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.}

<sup>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</sup>  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. ? Sol:  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

^{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}                

Percentages trick

Percentages trick

Find 7 % of 300. Sound Difficult?

Percents: First of all you need to understand the word “Percent.” The first part is PER , as in 10 tricks per list verse page. PER = FOR EACH. The second part of the word is CENT, as in 100. 

Like Century = 100 years. 100 CENTS in 1 dollar… etc. Ok… so PERCENT = For Each 100.

So, it follows that 7 PERCENT of 100, is 7. (7 for each hundred, of only 1 hundred).

8 % of 100 = 8. 35.73% of 100 = 35.73

But how is that useful??

METHOD 1

Back to the 7% of 300 question. 7% of the first hundred is 7. 7% of 2nd hundred is also 7, and 7% of the 3rd hundred is also 7. So 7+7+7 = 21.

If 8 % of 100 is 8, it follows that 8% of 50 is half of 8 , or 4.

Break down every number that’s asked into questions of 100, if the number is less then 100, then move the decimal point accordingly.

EXAMPLES:

8%200 = ? 8 + 8 = 16.

8%250 = ? 8 + 8 + 4 = 20.

8%25 = 2.0 (Moving the decimal back).

15%300 = 15+15+15 =45.

15%350 = 15+15+15+7.5 = 52.5

Also it’s useful to know that you can always flip percents, like 3% of 100 is the same as 100% of 3.

35% of 8 is the same as 8% of 35

METHOD 2

In other words, divide the 'percentage number'

15%500 =  (10%+5% =15%) now 10%500=50 and (10/2 = 5) so 50/2=25  now add both 50+25 = 75 is ur answer

20%490= (10% + 10%).. 49+49 = 98

27.5 %444 = (25% +2.5%).. you just need to shift '.' by one left.. 111+11.1 = 122.1

33% 312= (30% + 3%) or (10+10+10+ 1+1+1) or here its easy to find 3times of 312 = 936 

so that 936 will be 300% and 30% will be 93.6 and 3% will be 9.36

For answer add 93.6 and 9.36 =   104.66

3 digit multiplication

As you already saw 2-digit multiplication, now we see how to slove 3digit ones.

The Rule is    |  X  *  X  |



                                   a  b  c


                               x   d  e  f            
 and answer will be |  X  *  X  |


 same as the 2digit ones.. there must be a minimum of 5 digits in the answer when you multiply two 3-digit numbers.

 coming from right end to left end... (as it is the rule to solve multiplications)
first | denotes               c x f
next X denotes            bf + ec ie., (b x f) + (e x c)
then *  denotes        af + dc + be
next X denotes            ae + bd
 and last | denotes        a x d 



In every digit,, if you have a carry, then u can transfer it to the very next left digit.

and for this 2 digit or 3-digit multiplication, please don't remember  a x f,b x c like that, jut remember the method to solve it.. ie.,  for 2 digit |  X  |
                               and for 3 digit    |  X  *  X  |


here remember..  that star in middle for 3-digit  is like     \|/
                                                                                  .. /|\
Will be back to u soon
If u have any doubts in any topic such as percentages, train problems, and any thing. Please post here. I'll try to get back to you with shortcut and answers soon. 

A small start..

There are many contents in this Solving process. Basics are 'Simplifications'. In those Additions, Subtractions, multiplications and divisions take a key role. Additions and subtractions are tough for us.. in these 5-digit calculations are also very easy. Now, coming to multiplications and divisions, first of all u need to know tables very thoroughly. In these, 2 or 3 digit calculations are also tough. So now, we will see how to multiply 2-digit numbers.
 The Rule is " | X | "
  a     b
  |  X  |
  c     d
ie the answer will be of min. 3-digit number. so UNITS digit will be " | "  =  b*d or bd
                                                                       TENS digit will be  " X " = ad + bc
                                                                 Hundreds digit  will be  " | "  = ac
Sample
2   3               5     6
| X |                |  X  |
2   1               4     2      here 6 x 2 = 12, 5x4 = 20, ans 5x2 + 6x4 =  10 + 24 =34
4 8 3             20 12
                    + 34
                     2352


The above mentioned is the basic formula for any type of 2-digit multiplication


To know squares

(a+b)^2 = a^2 + 2ab + b^2

You know this formula right.. the same way 
to get square of a number UNITS digit will be --- b^2
                                        TENS digit will be --- 2ab
                               HUNDREDS digit will be -- a^2


34^2 =   9 4 6
       +    2 1                3x3 =9, 4x4 = 16 and 1 is carried to tens digit
       =  12 5 6              2x3x4 = 24, where 2 is carried to next digit





Will be back to u soon
If u have any doubts in any topic such as percentages, train problems, and any thing. Please post here. I'll try to get back to you in 2-3 days.