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# Unitary Method

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## Define Unitary Method

The unitary method is a fundamental concept of Mathematics and makes it convenient to solve various sums. This method generally involves finding the value of a unit in the given terms, using which the value of the given quantity of units can be calculated. The following example will help you understand the terms ‘unit’ and ‘value’. You buy 7 juicy apples from your local grocery shop. The shopkeeper puts up an offer of purchasing 10 apples for Rs 100. In this situation, the ‘units’ are the apples and the ‘value’ is the price of the apples.

It is important to recognize and familiarize the terms of units and values when using the unitary method in your sums.

A simple tip is to write the values to be calculated on the right-hand side and known values on the left-hand side. The problem present above clearly indicates the number and total price of apples as unknown. At this point, the use of ratio and proportions come into the picture.

### Concept Behind Unitary Method

The unitary method concept helps us to pick out a single value from a multiple set of values having different properties. For Instance, the price of 40 pens is Rs. 400, how do we calculate the price of a single pen?

This can be done using the unitary method. After finding the value of a single pen, we calculate the value of the required number of pens. This is done by multiplying the value of the single unit by the given number of units.

### Example of Unitary Method Problems

While going through a variety of sums to understand the unitary method, be sure to pay attention to the items in the given set of data. Given below are some of the common unitary method sums.

Let us consider the following example. A car runs 150 km on 15 liters of fuel, how many kilometers will it run on 10 liters of fuel?

First, we try to identify our quantities,

Here, the units are known however, the values are unknown.

Kilometer = Unknown (Place in the Right Hand Side)

No of liters of fuel = Known (place in the Left Hand Side)

Taking,

15 litres = 150 km

1 litre = $\frac{150}{15}$ = 10 km

10 litres = 10 х 10 = 100 km

Therefore, the car can run 100 kilometers on 10 liters of fuel.

### Unitary Method in Time and Work

The unitary method uses the properties of time and works in the relevant numerical problems.

Let us consider the following example. Desmond finishes his work in 15 days while Betty takes 10 days. Find the number of days it will take them to complete the same work together?

If Desmond takes 15 days to finish his work then,

Desmond’s 1 day of work = $\frac{1}{15}$

Betty’s 1 day of work = $\frac{1}{10}$

Now, the total work is done be Desmond and Betty in 1 day = $\frac{1}{15}$ + $\frac{1}{10}$

Taking the LCM (15, 10),

1 day’s work of Desmond and Betty = $\frac{2+3}{30}$

1 day’s work of (Desmond + Betty) = $\frac{1}{6}$

Thus, if Desmond and Betty work together, they can complete the work in 6 days.

### Ratio and Proportion in Unitary Method

The concepts of ratio-proportion and unitary method are inter-linked. The majority of the sums in 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.

To find the ratio of one quantity for another requires the use of the unitary method.

Consider the following example. The income of Ajay is Rs 12000 per month, and that of Bob is Rs 191520 per annum. If the monthly expenditure for each is Rs 9960 per month, express their savings in terms of ratios.

The savings of Ajay per month = Rs (12000 - 9960) = Rs 2040

In 12 months, Bob earns = Rs.191520

The Income of Bob per month = Rs $\frac{191520}{12}$ = Rs. 15960

The savings of Bob per month = Rs(15960 - 9960) = Rs 6000

Therefore, the ratio of savings of Ajay and Bob = 2040 : 6000 = 17 : 50

1. What are the Two Types of Unitary Methods?

Answer: The unitary method greatly relies on the concept of ratio and proportions. But, depending on the value and quantity of a unit to be primarily calculated, there are two types of variations found.

1. Direct Variation - There is an increase or decrease in one quantity and this reflects on the increase or decrease in the next quantity.

For example, the increase in the number of goods will hike up the price. The amount of work done by a group of men is relatively high than that of a single man.

1. Inverse Variation - The inverse of direct variation results in inverse variation. The quantity of two sets is inversely proportional that is, one increase results in the decrease of the other.

For instance, when speed increases, more distance is covered in less time.

2. What are the Applications of the Unitary Method?

Answer: The practical applications of the unitary method are vast. It is used for solving complex problems in Mathematics. This involves the sums of speed, distance, time, work, ratio, and proportions. To some extent, it aids in the realm of Finance as well. It can be used to calculate the cost of goods or to establish their pricing based on the local or global market trends.

It can be used to determine the profit and loss attained by a company. The most common problems solved by the unitary method uses the concept of distance covered and time taken by different transportation systems.