The unitary method is a way to solve problems by first finding the value of a single unit. Once we know the value of one unit, we can find the value of many units by multiplying it. This method helps us calculate any missing value in a problem. It is very useful in everyday life for solving questions related to cost, time, speed, distance, work, and quantity. We can also use the unitary method to compare two quantities and find their ratio. It makes problem-solving easier and faster. Let us study this topic in detail in this maths article.

What is Unitary Method?

Definition of Unitary Method : Unitary Method is a mathematical process to calculate the value of a single unit from the value of multiple units and the value of multiple unit rate from the value of a single unit.

• It is a numerical process that we use to calculate any value i.e., cost or speed or anything, of any desired quantity.
• The primary condition to use this method is that we should be provided with the value of certain quantities, and then we can use the Unitary Method to calculate the value of desired quantity.

Example: If we have been provided with data where it is said that 3 students can together complete a group project within 5hrs, and we need to calculate how many hours it is required to complete that project if the group has 5 students.

Here we have a value of a certain quantity i.e., 3 students need 5 hrs to complete, and we need to find the value for a desired quantity i.e., number of hours required by 5 students.

So we can use the Unitary Method here to get the desired value.

So, Number of hours taken by 1 student = 5/3 = 1.6667.

Therefore, Number of hours taken by 5 students = 1.6667×5 = 8.3335

Unitary Method Formula

The Unitary Method is a simple way to solve problems by first finding the value of one unit and then calculating the value of many units. It is used for problems related to cost, speed, distance, weight, time, and other real-life situations.

Steps and Formula

1. Find the value of 1 unit:
2. Value of 1 unit = Total value of given units ÷ Number of units
3. Find the value of multiple units:
4. Value of required units = Value of 1 unit × Number of required units

Types of Unitary Method

There are two types of variation that we see while using Unitary Method under coefficient of variation.

Direct Variation

• This is a direct relationship between the value of the given quantity and the value of the required quantity.
• In other words, if the value of one quantity increases, then the value of the other quantity also increases and vice versa. We can also say that one value is directly proportional to the other value.
• For example, with increase in speed of a car, the distance covered also increases and similarly with decrease with the speed of a car, the distance covered decreases.
• This shows Direct Variation between speed, time and distance.

Indirect Variation

• This is an indirect direct relationship between the value of the given quantity and the value of the required quantity.
• In other words, if the value of one quantity increases, then the value of the other quantity decreases and vice versa. We can also say that one value is inversely proportional to the other value.
• For example, with increase in speed of a car, the time taken to cover a distance decreases, whereas with decrease in speed of a car, the time taken to cover the same distance increases.
• This shows Indirect Variation between speed and time.

Steps to use Unitary Method

The steps to use the unitary method are mentioned below.

• Step 1: First we find the value of unit quantity by dividing the given value by the given quantity. Thus we obtain the value of unit quantity.
• Step 2: We then multiply the required quantity with the obtained value. Therefore we get the value of the required quantity.

Example:

We have been given the cost of buying 10 balls which is Rs 95, and we need to calculate the cost of 7 balls.

So according to the above mentioned step 1 we first find the value unit quantity i.e, cost of buying one ball = 95/10 = Rs 9.5

Now we can calculate the value of required quantity mentioned in step 2 by multiplying the cost of 1 ball with the obtained value, i.e., 9.5 x 7 = Rs 68.5

Thus we get the cost of 7 balls using unitary method which is Rs 68.5

Unitary Method in Ratio and Proportion

The Unitary Method is also used to find the ratio of one quantity with respect to another quantity. The concepts of ratio-proportion and unitary method are very much inter-related. The sums of ratio and proportion exercises are based on fractions. A fraction is represented as a:b. The terms a and b can be any two integers.

Example : The Income of Harish is Rs 20000 per month, and that of Shalini is Rs 191520 per annum. If the monthly expenditure of each of them is Rs 9000 per month, find the ratio of their savings.

Here,

• Savings of Harish per month = 20000-9000 = 11000.
• Income of Shalini per month = 191520/12 = 15960
• Savings of Shalini per month = 15960-9000 = 6960
• Therefore, ratio of saving of Harish with respect to savings to Shalini = 11000/6960 = 1.58046

Applications of Unitary Method

The unitary method is a simple and useful technique for solving everyday problems. It helps us find the value of one item or many items using basic multiplication and division. Here are some common applications explained in detail

1. Finding Cost of Items:

If you know the price of a single item, you can get the price of many items. Again, if you know the cost of many items, you can get the cost of an individual item. Example: If 1 notebook is Cost 3, then 15 notebooks are Cost 3 × 15 = 45.

2. Time, Speed, and Distance Problems:

The unitary method helps calculate how long it takes to travel a certain distance or the speed required to cover it. Example: If a car travels 60 km in 2 hours, the time to travel 90 km is (2 ÷ 60) × 90 = 3 hours.

3. Work and Labour Problems:

It can be used to find out how much work one person does in a day or how long a group will take to complete a task. Example: If A can complete a job in 10 days, A’s 1 day work = 1/10. If B can complete it in 5 days, B’s 1 day work = 1/5. Together, 1 day’s work = 1/10 + 1/5 = 3/10. Total days required = 10/3 ≈ 3.33 days.

4. Population and Quantity Problems:

The unitary method helps calculate quantities such as population growth, distribution of items, or production output. Example: If 4 boxes contain 48 chocolates, 1 box has 48 ÷ 4 = 12 chocolates.

5. Money and Finance Problems:

It is useful for calculating wages, interest, and other financial matters. Example: If a worker earns £120 in 8 days, his earnings for 1 day = 120 ÷ 8 = 15.

Unitary Method For Speed Distance Time

The unitary method for speed, distance, and time is a way to solve problems by first finding the time, distance, or speed for one unit. Once you know the value of one unit, you can easily calculate for any number of units. This method makes solving such problems simple and quick.

Example: A bike travels at a speed of 60 km/h and covers 180 km. How long will it take to cover 90 km?

Solution: First, find the time to cover 180 km:

Speed = Distance ÷ Time 60 = 180 ÷ T T = 3 hours

Using the unitary method: 180 km = 3 hours 1 km = 3 ÷ 180 hours 90 km = (3 ÷ 180) × 90 = 1.5 hours

Unitary Method For Time and Work

The unitary method for time and work helps find out how long it takes to complete a task by first calculating the work done in one day. Once the work done by one person or unit in a day is known, it is easy to find the time needed for any number of people or units. This method makes solving work-related problems simple.

Example: X can complete a task in 12 days, and Y can complete the same task in 8 days. How many days will it take if they work together?

Solution: X’s 1 day work = 1 ÷ 12 Y’s 1 day work = 1 ÷ 8

Total 1 day work = 1/12 + 1/8 = (2 + 3)/24 = 5/24

Time to complete work together = 24 ÷ 5 = 4.8 days

So, X and Y can complete the task in 4.8 days working together.

Properties of the Unitary Method

The Unitary Method is a simple technique used in mathematics to find the value of one unit first and then use it to calculate the value of multiple units. It is very helpful in solving problems related to cost, speed, distance, weight, and many real-life situations.

Properties of the Unitary Method

1. Finding One Unit First
   • In the unitary method, we always start by finding the value of 1 unit.
   • Example: If 5 pens cost Rs. 50, the cost of 1 pen = 50 ÷ 5 = Rs. 10.
2. Multiplying or Dividing for Multiple Units
   • After finding 1 unit, we can easily find the value of any number of units by multiplying or dividing.
   • Example: Cost of 7 pens = 10 × 7 = Rs. 70.
3. Works for Direct and Inverse Proportion
   • Direct proportion: If a quantity rises, the other also rises. More pens are more expensive.
   • Inverse proportion: As one amount goes up, the other goes down. Example: An increased number of workers finishes a job more quickly.
4. Simple and Easy to Use
   • The unitary method simplifies calculations and avoids complex formulas.
   • It is widely used in everyday problems like shopping, travel, and work.
5. Based on Basic Division and Multiplication
   • The method uses only division to find 1 unit and multiplication to find multiple units.

Important Notes on the Unitary Method

The unitary method is a simple way to solve problems involving quantities and their values. Here are the key points explained clearly:

1. Finding the Value of Many Quantities:

If you know the value of a single item, you can find the total value of many such items by multiplying the value of one item by the total number of items.Example: If 1 pen costs £5, then 10 pens will cost 5 × 10 = £50.

2. Finding the Value of One Quantity:

If you know the total value of many items, you can find the value of a single item by dividing the total value by the number of items.Example: If 12 pencils cost £24 in total, the cost of 1 pencil will be 24 ÷ 12 = £2.

This method helps solve a variety of problems in daily life, such as finding the price of products, calculating wages, or determining distances and time. By using simple multiplication and division, you can easily work out the value of one or many quantities.

Unitary Method Solved Examples

Problem 1: The cost of 2 notebooks is Rs. 90. Calculate the cost of 10 notebooks.

Solution:

We have the given quantity as 2 and the value of these 2 quantities is Rs. 90.

First we find the value of 1 quantity,

\( Cost\ of\ 1\ notebook=\frac{Cost\ of\ 2\ notebooks}{Given\ number\ of\ books}=\frac{90}{2}=Rs.\ 45 \)

Next we calculate the value of 10 notebooks,

\( Cost\ of\ 10\ notebooks=Cost\ of\ 1\ notebook\times Number\ of\ 10\ books\ =45\times10=Rs\ 450 \)

Thus we get the cost of 10 notebooks i.e., Rs. 450

Problem 2: Which of the following options is cost effective?(i) Bottle A costs Rs.55 for 2 Liters(ii) Bottle B costs Rs.70 for 3 Liters

Solution:

We can use the Unitary Method to choose the cost effective option. We can find the cost of 1 liter which will help us to identify the cost effective bottle.

(i) \( Cost\ of\ 1\ liter\ \ =\ \frac{55}{2}=Rs.\ 27.5 \)

(ii) \( Cost\ of\ 1\ liter\ \ \ =\ \frac{70}{3}=Rs.\ 23.3 \)

As the cost of 1 liter from bottle B is less than the cost of 1 liter from bottle A.

Thus bottle B is more cost effective.

Question 3: 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?

Solution: Length of shadow for 140 cm = 210 cm Length of shadow for 1 cm = 210 ÷ 140 = 1.5 cm Length of shadow for 175 cm = 1.5 × 175 = 262.5 cm

Answer: The shadow of the 175 cm tall tree is 262.5 cm.

Question 4: A steel rod of uniform thickness and length 6 m weighs 3 kg. What will be the weight of 4 rods of the same thickness, each 9 m long?

Solution: Weight of 6 m rod = 3 kg Weight of 1 m rod = 3 ÷ 6 = 0.5 kg Weight of 9 m rod = 0.5 × 9 = 4.5 kg Weight of 4 such rods = 4 × 4.5 = 18 kg

Answer: The total weight of 4 rods is 18 kg.

Question 5: A rubber band stretches 1.8 m when a 60 kg weight is suspended. How much will it stretch when a 45 kg weight is suspended?

Solution: Extension for 60 kg weight = 1.8 m Extension for 1 kg weight = 1.8 ÷ 60 = 0.03 m Extension for 45 kg weight = 0.03 × 45 = 1.35 m

Answer: The rubber band will stretch 1.35 m under 45 kg weight.

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