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July 24, 2013

Ratio and Proportion Shortcuts - Introduction

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When two things are of the same kind, we often compare them with one another. In initial days I used to compare the number of facebook likes of my blog with my opponents' blogs. My sister compares her height, my uncle compares his salary and my brother compares his backlog subjects with others. Often this comparison is expressed in phrases like "is greater than", "is less than", "is multiple of" etc.
Assume that in a cricket match, Ramesh scored 17 runs, while Suresh amassed 51 runs in an inning of cricket. Then you can say,
  1. Suresh scored 34 runs more than Ramesh or Ramesh scored 34 runs less than Suresh.
  2. Suresh scored three times as many runs as Ramesh or Ramesh scored only one third of the runs made by Suresh. 
We don't have any issues with the option 1. But in the option 2, we are just finding the ratio between the two given numbers.


In simple words, we are just trying to find out the ratio between the two given quantities (assume them X and Y).

Here the condition is, the value of Y should be always greater than zero (Y>0).

Ratio : A "ratio" is just a comparison between two different things.  To find the ratio of the first number to the second, we find "What multiple of the second number is the first number?". This is done by dividing the first number by the second one. 


For example, the ratio between 20 and 32 is

=  20 / 32  = 5/8

The ratio of 60/80 = 6/8

Here you can write the phrase "the ratio of X to Y" as  "X : Y". You should read it as "X is to Y"

While finding out the ratio of the given numbers, you should keep the following points in your mind :
  1. The quantities should be of same kind. You cant find out the ratio between 2 kg Iron and 1 liter Milk. 
  2. These quantities in the ratio are called Terms
    • The first term in ratio is called Antecedent and 
    • the second term is called Consequent
  3. We represent ratios in numbers. There are no units for this.
  4. If somebody says, the ratio of the two numbers is X : Y, then that doesn't means the first number should be X and the second number should be Y. The numbers may be CX : CY (here C is a non zero constant which is a multiple of X and Y).
  5. The ratio is always same. If you divide / multiple the numerator and denominator with the same number then the value should be same. 
  6. The ratio will change if you add or subtract a number from both numerator and denominator. 
  7. Percentage is a special kind of ratio. You can treat it as the ratio which is having its second term as 100.
  8. Two ratios a : b and c : d  (or a/b and c/d) are said to be equal  if a x d = b x c.
  9. If two ratios are X : Y and Y : Z, then you can simply write them as X : Y : Z. 
From the above description, we can derive some types of ratios. Those are :
  1. Duplicate ratio : The ratio of the squares of the two numbers.
    • 16 : 25 is the duplicate ratio of 4 : 5.
  2. Triplicate Ratio : The ratio of the cubes of the two numbers.
    • 64 : 125 is the triplicate ratio of 4 : 5
  3. Sub-duplicate Ratio : The ratio between the square roots of the two numbers.
    •  4 : 5 is the sub-duplicate ratio of 16 : 25.
  4. Sub-triplicate Ratio : The ratio between the cube roots of the two numbers.
    •  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.
    • 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.
    • The compound ratio of  3/4, 5/7, 4/5, 4/5 is 9/35


Proportion : 

Proportion is nothing but the equality of two ratios (fractions). As we have mentioned above, If X : Y = Z : K, we write X : Y :: Z : K and we say that X, Y, Z and K are in proportion.

Confused ? Lets try to understand this with a simple example. You went to market to buy some eggs. The person who sells eggs tells you the price of eggs is Rs. 48 a dozen (12 eggs). But you don't want 12 eggs. You just need 6. So, you just will pay Rs. 24 to buy Half of the dozen. Right?
Hey wait, how did you determine their cost?

As you need half of the eggs, you just reduced the cost to the half. So, the cost of 6 Eggs is half of 48. i.e., 24. That's it. This is called Proportion :)

Technically you can say that the numbers a, b, c and d are in proportion if a/b = c/d.

Here a, b, c and d are called proportionals (first, second, third and fourth).

Ex : 

The numbers 5, 6, 10 and 12 are in proportion. Because 5:6 = 10:12

you can write them as 5:6::10:12

Types of Proportions are :

  • 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. So we can say that the  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. So we can say that the  third proportion of a, b =  b2/a.  
  • Second or Mean proportion : If a : x = x : b , then x is called second or mean proportion of a and b. So we can say that the mean proportion of a and b =  root(ab)
Some important points you should keep in mind while dealing with Proportions :
  1. If A, B, C, D are in proportion then 
    • A and D are called Extremes
    • B and C are called Means. 
    • As you know, 
      • A / B = C / D
        • Then the product of the extremes is equal to the product of means. 
  2. The concept of proportion need not be restricted to only two equal ratios. It may be extended thus,
    1. If A/B = C/D = E/F = .... then A, B, C, D, E, F ... are said to be in proportion. 
  3. If two proportions are equal, we can say that the numbers are in Continued Proportion. 
    • 25 : 20 and 20 : 16. Here the ratios and proportions are equal. So the numbers 25, 20, 20, 16 are in continued proportion. 
  4. From the above example, we can get the following conclusion.
    • b2= a x c

9 comments:

  1. Dear Madam,
    It is good lesson but if you solve it with example it will be more useful to us to know how to solve it

    ReplyDelete
  2. not useful for us.... give any example... it is just like theory.....

    ReplyDelete
    Replies
    1. Check again friend. There are examples too. At the bottom of the post :)

      Delete
    2. Thanks Sir / Mam. Its useful

      Delete
  3. Its very easy to under stand thank you

    ReplyDelete
  4. It's just like theory...better to add short cuts for solving the problems also..

    ReplyDelete
  5. can u explain compound ratio eg

    ReplyDelete

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