Unitary method: the power of one


Unitary Method is one of the basic topic in mathematics. I have chosen this topic because it has a lot of applications. It is used in many areas where the change in one quantity is linearly dependent on other quantity. In unitary method we use the change due to unit quantity. Some of the applications of unitary method are
  1. To find distance

    When we know the distance traveled in unit time then we can find the distance traveled in different time interval.

  2. To find cost

    When we know the cost of one unit then we can find the cost of different units.

  3. To find materials required

    When we know the amount of paint required to paint on 1 unit surface then we can find the amount of paint required on different amount of surfaces.

Similarly, there are many uses of unitary method. Now, let us look at unitary method in detail. The description can be understood by anyone who has studied slope intercept form of equation of a line. The equation of a line in slope intercept form is
y = mx + c. The value of many quantities increases with respect of other quantity and if it increases linearly then unitary method can be used. The value of most of the quantities is zero when the other quantity is zero. It sets the c i.e. the y-intercept to be zero. The equation of the line becomes as y = mx. Now y is the value of one quantity and x is the value of the other quantity. x is the independent quantity and y is the dependent quantity. We can find m if we are given the value of y and the corresponding value of x. After it we can use the relation y = mx to find value of y at different x.

Now let us take an example of amount of paint required to paint a wall.

To paint a wall we must consider the surface area of the wall and the volume of paint required to paint. As in unitary method we need to know the amount of paint required for unit surface area. Let the length of wall be l units and breadth of the wall be b units. Now Area of the wall is l×b unit2. And the volume of paint required is l×b×t unit3. Where t is the thickness of paint. To paint a unit surface area the amount of paint required is t unit3 (l×b×t/l×b). Now to paint a different amount of surface area with length l1 and breadth b1, we need l1×b1×t. Here l1×b1 is the surface area and t is the rate. Here we see that the amount of paint required is linearly dependent. Hence the unitary method can be used.

Q:   To paint 30 square units of area we need 60 cubic units of paint. Find how much paint is required to paint 90 square units  of surface area.

Solution:
30 sq. units of area can be painted with 60 cu. units of paint.
1 sq. units of area can be painted with (60/30) cu. units of paint.
90 sq. units of area can be painted with (60/30)×90 cu. units of paint.
i.e. 180 cu. units of paint.

Main Points for unitary method

  • Find the value of unit quantity to find the value of any required quantity.

  • Speed (S) = Distance (D)/ Time (T),
    Speed(S) is distance traveled in unit time.

  • Distance (D) = Speed (S) × Time (T)
    Distance is speed times the given time.

  • Time (T) = Distance (D)/ Speed (S),
    Time(T) is distance traveled at unit speed in unit time.

  • Average Speed (AS) = Total Distance (TD)/ Total Time (TT),
    Average Speed (AS) is distance traveled in unit time.

  • Total Distance (TD) = Average Speed (AS) × Time (TT)
    Total Distance is average speed times the given time.

Unitary Method and Rate

The value of unit quantity can be called as "rate for the whole quantity as per one unit".


Unitary method is based on the fact that the value of required quantity depends on the value of one quantity. So, in unitary method we find the value of one quantity to find the value of required quantity.

Example


Q 1: A person can fit 10 books in 2 box. How much boxes he need to fit 100 books?

Solution:
10 books need 2 boxes.
1 book will need 2/10 boxes.
So, 100 books need (2/10)×100 = 20 boxes.

Q 2: A milkman takes 10 vessels for 500 liters of milk. How much will he take for 100 liters of milk?

Solution:
500 liters fills 10 vessels.
1 liter will fill 10/500 vessels.
So, 100 liters will fill (10/500)×100 = 2 vessels.




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