties
• Ratio and Proportion Formula
• Difference between Ratio and Proportion
• Applications of ratio and proportion
• Frequently asked questions

Ratio and Proportion Definition

When a fraction a/b is written a: b then it is termed as a ratio b. When two ratios let’s say a: b and c: d
are equal a: b and c: d are said to be proportional to each other. Two proportional ratios are written as
a: b:: c: d.
Ratio Definition
Ratio is a comparison of two quantities of the same unit.

The ratio of two quantities is given by using the colon symbol (:). The ratio of two quantities a and b is
given as

a:b

Where,

• a is called Antecedent

• b is called Consequent

The ratio a: b means ak/bk where k is the common factor. k is multiplied to give equivalent fractions
whose simplest form will be a/b. We can read a:b as ‘a ratio b’ or ‘a to b’.

Ratio Properties :
Key properties of Ratio are:

•If a ratio is multiplied by the same term both in antecedent and consequent then there is no change in
the actual ratio. For example,

•If the antecedent and consequent of a ratio are divided by the same number then also there is no
change in the actual ratio. For Example,

•If two ratios are equal then their reciprocals are also equal i.e. if a: b = c: d then b: a = d: c

•If two ratios are equal then their cross-multiplications are also equal. For Example

•The ratios for a pair of comparisons can be the same but the actual value may be different. For
Example, 45:60 = 5:6 and 100:120 = 5:6 hence ratio 5:6 is same but actual value is different
Proportion Definition
Proportion refers to the comparison of ratios. If two ratios are equal then they are said to be
proportionate to each other.

Two proportional ratios are represented by a double colon (::). If two ratios a: b and c: d are equal then
they are represented as

a: b:: c: d

where

• a and d are called extreme terms

• b and c are called mean terms

Proportion Properties :
Key properties of Proportions are:

• For two ratios in proportion i.e. a/b = c/d, a/c = b/d holds true.

• For two ratios in proportion i.e. a/b = c/d, b/a = d/c holds true.

• For two ratios in proportion i.e. a:b::c:d, the product of mean terms is equal to the product of extreme
terms i.e. ad = bc

• For two ratios in proportion i.e. a/b = c/d, (a + b)/b = (c + d)/d is true.

• For two ratios in proportion i.e. a/b = c/d, (a – b)/b = (c – d)/d is true.

Types of Proportions
There are three types of Proportions:

• Direct Proportion

• Inverse Proportion

• Continued Proportion

Let’s learn about them in detail.

Direct Proportion
When two quantities increase and decrease in the same ratio then the two quantities are said to be in
Direct Proportion. It means if one quantity increases then the other will also increase and if one will
decrease then the other will also decrease.

It is represented as a ∝ b. For Example, Momentum is a product of mass and velocity hence, momentum
is directly proportional to mass and velocity (p∝m or p∝v) which means if mass is constant and velocity
is high then momentum will also be high and if velocity is constant then momentum will increase with
an increase in mass.

Inverse Proportion
When two quantities are inversely related to each other i.e. increase in one leads to a decrease in the
other or a decrease in the other leads to an increase in the first quantity then the two quantities are said
to be Inversely Proportional to each other.

For Example, Pressure is given by force divided by area (P = F/A). Here Pressure is inversely related to
area which means a decrease in area will lead to an increase in pressure and an increase in area will lead
to a decrease in pressure.

Continued Proportion
If the ratio a:b = b:c = c:d, then we see that the consequent of the first ratio is equal to the antecedent of
the second ratio, and so on then the a:b:c:d is said to be in continued proportion.

• If the consequent and antecedent are not the same for two ratios then they can be converted into
continued proportion by multiplying.

• For Example, in the case of a:b and c:d consequent and antecedent are not same then the continued
proportion is given as ac:cb:bd.

• In the continued proportion a:b:c:d., c is called the third proportion, and d is called the fourth
proportion.

Check: Ratio and Proportion

Ratio and Proportion Formula
Let’s discuss the formulas on ratio and proportion in detail.

Ratio Formulas
The ratio of any two quantities a:b is given by a/b where a/b must be in simplest form. In case, you are
not able to find the common factor between a and b then find the HCF or a and b and divide a and b
with HCF to reduce it to the simplest form.

There are more formulas that we will discuss below:

Compound Ratios
When two ratios are multiplied then the new ratio is called the compound ratio. For example a:b and c:d
are two ratios then ac:bd is a compound ratio.

Duplicate Ratios
For a:b, a2:b2 is called duplicate ratios

For a:b, √a:√b is called sub-duplicate ratios

For a:b, a3:b3 is called triplicate ratios

Proportion Formulas
These are the formulas used to solve problems of proportion:

• If a:b = c:d, then we can say that (a + c):(b + d), it is also called Addendo.

• If a:b = c:d, then we can say that (a – c):(b – d), it is also called Subtrahendo.

• If a:b = c:d, then we can say that (a – b):b = (c – d):d, it is also known as Dividendo.

• If a:b = c:d, then we can say that (a + b):b = (c + d):d, it is also known as Componendo.

• If a:b = c:d, then we can say that a:c= b:d, it is also known as Alternendo.

• If a:b = c:d, then we can say that b:a = d:c, it is also called Invertendo.

• If a:b = c:d, then we can say that (a + b):(a – b) = (c + d):(c – d), it is also known as Componendo
and Dividendo.

• If a is proportional to b, then it means a = kb where k is a constant.

• If a is inversely proportional to b, then a = k/b, where k is a constant.

• Dividing or multiplying a ratio by a certain number gives an equivalent ratio.

Mean Proportion
Consider two ratios a:b = b:c then as per the rule of proportion product of the mean term is equal to the
product of extremes, this means b2 = ac, hence b = √ac is called mean proportion.

Difference between Ratio and Proportion

The comparison between Ratio and Proportion is tabulated below:

RATIO PROPORTION
Ratio is used to compare two quantities of the Proportion is used to express the relation of two

Proportion is represented using (::), a:b = c:d ⇒

same unit ratios. ratios.
Ratio is represented using (:), a/b = a:b.
a:b::c:d.
Ratio is an expression. Proportion is an equation it equates two ratios.

Applications of ratios and proportion

Ratio

1. Business and Finance:

Ratios are fundamental in analyzing business performance. Financial ratios such as profit margin, debt-to-

equity, and current ratio help assess a company's profitability, solvency, and liquidity. Investors use these

ratios to evaluate financial health and make informed investment decisions, aiding in risk management and

strategic planning.

2. Cooking and Recipes:

Ratios are used in cooking to maintain consistency in recipes. For instance, a 2:1 ratio of flour to water

ensures the correct dough consistency. Adjusting ingredient ratios according to the number of servings allows

precise scaling of recipes, ensuring the dish tastes the same each time.
3. Engineering and Design :

In engineering, ratios like the gear ratio or aspect ratio are critical for machinery design. For example, the

gear ratio determines the mechanical advantage, while aspect ratios in design ensure optimal functionality

and aesthetics. Engineers rely on these ratios to balance performance, durability, and design efficiency.

4. Construction and Architecture:

In architecture, ratios are used to create proportionate and aesthetically pleasing designs. The golden ratio,

for example, is used to achieve harmony in structures. Additionally, ratios help in scaling blueprints and

calculating material requirements. Builders use them to ensure proper measurements and avoid errors in

construction.

5. Pharmacology:

Ratios play a vital role in pharmacology, particularly in drug formulation and dosage calculations. A common

application is the ratio of active ingredients to excipients in a medication. Correct ratios ensure the right

concentration for therapeutic effects, while adjusting doses according to body weight requires precise ratio

calculations for safety.

6. Sports and Performance Analysis:

In sports, ratios are used to analyze performance metrics like shooting accuracy, goals-to-attempts, or assists-

to-turnovers. Coaches use these ratios to assess player effectiveness and adjust strategies. For example, a

higher shooting-to-attempts ratio indicates good shooting performance, helping coaches optimize player

skills and team strategy.

7. Education and Grading Systems:

Educational institutions use ratios in grading systems, like the ratio of correct answers to total questions,

which determines a student's score. The student-to-teacher ratio is another critical metric, influencing the

quality of education and resource allocation. This helps in evaluating classroom efficiency and identifying

areas for improvement.

8. Economics and Market Analysis:

In economics, ratios like the price-to-earnings (P/E) ratio and GDP growth rates provide insights into market

health. The P/E ratio helps investors assess whether a stock is undervalued or overvalued. Economic ratios

help policymakers understand inflation, employment, and growth, driving decisions on monetary policy and

fiscal strategies.

9. Chemistry and Laboratory Work :

Chemists use ratios in chemical reactions to maintain correct proportions of reactants, ensuring desired

outcomes. For example, the stoichiometric ratio ensures complete reactions without excess waste. Ratios are

also crucial in dilutions, where precise amounts of solute and solvent are mixed to achieve the correct

concentration for experiments.

10. Agriculture and Farming:

In agriculture, ratios are crucial in crop planting, irrigation, and fertilization. For instance, the seed-to-soil

ratio determines optimal planting density. Fertilizer application often follows specific nutrient ratios, ensuring

plants receive the right nutrients for growth. Farmers use these ratios to maximize crop yield and minimize

resource wastage.
11. Population Studies:

Demographers use ratios in population studies, such as birth-to-death ratios or male-to-female ratios, to

analyze population dynamics. These ratios provide insights into demographic trends, helping governments

plan for future infrastructure needs, healthcare services, and educational facilities. They are essential in

understanding the growth and aging of populations.

12. Transportation and Logistics:

Ratios in transportation help determine fuel efficiency and load capacity. For example, the weight-to-volume

ratio of goods ensures optimal space usage in trucks or ships. Logistics managers use these ratios to plan

routes and manage inventory, improving delivery efficiency and reducing operational costs in supply chain

management.

13. Photography and Cinematography:

Ratios like the aspect ratio are essential in photography and cinematography. The aspect ratio (width to

height) of an image affects composition and visual appeal. In video production, understanding the correct

aspect ratio ensures that the content fits screens of different sizes without distortion, enhancing the viewer’s

experience.

14. Healthcare and Medical Research:

In healthcare, ratios are used in diagnostic tests and research. For example, the sensitivity-to-specificity ratio

of a test helps assess its reliability in detecting diseases. Additionally, ratios like the heart rate-to-oxygen level

ratio can indicate the patient's cardiovascular health. Researchers rely on ratios to interpret medical data

accurately.
15. Environmental Science:

Ratios are important in environmental science, particularly when studying pollution levels and resource

usage. The carbon-to-nitrogen ratio in soil is essential for determining soil fertility. Similarly, pollutant-to-

water ratio measurements help evaluate water quality. These ratios guide environmental policies and

conservation efforts aimed at sustainability and ecosystem balance.

Proportion

1. Cooking and Recipe Scaling:

Proportions are used in cooking to adjust ingredient quantities based on desired servings. For example, if a

recipe calls for 2 cups of flour for 4 servings, a proportion helps scale it for 10 servings. This ensures the

consistency and taste of the dish remain intact when adjusted.

2. Financial Planning :

Proportions are used in budgeting and financial planning to allocate resources effectively. For instance, the

proportion of income spent on necessities like housing, food, and transportation helps create a balanced

budget. Properly applied, proportions ensure individuals or businesses manage expenses relative to their

income or earnings.

3. Construction and Architecture :

In architecture, proportions are used to maintain aesthetic balance and structural integrity. The Golden Ratio,

for example, is often applied in the design of buildings to create pleasing proportions. Proportions also help

scale blueprints to real-life dimensions, ensuring accuracy when constructing buildings and structures.
4. Medicine and Drug Dosage:

Proportions are crucial in calculating drug dosages, ensuring that medications are given in the correct

amounts based on a patient’s weight or age. For instance, if a standard dosage is 10 mg per 20 kg of body

weight, proportions help determine the correct dosage for any patient, ensuring safety and efficacy.

5. Carpentry and Woodworking:

In carpentry, proportions are used to ensure that furniture or structures are built to the right scale and

dimensions. For example, when making a table, proportions ensure the correct height-to-width ratio for

comfort and aesthetic appeal. Proportions help create balanced designs and avoid structural imbalance in

wooden objects.

6. Sports and Training:

Proportions are applied in sports training to adjust exercise routines based on an athlete’s body type, age,

and skill level. For example, a coach may increase the intensity of an athlete’s workout in proportion to their

performance, helping them progress safely without risking injury or overtraining.

7. Photography and Image Scaling:

Proportions are used in photography when cropping or resizing images. If an image is enlarged or reduced,

maintaining the original proportions (aspect ratio) ensures that the image doesn’t become stretched or

distorted. This principle is key to maintaining visual quality across different mediums, like print and digital

displays.

8. Marketing and Advertising:

In marketing, proportions are used to determine the right balance of spending across different channels, such

as digital ads, TV, or print. Proportions help marketers optimize budgets to ensure that they allocate enough

resources to high-return strategies while avoiding overspending on underperforming channels.

9. Population Studies:

Proportions are used in demography to analyze population characteristics. For example, the proportion of

males to females in a specific region helps understand gender distribution. These proportions provide

valuable data for government planning in areas like healthcare, education, and workforce development,

ensuring appropriate allocation of resources.

10. Environmental Science and Sustainability:

Proportions are used in environmental science to study and manage natural resources. For instance, the ratio

of renewable to non-renewable energy sources in a country’s energy mix is a proportion. Maintaining

balanced proportions helps guide sustainability efforts and ensures that future generations have access to

essential resources.

11. Economics and Business Analytics:

In business analytics, proportions are used to calculate profitability, revenue shares, and market trends. For

example, a company may analyze the proportion of its revenue coming from various product lines. These

proportions help businesses make data-driven decisions to maximize profitability and allocate resources

more effectively.

12. Nutrition and Diet Planning:

In nutrition, proportions help design balanced meals and diets. For instance, the proportion of carbohydrates,

proteins, and fats in a meal is essential for a well-rounded diet. Nutritionists use proportions to create meal

plans that align with a person’s health goals, like weight loss, muscle gain, or managing chronic conditions.

13. Geography and Cartography:

In cartography, proportions are used to create scale maps. A map’s scale is a proportion that shows the

relationship between a distance on the map and the actual distance on the ground. This proportionality helps

travelers and planners make accurate decisions based on map readings, whether for navigation or planning.

14. Physics and Engineering:

Proportions are used in physics and engineering to understand relationships between variables. For example,

Ohm’s Law in electricity states that voltage is directly proportional to current and resistance. Engineers use

these proportions to design circuits, predict behaviors, and optimize devices by applying known ratios

between different physical properties.

15. Fashion Design:

Proportions play a vital role in fashion design, particularly when creating patterns and garments. Designers

use proportions to determine the appropriate measurements for clothing items based on body types,

ensuring that garments fit well and are comfortable. Proportions help maintain aesthetic balance and

functional design in clothing construction.

Frequently Asked Questions on Ratios and Proportion

Q1. What is the ratio? Give an example.
A ratio is a mathematical expression written in the form of a:b, where a and b are any integers. It
expresses a fraction. For example. 2:3 = ⅔.

Q2. What is a proportion? Give example

A proportion is a statement where two or more ratios are equivalent. For example, ⅔ = 4/6 = 6/9.

Q3. How to solve proportions with examples?

If a:b::c:d is a proportion, then;
a/b=c/d
ad=bc
Example: If ⅔=4/6, then,
2x6=3x4
12 = 12

Q4. What is the concept of ratios?

The concept of ratio defines us to compare two quantities while the proportion is an equation
that shows that two ratios are equivalent.

Q5. What are the two different types of proportions?

The two different types of proportions are:
Direct Proportion
Inverse Proportion

Q6. Can we express ratio in terms of fractions?

Yes, we can express ratio in terms of fractions. For example, 3: 4 can be expressed as 3/4.

Q7. What is the formula for ratio and proportion?

The formula for ratio is:
x:y ⇒ x/y, where x is the first term and y is the second term.
The formula for proportion is:
p: q :: r : s ⇒ p/q = r/s,
Where p and r are the first term in the first and second ratio
q and s are the second term and in the first and second ratio.

Q8. Find the means and extremes of the proportion 1: 2 :: 3: 4.

In the given proportion 1: 2 :: 3: 4,
Means are 2 and 3
Extremes are 1 and 4.