Class 7 - Unitary Method

The unitary method is one of the most widely used concepts in mathematics, especially for solving everyday problems involving ratios, proportions, and comparisons. It is based on a simple idea: first find the value of a single unit, and then use it to determine the value of multiple units. The unitary method provides a clear and logical approach for calculating the cost of items, wages, time, distance problems, etc. In this guide, you will learn the concept, steps, and examples of the unitary method in a simple and easy-to-understand way.

Table of Contents

• What is Unitary Method?
• Formula of the Unitary Method
• Types of Unitary Method
• Unitary Method in Ratio and Proportion
• Real-Life Applications of the Unitary Method
• Solved Examples on Unitary Method
• Practice Questions on Unitary Method

What is Unitary Method?

The unitary method is a method in which we first determine the value of one unit from the value of several units and then use it to find the value of any required number of units. This method is a mathematical technique used to solve problems involving ratios and proportions.

The unitary method is a mathematical technique where we:

1. First, find the value of one unit

2. Then use it to find the value of multiple units

For example: If 5 apples cost ₹50, then each apple has equal value. So, the cost depends directly on quantity. This idea forms the basis of direct proportionality, which is central to the unitary method.

Formula of the Unitary Method

Value of 1 unit =  Number of unitsTotal value​

Required value=Value of 1 unit × Required units

Let’s break it down logically:

If: A units = B value
Then: 1 unit = BA
And: x units=  BA×xunits

Types of Unitary Method

1. Direct Variation (Direct Proportion)

In this case, when one quantity increases, the other also increases.
For example,

• More items → More cost

• More distance → More time (at constant speed)

Example 1: If 6 notebooks cost ₹72, what is the cost of 15 notebooks?
Solution: Cost of 1 notebook = 72 ÷ 6 = ₹12
Cost of 15 notebooks = 12 × 15 = ₹180

2. Inverse Variation (Inverse Proportion)

In this case, when one quantity increases, the other decreases.
For example,

• More workers → Less time

• Higher speed → Less time

Example 2: If 8 workers complete a task in 10 days, how many days will 4 workers take?
Solution: Total work = 8 × 10 = 80 worker days.
Days for 4 workers = 80 ÷ 4 = 20 days. 

Unitary Method in Ratio and Proportion

Instead of directly solving proportions, we first find the value of one unit, and then calculate the required value.

Let us take a look at an example. If 5 books cost ₹75, what will be the cost of 8 books?

Here, the proportion is 575=8x

Using the unitary method: The cost of 1 book = 75 ÷ 5 = ₹15
Cost of 8 books = 15 × 8 = ₹120 

So, with the help of the unitary method, we can also find the missing value in the given proportion of two quantities. 

Real-Life Applications of the Unitary Method

The unitary method is widely used in everyday situations to find the value of one unit and then calculate the value of multiple units using simple multiplication or division. Some of the key applications are:

• Finding Cost of Items: Helps calculate the cost of one item or many items.
  Example: If 10 items cost ₹50, ⇒ 1 item = ₹5 ⇒ 8 items = ₹40

• Time, Speed, and Distance: Used to find time taken, distance covered, or speed.
  Example: If 60 km takes 2 hours → 90 km takes 3 hours

• Work and Labour Problems: Helps determine work done per day and total time required.
  Example: If a job takes 10 days, ⇒ 1 day's work = 1/10

• Population and Quantity Distribution: Used to divide or distribute quantities evenly.
  Example: 48 chocolates in 4 boxes  ⇒1 box = 12 chocolates

• Money and Finance Calculations: Useful for wages, earnings, and financial planning.
  Example: ₹120 in 8 days  ⇒ ₹15 per day

• Finding Speed from Given Data: Helps calculate speed when distance and time are given in different forms

• Area and Measurement Problems: Used to find dimensions like area or side length based on given ratios

• Percentage Calculations: Helps compute percentages easily by first finding 1%
  Example: 1% of 150 = 1.5 ⇒ 20% of 150 = 1.5 × 20 = 30
  
  

Solved Examples on Unitary Method

Example 1: A car travels 180 km in 3 hours. How much distance will it travel in 5 hours?
Solution: Distance in 3 hours = 180 km
Distance in 1 hour = 180 ÷ 3 = 60 km
Distance in 5 hours = 60 × 5 = 300 km
The distance the car will travel in 5 hours is 300 km

Example 2: If 6 machines can produce 300 units in a day, how many units can 1 machine produce in a day?
Solution: Units produced by 6 machines = 300
Units produced by 1 machine = 300 ÷ 6 = 50
In 1 day, the machine can produce 50 units.

Example 3: The cost of 24 cupcakes is ₹480. Find the cost of 36 such cupcakes.
Solution: Cost of 24 cupcakes = ₹480
The cost of 1 cupcake = 480 ÷ 24 = ₹20
Cost of 36 cupcakes = 20 × 36 = ₹720
The cost of 36 cupcakes is ₹720.

Example 4: A bike consumes 32 litres of petrol to cover 2304 km. How much petrol is needed for 6480 km?
Solution: Petrol for 2304 km = 32 litres
Petrol for 1 km = 32 ÷ 2304 = 1/72 litre
Petrol for 6480 km = 6480 × (1/72) = 6480 ÷ 72 = 90 litres
90 litres of petrol is needed for 6480 km

Example 5: If 8 workers can complete a piece of work in 12 days, how many days will 6 workers take to complete the same work?
Solution: Work done by 8 workers in 12 days  = 8 × 12 = 96 worker-days
Number of days required by 6 workers = Total work ÷ Number of workers
= 96 ÷ 6 = 16 days.
6 workers take 16 days to complete the same work

Practice Questions on Unitary Method

1. A man earns ₹4900 by working for one week. How much will he earn at the same rate if he works for 19 days?

2. A restaurant used 35 kg of rice in 7 days. How much rice will be needed for 45 days?

3. A factory manufactures 2410 bottles in 5 days. How many bottles will be manufactured in 7 days?

4. The length of the shadow of a 140 cm tall tree at a particular time of day is 210 cm. What will be the length of the shadow of a 175 cm tall tree at the same time?

5. 80 men can finish a piece of work in 52 days. In how many days can 64 men finish it?