## The Unitary Method

‘Unitary Method’ is not a very familiar name to you till now. But this is the easiest as well as the most convenient method to solve our day-to-day problems. In this lesson, you will learn this complete new method named ‘Unitary Method’ or ‘Method of Reduction to the Unit’. To give you the proper idea of this process, we will cite you the following examples:

### < A > The Cost of any Number of Things (Unit Cost is Given)

You have already learnt that if the length, weight, value etc. of a particular material or thing is known to you, you can find out the length, weight, value etc. of any number of such materials or things by multiplying the given length, or weight, or value by the number of things.

#### Solution:

According to the problem,
The weight of 1 bag of sugar = 2 Kg. 11 grams
Hence, the weight of 6 bags of sugar = 2 Kg. 11 grams x 6= 12 Kg. 66 grams.
So, the weight of 6 such bags of sugar is 12 Kg. 66 grams.

In the above example, the weight of 1 bag of sugar is given, and you need to calculate the weight of such 6 bags. Hence, you multiply the weight of 1 bag of sugar, i.e.; 2 Kg. 11 grams by the number of the bags, i.e.; 6 to find out the total weight of those 6 bags of sugar.

### < B > Unit Cost (Cost of a number of Things is Given)

Similarly, we can say,

If the length, weight, value etc. of a number of things is given, the length, or weight, or value of one such thing can be found by dividing the value by that number of things.

#### Solution:

According to the problem,
5 books cost = \$125
Hence, 1 book costs = \$125 ÷ 5 = \$25
Therefore, the cost of 1 book is \$25.

In the above example, the cost of 5 books is given, and you need to find out the cost of 1 book. Hence, you divide the cost of 5 books, (i.e.; \$125) by the number of books (i.e.; 5) to find out the cost of 1 book.

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### < C > The Cost of any number of things (Unit Cost is Not Given)

You have already learnt about the processes of finding out the length, weight, value of one or more things, when length, weight, or value of more things or one thing is given respectively. However, the length, weight, or value of any number of things can be determined from the given lengths, or weights, or values of a given number of things.

At first, you need to calculate the length, or weight, or value of one thing by dividing the given length, or weight, or value by the given number of thing, as shown in the example 2. After that, the length, or weight, or value of the required number of things is determined by multiplying the length, or weight, or value of one thing (which is already found by the previous calculation) by the total number of things whose length, or weight, or value is to be found.

Check out the following examples for more clarification:

#### Solution:

As you can see in the problem,
75 bags can hold = 6150 Kg. Hamburger.
Then, 1 bag holds = (6150 Kg. ÷ 75) Hamburger.
And, 42 bags hold = [(6150 Kg. ÷ 75) x 42] Hamburger.
= (82 x 42) Kg. Hamburger.
= 3444 Kg. Hamburger.

Hence, 42 bags can hold 3444 Kg. Hamburgers.

#### Solution:

This problem can be solved just in the previous way.
Given, the cost of 46 pens = \$690
So, the cost of 1 pen = \$690 ÷ 46
Similarly, the cost of 85 pens = (\$690 ÷ 46) x 85
= \$15 x 85 = \$1275.

Hence, 85 pens will cost \$1275.

If you notice the last two examples carefully, you will see that while proceeding from a certain number of things to another, we have passed through the unit value of that thing. In the example 3, we have passed through the value of 1 bag while proceeding from 75 bags to 42 bags. In the same way, we have determined the cost of 1 pen at first while trying to find out the cost of 85 pens from that of the 46 pens, in the example 4.

Hence, we need to go through the length, or weight, or value of ‘one’ (unit) while changing from what is given to what is required. For this reason, this particular method is named as the ‘Unitary Method’ or the ‘Method of Reduction to the Unit’.

Moreover, it can be said from the above two examples,

The greater the number of things, the greater is the required value, and the less the number of things, the lesser is the value. In additions of this type of unitary method, we need to maintain another basic rule that says, as one of the factors increases, the other factor also increases accordingly.

Now, go through the following example:

#### Solution:

Generally, when we need to perform addition in this type of unitary method, the previous rule becomes reverse in nature.

In additions of this type of unitary method, we need to maintain the basic rule that says, as one of the factors increases, the other factor decreases accordingly.

According to the given problem,
15 men can do the work in = 12 days.
So, 1 man can do the work in = (12 x 15) days.
And similarly, 18 men can do the work in = [(12 x 15) ÷ 18] days.
= 180 ÷ 18 days = 10 days.

Hence, 18 men will do that work in 10 days.

Here, at first we need to find out the time 1 man will take to finish the work. From the above said basic rule, we can say that 1 man will take the time 15 times the number of days taken by those 15 men to execute the whole work. Therefore, 1 man takes = (12 x 15) days. And, if 1 man takes (12 x 15) days, 18 men will do it in lesser time. Hence, 18 man will take (1 / 18) of the time taken by 1 man, means they will take [(12 x 15) / 18] days to complete the whole work.

So, from the above example we come to the conclusion,

If the number of men is increased, they will take less time to execute the same work at the same rate. On the other hand, if the number of men is decreased, they will take more time to execute the same work at the same rate.
N.B.: You must carefully note that the conditions or factors of the problem are so stated as to have the quantity, in terms of which the answer is to be stated, always on the right hand side, or in other words, at the end of the line. Again, when solving the problems based on the unitary method, start by asking yourself whether you expect a greater or lesser result.

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