Ratio & Proportion

Ratio:-Ratio is a comparison between two quantities in same unit by using division.

              Ratio of two values x and y is written as x:y or x/y or x to y.

Type  of ratio:-

(1).Duplicate ratio:-The ratio of square of two numbers is called duplicate ratio.

                                 The duplicate ratio of x and y is x²/y²_.

(2)Triplicate ratio:-The ratio of cube of two numbers is called triplicate ratio.

                                The triplicate ratio of x and y is x³/y³_.

(3).Sub duplicate ratio:-The ratio of square root of two numbers is called sub duplicate ratio.

                                      The sub duplicate ratio of x and y is √x/√y_.

(4) Sub triplicate ratio:-The ratio of cube root of two numbers is called sub triplicate ratio.

                                      The sub triplicate ratio of x and y is ∛x/∛y_.

(5) Inverse ratio:-The ratio between the reciprocals of two quantities is called inverse ratio.

                            The inverse ratio of x and y is y/x.

(6) Compound ratio:-The product of two or more ratios; thus xz:yw is a ratio compounded 

                                   of the simple ratios x:y and z:w.

Proportion:-Proportion is the equality of two ratio.

                         If a:b=c:d we write a:b::c:d.

                         Product of means = Product of extremes

                         ⟹ad = bc

Type  of Proportion:-

(1).Fourth Proportional:-If a:b = c:d then d is called fourth proportional.

(2).Third Proportional:-If a:b = b:c then b is called third proportional.

(3).Mean Proportional:-If a:x = x:b then x is called third proportional.

                                      It can also written as x =√(ab)

(3).Componendo and Dividendo rule:-If a:b = c:d

                                       Then (a+b)/(a-b)=(c+d)/(c-d) is componendo and dividendo rule.

Example 1.:-If a:b = 5:7 and b:c = 6:11 then find a:b:c.
                    
Explanation:-   a       :        b          :            c

                         5                7
                                           6                       11

                        a       :         b          :            c

                        5*6            7*6
                                          6*7                   11*7
                         30             42                     77
                        Hence  a:b:c = 30:42:77
Example 2.:-Find the fourth  proportional to the numbers 60,48,30.
Explanation:- Let fourth proportional is x.
                       ⟹ 60:48::30:x
                       As  Product of means = Product of extremes
                       ⟹ 60*x = 48*30
                       ⟹ x = 48*30/60 = 24
                      Hence fourth proportional is 24.
Example 3.:-Find the third  proportional to the numbers 16,36.  
Explanation:- Let third proportional is x.
                      Thus   16:36::36:x
                      As  Product of means = Product of extremes
                       ⟹ 16*x = 36*36
                       ⟹ x = 36*36/16 = 81
                      Hence third proportional is 81.    
Example 4.:-If 76 is divided into four parts proportional to 7,5,3,4.
                       Then find the smallest part.  
Explanation:- Given ratio =7:5:3:4.
                      Sum f the ratio is 19
                      Smallest part is (3/19)*76
                                                 = 12
                      Hence smallest part is 12.
Example 5.:-A bag contain 50 paise,25 paise,10 paise coins in the ratio 5:9:4.
                       Amounting to be Rs. 206 Find the number of coin of 50 paise.  
Explanation:- Let the number of 50 paise,25 paise,10 paise coins be 5x:9x:4x.
                       We know value of 50 paise coin is Rs. 0.5
                                        value of 25 paise coin is Rs. 0.25
                                        value of 10 paise coin is Rs. 0.1
                       Then 5x*0.5 + 9x*0.25 + 4x*0.1 = 206
                             2.5x + 2.25x + 0.4x = 206
                             5.15x = 206
                                    x = 40
                      Hence the number of coin of 50 paise is 5*40 = 200 
Example 6.:-The present ratio of ages of ages of two brothers is 4:5.
                      18 years ago their ages ratio was 11:16.Find the age of elder brother.  
Explanation:- Let the present ages are 4x:5x
                      Now given  

                      (4x-18)/(5x-18) = 11/16
                       64x-288 = 55x-198
                       9x = 90
                        x = 10

                       Hence the present age of elder brother is 5x = 5*10 = 50.

CALENDAR

CALENDAR
Calendar is a chart or series of pages showing the days, weeks, and months of a particular year, or giving particular seasonal information.
We follow the Gregorian calendar which is also called as the Christian Calendar or the Western calendar 
The Western calendar is the widely used in  international.
Note:-The basic thing to remember to solve calendar problems.

Odd Days:- In a given time period, the number of remaining days more than complete  weeks are                           called odd days.  

Ordinary year:-An ordinary year has 365 days.

Leap year :- A leap year has 366 days.

(1) Every year divisible by 4 is a leap year,if it is not century.

(2) Every century divisible by 400 is a leap year.

      As 400,800,1200,1600,2000 etc.

Counting of odd days:-

(1). One ordinary year = 365 days = 52*7+1 days

                                      =52 weeks + 1 day.

       Hence 1 ordinary year has 1 odd day.

(2). One leap year = 366 days = 52*7+2 days

                                      =52 weeks + 2 days.

       Hence 1 leap year has 2 odd days.

(3). 100 years = 76 ordinary years + 24 leap years

                       =76*1 odd days  + 24*2 odd days

                       = 126 odd days = 17*7 + 5 days.

       Hence 100 years has 5 odd days.

(4). 200 years = 5*2 odd days.=10 odd days

                       =1 week + 3 odd days.

(5). 300 years = 5*3 odd days.=15 odd days

                       =2 weeks + 1 odd day.

(6). 400 years = 5*4 odd days.+1 odd day=21 odd days

                       =3 weeks = 0 odd day.

Table of week related to odd day

Number of odd days	Day
0	Sunday
1	Monday
2	Tuesday
3	Wednesday
4	Thursday
5	Friday
6	Saturday

Example.1.:-What as the  day of the week on 15^th August 1947( Independence day).

Explanation:-Counting of odd day
                     Number of odd day in 1600 years = 0
                     Number of odd day in 300 years = 5*3 = 15 days = 2 weeks + 1 
                     Number of odd day in 46 years = 11 leap years + 35 Ordinary years
                                                                       =  11*2 + 35*1 = 57 odd days
                                                                       =8*7 + 1 = 1 odd day
                    Odd days in 1946 years = 0+1+1 = 2.
In 1947^th year
                    January = 31 days = 3 odd days

                    February = 0 odd days

                    March = 31 days = 3 odd days

                    April = 30 days = 2 odd days

                    May = 31 days = 3 odd days

                    June = 30 days = 2 odd days

                    July = 31 days = 3 odd days

                    August = 15 days = 1 odd day

Total number of odd days= 2+3+0+3+2+3+2+3+1=19 = 2*7 + 5

                                         = 5

Hence the required day was Friday.

Example.2.:-What as the  day of the week on 26^th January 1950.

                                                 ( Constitutional Implemented day)

Counting of odd day
                     Number of odd day in 1600 years = 0
                     Number of odd day in 300 years = 5*3 = 15 days = 2 weeks + 1 
                     Number of odd day in 49 years = 12 leap years + 37 Ordinary years
                                                                       =  12*2 + 37*1 = 61 odd days
                                                                       =8*7 + 5 = 5 odd days
                    Odd days in 1946 years = 0+1+5 = 6.
In 1950^th year
                    January = 26 days = 5 odd days

                    

Total number of odd days= 6 + 5=11 = 1*7 + 4

                                         = 4

Hence the required day was Thursday.

Example.3.:-What as the  day of the week on 2^nd October 1869 ( Bapu Jayanti).

Counting of odd day
                     Number of odd day in 1600 years = 0
                     Number of odd day in 200 years = 5*2 = 10 days = 1 week + 3 
                     Number of odd day in 68 years = 17 leap years + 51 Ordinary years
                                                                       =  17*2 + 51*1 = 85 odd days
                                                                       =12*7 + 1 = 1 odd day.
                    Odd days in 1869 years = 0+3+1 = 4.
In 1869^th year
                    January = 31 days = 3 odd days

                    February = 0 odd days

                    March = 31 days = 3 odd days

                    April = 30 days = 2 odd days

                    May = 31 days = 3 odd days

                    June = 30 days = 2 odd days

                    July = 31 days = 3 odd days

                    August = 31 days = 3 odd days

                    September = 30 days = 2 odd days

                    October = 2 days

Total number of odd days= 4+3+0+3+2+3+2+3+3+2+2=27 = 3*7 + 6

Alternative short trick to find calendar

Note:-Remember these things

Century code table

5th	6th	7th	8th	9th	10th	11th	12th	13th	14th	15th	16th	17th	18th	19th	20th
4	2	0	6	4	2	0	6	4	2	0	6	4	2	0	6

Month code

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
1	4	4	0	2	5	0	3	6	1	4	6

Month code

Sun	Mon	Tues	Wed	Thurs	Fri	Sat
1	2	3	4	5	6	7

Formula

Remaining odd day=[Date+Month+Year+Leap Year+Century]/7

Example.1.:-What as the  day of the week on 15^th August 1947( Independence day).

Remaining odd day=[15+3+47+11+0]/7 = 76/7 = 6

Hence the required day was Friday.

e required day was Saturday.

Average

Average 

Average Concept:-

      The average of a number is a measure of the central tendency of a set of numbers.

       In other words, it is an estimate of where the center point of a set of numbers lies.

Average:-It is defined as the ratio of sum of observations to Number of observations.

          

Average =	Sum of observations
	Number of observations

Some Important Results:- 

(1) Average of n natural numbers = (n + 1) / 2

(2) Average of even n natural numbers = (n + 1)

(3) Average of odd n natural numbers = n
(4)  If value of each term increases/decreases by k, then the average of the group also

      increases/decreases by k. 

(5) If we know average of two groups individually, then the average of combined group 
      cannot be determined.
(6) There are two groups A and B .If A has n₁ observations B. has n₂ observations and
     their averages are x₁ and x₂ respectively. If we have to find the average of the 
    Combined group of A and B use the formula
      
     

Average of whole group =	(n1x1 + n2x2)
	(n1 + n2)

Example 1 :-Average of 4 consecutive number is 5.5 find the largest number.

                   Let numbers are x,x+1,x+2,x+3.
 Explanation     Average =  x+(x+1)+(x+2)+(x+3)/4 = 5.5

                    4x + 6 = 22

                    4x = 16

                     x = 4

                   largest number is x+3 = 4+3 = 7

Example 2 :-Average  of 4 consecutive odd number is 24 find the smallest number.
 Explanation    Let numbers are x,x+2,x+4,x+6.

                   Average =  x+(x+2)+(x+4)+(x+6)/4 = 24

                    4x + 12 = 96

                    4x = 84

                     x = 21

                   Smallest number = x = 21

Example 3 :-The average age o three boys is 15 years.If their age are in ratio 3:5:7
                      what is the age o youngest boy.
 Explanation    Let the ages of three boys be 3x,x,7x
                      Average age =(3x+5x+7x)/3 = 15
                       15x=45
                        x=3
                       The age o youngest boy is 3x=3*3=9

Example 4 :-There were 35 students in a hostel. due to the admission 

                      of 7 new students the expenses for the mess were increased 

                      by 42/- per day while the average expenditure per head diminished

                      by Rs 1 .what was the original expenditure of the mess.
 Explanation    After the  7 new students, number of students will become 42

                     average expenditure per head = (x-1)

                     New total expenditure = 42(x-1)

                     Given that Expenses for the mess were increased by 42

                     42(x-1) -35x =84

                   42x-42-35 =84

                   7x = 84

                   x=12
Example 5 :-There are two sections A and B of a class, consisting of 36 and 44 students’ 
                    respectively.If the average weight of section A is 40kg and the average weight
                   of section B is 35kg, find the average weight of the whole class.
Explanation     Now n₁=36,n₂=44,x₁=40,x₂=35
                 Average of whole group =(n₁x₁ + n₂x₂)/(n₁ + n₂)
                                                          = (36*40 + 44*35)/(36+44)
                                                         =2980/80
                                                         = 37.25
Example 6 :-In the first 10 overs of a cricket game, the run rate was only 3.2

                     What should be the run rate in the remaining 40 overs to reach

                    the target of 282  runs.

Explanation   the first 10 overs  the run  was  32

                   the first 50 over’s  the run  was  282

                    the  remaining 40 overs  the run  was  282-32 = 250

                    the run rate= 250/40 = 6.2
Example 7 :-The average of 25 results is 18.The average of first twelve of them is 14
                        and that of last  twelve is 17.Find the thirteenth result.

Explanation 

1_2_3_4_5_6_7_8_9_10_11_12_13_14_15_16_17_18_19_20_21_22_23_24_25

                       Sum of 25 results is 18*25 = 450
                       Sum of first twelve results is 14*12 = 168
                       Sum of last twelve results is 17*12 = 204
                       Thirteenth result = Sum of 25 results
                                                        - (Sum of first twelve results + Sum of last twelve results)
                                                     = 450-(168 + 204)
                                                     =  450 - 372 = 78
Example 8:-The average of  11 results is 60.The average of first six result  is 58
                        and that of last  six is 63.Find the seventh result.
Explanation   1_2_3_4_5_6_7_8_9_10_11

Sum of 11 results is 11*60 = 660
                       Sum of first six results is 6*58 = 348
                       Sum of last six results is 6*63 = 378
                       Seventh result = (Sum of first six results +Sum of last six results)
                                                  -Sum of 11 results   
                                               = (348 + 378) - 660
                                               =  726 - 660 = 66