Wednesday, 29 May 2013

RATIO AND PROPORTION


RATIO
  Ratio means the number of times one quantity contains another quantity of the same kind. The comparison is made by considering what part or multiple the first quantity of the second.
       Thus the ratio between Rs 50 and Rs 200 can be possible, but not between Rs 50 and 200 marks.
       The ratio between one quantity to another is measured by  a : b or a/b

Ex: 8:9   or   5:7 etc.
       The two quantities in the ratio are called its terms. The first is called the antecedent and the second term is called consequent.
 
Types of Ratios:
 
1. Duplicate ratio: The ratio of the squares of the two numbers.
    
Ex: 9 : 16 is the duplicate ratio of 3 : 4.
 
2. Triplicate Ratio: The ratio of the cubes of the two numbers.
    
Ex: 27 : 64 is the triplicate ratio of 3 : 4.

3. Sub-duplicate Ratio: The ratio between the square roots of the two numbers.
      Ex: 4 : 5 is the sub-duplicate ratio of 16 : 25.
 

4. Sub-triplicate Ratio: The ratio between the cube roots of the two numbers.
      Ex: 4 : 5 is the sub-triplicate ratio of 64 : 125.
 

5.Inverse ratio: If the two terms in the ratio interchange their places, then the new ratio is inverse ratio of the first.
      Ex: 9 :5 is the inverse ratio of 5 : 9.
 

6. Compound ratio: The ratio of the product of the first terms to that of the second terms of two or more ratios.
      Ex: The compound ratio of 
3/4, 5/7, 4/5, 4/5 is 9/35
 




PROPORTION 
If two ratios are equal, then they make a proportion.
 Thus 
4/5 = 8/10 or 4:5 = 8:10
Each term of the ratios 4/5 and 8/10  is called proportional.
The middle terms 5 and 8 are called means and the end terms 4 and 10 are called extremes.


Product of Means = Product of Extremes

Continued Proportion: In the proportion  8/12= 12/8  8, 12, 18 are in the continued proportion.
 
Fourth proportion: If a : b = c : x, then x is called fourth proportion of a,b and  c.
There fore fourth proportion of  a, b, c  =
b x c/a
 
Third proportion: If a : b = b : x, then x is called third proportion of a and b.
Therefore third proportion of a, b =  b^2/a

 
Second or mean proportion: If a : x = x : b , then x is called second or mean proportion of a and b.
Therefore mean proportion of a and b =  root(ab)

EXAMPLES
Example 1: Find out the two quantities whose difference is 30 and the ratio between them is 5/11.
Sol: The difference of quantities, which are in the ratio 5:11, is 6. To make the difference 30, we should Multiply them by 5.
        Therefore 
5:11 = 5x5 : 11x5 = 25 : 55
 

Example 2: A factory employs skilled workers, unskilled workers and clerks in the ratio 8:5:1 and the wages of a skilled worker, an unskilled worker and a clerk are in the ratio 5:2:3 when 20 unskilled workers are employed the total daily wages fall amount to Rs. 318. Find out the daily wages paid to each category of employees.
Sol: Number of skilled worker: unskilled worker: clerks = 8:5:1 and the ratio of their respective Wages = 5:2:3
       Hence the amount will be paid in the ratio 8 × 5 : 5 × 2 : 3 × 1 = 40 : 10:3
          
       Hence total amount distributed among unskilled workers
                                   318  x 10 / (40+10 +3)
= Rs 60.

But the number of unskilled workers is 20, so the daily wages of unskilled worker
                                                     
60/20 = rs.30
The wages of a skilled worker, an unskilled worker and a clerk are in the ratio = 5:2:3
      Multiplying the ratio by 
(5/2) and (3/2)we get = 7.50 : 3 : 4.50
      So, if an unskilled worker gets Rs.3 a day then a skilled worker gets Rs. 7.50 per day a clerks Rs. 4.50 a day

 

Example 3: Two numbers are in the ratio of 11:13. If 12 be subtracted from each, the remainders are in the ratio of 7:9 Find out the numbers.
Sol: Since the numbers are in the ratio of 11:13. Let the numbers be 11x and 13x. Now if 2 is subtracted from each, the numbers become (11x -12) and (13x-12).  As they are in the ratio of 7:9              (11x-12): (13x-12):: 7: 9
                (11x – 12) 9 = (13x – 12) 7
                99x – 108 = 91x – 84
                9x = 24 or x = 3
  Therefore the numbers are 11 x 3 = 33 and 13 x 3 = 39



Example 4:  In what ratio the two kinds of tea must be mixed together one at Rs. 48 per kg. and another at Rs. 32 per kg. So that the mixture may cost Rs. 36 per kg. ?
SolRatio of inferior quality to superior quality=(rate of superior - rate of mix)/(rate of mix- rate of inferior)

= (48-36)/(36-32)
=12/4
=3:1

Example 5:  If Rs. 279 were distributed among Ram, Mohan and Sohan in the ratio of 15:10:6 respectively, then how many rupees did Mohan obtain?
Sol: Ratio in which Ram, Mohan and Sohan got  = 15 :10 : 6
                        Sum of ratios  = 15 + 10 + 6 = 31
                        Share of Mohan
= (10 x 279)/ 31
                                                         = Rs. 90
 

Example 6:  A bag contains of one rupee, 50 paise and 25 paise coins. If these coins are in the ratio of 2:3:10, and the total amount of coins is Rs288, find the number of 25 paise coins in the bag.
Sol:   Ratio of one rupee, 50 paise and 25 paise coins
                                                                 = 2:3:10
                        Ratio of their values  = 8:6:10 = 4:3:5
 And sum of the ratios of their values = 4 + 3 + 5 = 12
                     Value of 25 paise coins (5x288)/12
= Rs. 120
                      No. of 25 paise coins   = 120 × 4 = 480                


6 comments:

  1. Nice way of solving problems...Thanx a lot..

    ReplyDelete
  2. the answer to question 4 is not appropriate.. kindly provide correct solution..

    ReplyDelete
  3. solution to question 4:

    Ratio of inferior quality to superior quality=(rate of superior - rate of mix)/(rate of mix- rate of inferior)

    = (48-36)/(36-32)
    =12/4
    =3:1

    ReplyDelete
    Replies
    1. This comment has been removed by the author.

      Delete
  4. thank you samuel.. i didn't see question before and checked only the answer and left it.
    i just checked it now.. both question and answer.. :P

    ReplyDelete
  5. Answer to Question 1 is not 25:25 ( Therefore 5:11 = 5x5 : 11x5 = 25 : 25)
    Correct Answer is 25:55 ( Therefore 5:11 = 5x5 : 11x5 = 25 : 55)

    ReplyDelete