Written By Sushma_P
Last Modified 14-03-2024

Unitary Method: Formula and Solved Examples

The unitary method, in simple terms, is used to calculate the value of a single unit from a specified multiple. If the cost of 100 pens is Rs. 400, how do you calculate the cost of one pen? The unitary approach can be used to achieve arrive at the solution. Moreover, once the value of a single unit has been determined, the value of the needed units may be calculated by multiplying the single value unit. This approach is mostly used to calculate ratios and proportions.

By using the unitary method, we can find the missing value. For example, if 1 packet of milk costs ₹5, then what would be the cost of 3 such packets? We can easily calculate that, the cost of 3 packets is ₹15. Let us understand the concept in detail in this article.

What is Unitary Method?

The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value. In essence, this method is used to find the value of a unit from the value of a multiple. Unitary method in Hindi is known as ऐकिक नियम.

Ratio Proportion and Unitary Method

We use the unitary method to find the ratio of one quantity with respect to another quantity. To understand the concept of the unitary method formula in ratio and proportion, let us take an example.

Example: Akash’s income is $₹20000$ per month, and that of Arjun is $₹360000$ per annum. If the monthly expenditure of each of them is $₹10000$ per month, find the ratio of their savings.

Solution: Savings of Akash per month $ = ₹(20000 – 10000) = ₹10000$
Earning of Arjun in $12$ months $= ₹360000$
Income of Arjun per month $=\frac{360000}{12} = ₹ 30000$
Savings of Arjun per month $= ₹(30000-10000) = ₹ 20000$

Therefore, the ratio of savings of Akash and Arjun $= 10000 : 20000=1: 2$

Types of Unitary Method

In the unitary method, the value of a unit quantity is found first to calculate the value of the different number of units. It has two types of variations.

Direct Variation
Inverse Variation

Direct Variation

In the direct variation, an increase in one quantity will cause an increase in another quantity; similarly, a decrease in one quantity will cause a decrease in another quantity. For instance, if the number of goods increases, their cost also increases.

Also, the amount of work done by one person will be less than the amount of work done by a group of people. Hence, if we increase the number of people, the work done will also increase.

Indirect Variation

It is the inverse of direct variation. Similar to inversely proportional. If the value of one quantity increases, then the value of another quantity decreases. For example, if we increase the speed, then we can cover the distance in less time. So, with an increase in speed, the time will decrease.

Unitary Method for Time and Work

Let’s take an example to understand the unitary method formula for time and work.

$“A”$ finishes his job in $15$ days, while $“B”$ takes $10$ days. In how many days will the same job be done if they work together?
If $A$ takes $15$ days to finish his job then,
$A’$s $1$ day of job $= \frac{1}{{15}}$

Similarly, $B’$s $1$ day of job $ = \frac{1}{{10}}$
Now, job is done by $A$ and $B$ in $1$ day $ = \frac{1}{{15}} + \frac{1}{{10}}$
Taking LCM $(15, 10)$, we have,
$1$ day’s job of $A$ and $B = \frac{{(2 + 3)}}{{30}}$
$1$ day’s job of $(A + B) = \frac{1}{6}$

Thus, $A$ and $B$ can finish the job in $6$ days if they work together.

Unitary Method Questions

$10$ workers finish a job in $20$ hours. How many workers are required to finish the same work in $15$ hours?
If the annual rent of a house is $₹294000$, calculate the rent of $8$ months.
If $85$ pages weigh $17\,{\rm{g}},$ calculate the weight of $180$ pages.
If $5$ buses can carry $300$ people, find out the total number of people which $8$ buses can carry.
Rakesh completes $\frac{3}{4}$ of a job in $8$ days. How many extra days will he take to finish the job at his current rate?

Worksheet on Word Problems on Unitary Method

Word problems on the unitary method formula with the combination of questions on direct and indirect variation.

1. $6$ farmers harvest the crops in the field in $10$ hours. How many workers are required to do the same amount of work in $18$ hours?

2. $3$ men or $2$ women can earn $$192$ in a day. Find the earning of $7$ men and $5$ women in a day?

3. The weight of $56$ pages is $8\,{\rm{g}}.$ What is the weight of $150$ such pages? How many such pages weigh $5\,{\rm{g?}}$

4. Meera types $400$ words in $30$ minutes. How many words would she type in $7$ minutes?

5. A man is paid $₹750$ for $6$ day’s work. If he works for $28$ days, how much will he get?

6. A tank of water can be filled in $7$ hours by $5$ equal-sized pumps working together. How much time will $8$ pumps take to fill it up?

7. $15$ workers can build the wall in $20$ days. How many workers will build the wall in $12$ days?

8. $76$ men can complete the work in $42$ days. In how many days will $56$ men do the same work?

9. In a camp, there are provisions for $400$ persons for $23$ days. If $60$ more persons join the camp, find the number of days the provision will last?

10. If $10$ workers working for $4$ hours complete the work in $12$ days, in how many days will $8$ workers working for $6$ hours complete the same work?

11. The freight for $75$ quintals of goods is $₹375$. Find the freight for $42$ quintals.

12. A truck travels ${\rm{150\,km}}$ in $3$ hours.

(a) How long will it take to travel $912\,{\rm{km?}}$

(b) How far will it travel in $10$ hours?

Real-Life Applications of Unitary Method

The unitary method is very much helpful in solving various problems that we come across in our day-to-day life. Some of the real-life applications of the unitary method are:

To find the speed of a vehicle for a given distance, if the speed and distance are given in different quantities.
To find the number of men required to complete a given amount of work.
To find the area of a square of a given length if the ratio of its area and side is given.
To find the cost of a certain number of objects, if the cost and number of objects are given in different quantities.
To find the percentage of a quantity.

Solved Examples – Unitary Method Formula

(Includes Unitary Method Problems and Unitary Method Questions for Competitive Exams)

Q.1. Rupali can type $540$ words in $30$ minutes. How many words will she able to type in $20$ minutes with the same efficiency?
Ans: Number of words typed in $30\,\min = 540.$
The number of words typed in $1\;{\rm{min}} = \frac{{540}}{{30}} = 18$
Therefore, the number of words typed in $20\,\min = 20 \times 18 = 360.$
Hence, Rupali will be able to type $360$ words in $20$ minutes.

Q.2. If $45$ students can consume a stock of food in $2$ months, then for how many days the same stock of food will last for $27$ students?
Ans: The food will last for $100$ days. Let us see how. According to the first condition, students consume food in $2$ months, i.e., in $60$ days. So this ratio would be $45 : 60$
According to the second condition, the same amount of food is consumed by $27$ students in $x$ days. So that ratio would be $27: x $. By Inverse Proportion,
$45×60=27×x$
$ \Rightarrow x = \frac{{\left( {45 \times 60} \right)}}{{27}}$
$ \Rightarrow x = 100$

Q.3. The cost of $8$ apples is $₹ \, 120$. Find the number of apples that can be purchased with $₹ \, 240$.
Ans: Given, the cost of $8$ apples $=₹ \, 120$. Hence the cost of $1$ apple $=₹\, 15.$
Now, the number of apples that can be purchased in $₹\, 15=1.$
The number of apples that can be purchased with $₹1 = \frac{1}{{15}}$.
Hence, the number of apples that can be purchased with $₹240 = \frac{1}{{15}} \times 240 = 16$ apples.

Q.4. Rohan goes to a stationery shop to buy some books. The shopkeeper informs him that $2$ books would cost $$90$. Can you find the cost of $5$ books with the help of the unitary method?
Ans: The number of books corresponds to the “unit”, and the cost of the books corresponds to the “value”. Let’s solve it step-wise.
Step 1: First, we will find the cost of $1$ book.
Cost of $1$ book $= \frac{{{\rm{ Total \, cost \, of \, books }}}}{{{\rm{ Total \, number \, of \, books }}}} = \frac{{90}}{2} = 45.$
Step 2: Now, we will find the cost of $5$ books.
Cost of $5$ books $=$ Cost of $1$ book $\times $ Number of books $=45×5=225.$
Therefore, the cost of $5$ books is $$225$.

Q.5. A car travelling at a speed of ${\rm{140\,kmph}}$ covers ${\rm{420\;km}}$. How much time will it take to cover ${\rm{280\;km}}$?
Ans: First, we need to find the time required to cover ${\rm{420\;km}}.$
${\rm{Speed = }}\frac{{{\rm{ Distance }}}}{{{\rm{ Time }}}}$
$ \Rightarrow 140 = \frac{{420}}{T}$
$ \Rightarrow T = 3$ hours
Applying the unitary method, we get,
To travel ${\rm{420\;km}},$ the time required is $3$ hours
So, to travel $1\,{\rm{km}},$ the time required is $\frac{3}{{420}}$ hours
Hence, to travel ${\rm{280\;km}}$, the time required is $\frac{3}{{420}} \times 280 = 2$ hours

Summary

Unitary method in maths is a method by which we can find the value of one unit from the value of many units and the value of many units from the value of one unit. It is a method that we use for the majority of the calculations in Mathematics. You will find this method helpful while solving problems on ratio and proportion, geometry, algebra etc.

Here, in this article, we learnt the definition of the unitary method with an example. We understood the technique involved in solving the problems related to the unitary method of converting the value of many to a single unit and the value of a single unit to the value of many units. We came through the types of unitary methods and understood direct and inverse variation.

We also studied the relation between ratio proportion and the unitary method. This article provides a worksheet related to the unitary method that helps understand different examples applicable in daily life. Solved examples mentioned in this article will be helpful in taking competitive examinations.

FAQs on Unitary Method

Following are the frequently asked questions on unitary method in maths:

Q.1. What is the formula of the unitary method? Or what is unitary law?
Ans: The formula of the unitary method is to find the value of a single unit and then multiply the value of a single unit by the number of units to get the necessary value.

Q.2. How do you solve a unitary problem?
Ans: The unitary method is a method for solving a problem by first finding the value of one unit and then finding the unknown value by multiplying the one unit value.

Q.3. What are the applications of the unitary method?
Ans: Practical applications of the unitary method are many. It is applicable in problems related to distance, time, work, speed, ratio and proportion. It can be used to calculate the cost of goods or to establish their pricing based on local and global market trends.

Q.4. What are the two types of unitary methods?
Ans: The unitary method is mainly dependent on ratio and proportion. But depending on the value and quantity of a unit to be primarily calculated, there are two types of variations:
1. Direct variation
2. Inverse variation

Q.5. How to teach the unitary method to students?
Ans: Let us take an example in which we need to find the price of $50$ shirts if the price of $8$ shirts is given to us as $₹800$. In this case, we will first find the price of $1$ shirt and multiply it by $50$.
Therefore, price of $1$ shirt $ = \frac{{800}}{8} = ₹100$
Price of $50$ shirts $=50×100=₹ 5000.$
Hence, we got the price of $50$ shirts using the unitary method.

We hope this article on the unitary method has provided significant value to your knowledge. Embibe wishes you the best of luck!