Monday, 4 April 2016

Aptitude Questions with Answers - Ratio and Proportion Concept

        Ratio and Proportion Concept

Ratio : The comparison between two quantities in terms of magnitude is called ratio i.e, it tells that the one quantity is how many times the other quantity.
For example, Sunil has 750 rupees and Ramesh has 1400 rupees. It means that ratio of the money Sunil and Ramesh 750 is to 1400. It can be expressed as 750: 1400
Important note : In a ratio , the order of terms is very important . For example, in the above example, the required ratio is 750: 1400 while 1400:750 is wrong .
 So ratio of any two quantities is expressed as a :b or
a / b
. The two quantities that are being compared are called terms.
In a:b , the first term a is called antecedent and second term b is called  consequent .

Properties  of Ratio :
    1.    A ratio is a number , so to find the ratio of two quantities , they must be expressed in the same units (weight , length , volume or currency etc)
Example: We cannot compare 8 kg of rice and 3 toys. 
2.    Ratio has no units
3.    A ratio does not change if all of its terms are multiplied or divided by the same number, thus    2 : 3  = 4 : 6 = 6 :9
4.    The ratio of two fractions can be expressed in ratio of integers.
     Example:  (
2 / 3
): (
5 / 3
) =
2 / 3
   x 
3 / 5
  = 2 : 5
5.    The terms of the ratio are always expressed is smaller values.

Types of Ratios :
1.    Duplicate Ratio: The ratio of the squares of two numbers is called the duplicate ratio of the two numbers or When the ratio is compounded with itself, it is called as duplicate ratio.
Duplicated Ratio of a : b => a2 : b2
The duplicate ratio of 3 : 4  is  32 : 42  = 9 :16

2.    Sub Duplicate Ratio: The ratio of the square roots of two numbers is called the sub-duplicate ratio of two numbers .
Duplicate Ratio of a : b => $ \sqrt{a}$ : $\sqrt{b}$
Duplicate Ratio of 16: 49 => 4 :7

3.    Triplicate Ratios : The ratio of the cubes of two numbers is called the triplicate ratio of the two numbers .
Triplicate Ratio of a : b => a3 : b3
The Triplicate ratio of 3 : 4  is  33 : 43  = 27 :64

4.    Sub Triplicate Ratio : The ratio of the cube roots of two numbers is called the sub-duplicate ratio of two numbers .
Sub - Triplicate Ratio of a : b =>Cuberoot
Sub Triplicate Ratio of 64: 125 => 4 :5

5.    Inverse Ratio or Reciprocal Ratio:  If first term (antecedent) and second term      (consequent) of a ratio interchange their places , the new ratio is called the inverse ratio of the given ratio .
If a :b be the given ratio , then
1 / a
:
1 / b
or b:a is its inverse ratio
Example: If 3 : 5 is the ratio, then its inverse ratio => 5 : 3

6.    Compound Ratio: The ratio of the product of the first terms ( antecedent) to the product of the second terms (consequent) of two or more ratios is called the compound ratio.
Thus a:b , c:d  and e:f are three given ratios , then a x c x e : b x d x f is the compound ratio of the given ratios.
Example: The compound ratio of 2 : 3,  4 : 5 and 6:1 is
ð   2 x 4 x 6 : 3 x 5 x 1
ð  48 : 15
ð  16 : 5
Proportion:
The equality of two ratios is called a proportion and we say that the four numbers are in proportion.
If
a / b
=
c / d
, then a,b,c and d are said to be in proportion  and we write  a:b:: c:d . This is read as a is to b as c is to d
Here all terms a,b, c and d are called proportional’s . a,b,c and d respectively called  first , second (mean), third and fourth proportional.
Here a,d are known as extremes and b and c called means.

For example
3 / 5
=
6 / 10
, we write 3 :5 :: 6 : 10 and say  3,5 , 6 and 10 are in proportion .

Properties of Proportion :
1)    If four numbers are in proportion then product of the extremes is equal to the product of the means
If these are not in proportion, then product of extremes is not equal to the product of means.
If a:b::c:d   => a x d = b x c
2)    Continued Proportion : If a,b and c are three numbers such that a:b = b:c , then these numbers a,b and c are said to be in continued proportion
3)    Fourth Proportion : The fourth proportion of a, b and c is $ \frac{bc}{a}$
Example: What is the fourth proportion of 6,12  and 8
                  Here a =6 , b = 12 and c=8
                   Fourth proportion = $\frac{bc}{a}$=
12 x 8 / 6
=16
4)    Third Proportion : The third proportion of a and b is      
b2 / a
Example: What is the third proportion of 12 and 6
                Here a= 12 and b =6
        Third proportion =
b2 / a
   =
62 / 12
= 3                 
5)    Mean Proportion : The mean proportion of a and b is  a x b  
Example: Find the mean proportion of 81,144
                         A= 81 and b=144
      Mean Proportion =  81 x 144 

Variation:
Two quantities a and b are said to be varying with each other if there exists some relationship between a and b.

Direct Proportion:
Two quantities are said to be directly proportional, if the increase (or decrease) in one quantity causes increase (or decrease) in the other quantity by same the proportion.

In other words, the ratio of a and b is a constant. The statement b varies as a is written as b ∞ a .
Examples:
             I.        The price of articles varies directly to the number of articles. More articles more cost and less articles less cost
           II.        The word done varies directly to the number of men at work. Fewer men at work, less work is done in same time. More men at work, more done in same time.
Inverse Proportion:
Two quantities a and b are said to be vary inversely if the increase (or decrease) in one quantity causes decrease (or increase) in the other quantity by the same proportion.
The statement b varies inversely as a is symbolically written as b ∞
1 / a
Example
        I.        The price of an article is varies inversely to the demand of the article. When the price of the articles goes up , the demand for the article comes down
      II.        The time taken to finish a work varies inversely to the number of men at work
More men at work, less time taken to finish the work

Fewer men at work, more time taken to finish the work.