AVERAGE:
Average = sum/n
n = sum/average
EG:
Find the average of first 5 prime numbers ?
2+3+5+7+11 = 28/5 = 5.6
EG:
Find the average of first 5 even numbers?
2+4+6+8+10 =30/5 =6
if it is a arthmetic progression
constant increase
avg : (first term + last term)/2
(2+10)/2 = 6
or
exact middle number
EG:
find average of 10,12,14,16
(10+16)/2 = 13
or
(10+12+14+16)/4 = 52/4 =13
EG:
find the average on first 15 multiples of 3?
(3 + (3*15))/2 = 48/2 = 24
EG:
Find the average on first 20 multiples of 6?
(6+(20*6))/2 = 126/2 = 63
EG:
average of 5 consecutive even numbers is 36 then find numbers?
32 34 36 38 40
EG:
Average of 6 consecutive odd numbers is 40
35 37 39/41 43 45
EG:
sum of 3 consecutive rational numbers is 45 find sum?
x+x+1+x+2 = 45
14,15 16
note:
if the total sum of items is share equally in between each and every item then per head value is called average
pre head change = average change
EG:
average of sum of numbers is 50 each number is increased by 2
then find new average
ans : 52
EG:
average sum of numbers is 25 if each number is multiplied by 2 and then deminished by 5
then find the new average
(25*3) - 5 = 75-5 = 70
MODEL:
EG
in a class average age of 20 students is 12 years if teacher age is included age increased by 2 years then find
age of teacher
20*12 = 240
21*14 = 294
......
54
or
12 +(21*2)
54
EG:
average weight of 11 players is 80 kg if coach weight is included average decreased by 2 kg then find coach age?
11*80 = 880
12*78 = 936
......
56
EG :
The score of a batsman for 12 matches is 45 rums howmany runs he has to score in his next match to increase his vaerage by 3 runs?
12 matchs ave45 = 12*45 624
13 match ave increased by 3 13*48 540
624- 540 = 84
EG:
average score of a batsman for 15 matches is 40 runs how many runs he has to score in his nextmatch to decrease his average by 1 run?
15*40 - 16 *39 40
16
.........
24
MODEL 2:
EG
OUT OF SIX MEMBERS IN A GROUP ONE PERSON WHOSE WEIGHT IS 40 KG IS REPLACE BY A NEW PERSON THEIR BY AVERAGE IS INCREASED BY 2 KG THEN FIND THE WEIGHT OF NEW PERSON?
40 +(6*2) = 52
EG:
out of 10 members in a group one person whose weight is 80 kg is replaced by a new person thereby average is decreased by 3kg
find thw weight of new person?
80 - (10*3) = 50
EG:
our of 10 members 2 members whose weight is 50 kg and 60 kg respectivilt are replace by 2 new persons thereby average is increased by 2 kg then find the avrage
weight of 2 new persons?
110 + (10*2) = 130
130/2 = 65
EG:
out of 9 observations one record was wrongly taken as 36 instead of 63 therefore average
is 50 find correct average?
n = 9
ave = 50
actual 63 wrong 36
so 63-36 =27
avg + 27/9 = 3
actual average = 50 +3 =53
EG:
out of 10 observations 1 reading was wrongly taken as 80 instead of 30 therefore the average is 28
find the correct average?
n = 10
average = 28
30
80
50 decreased
50/10 =5
correct average = 28 -5 = 23
EG:
Out of some numbers two numbers wrongle taken as 23 & 76 instead of 32 & 67 find the change in average?
99 99
no change
Model 4:
EG:
4 years ago average age of 3 members in a family is 14 years in the meanwhile a baby has been born
therefore the present average of family remains same find present age of baby?
3*14 = 42
12
..........
54
4*14 = 56
56
54
........
EG:
Five years ago average age of 4 members in a family is 22 years in the meanwhile a baby has been born
therefore present average age of a family remain same find the age of baby?
4*22 = 88
4*5 = 20
......
108
5*22 = 110
110
108
......
2 years
EG:
6 years ago average age of a,b,cd is 44 years 2 years ago c expire present average age of a,bd is 48 years find the age of c when he expire?
4*44 = 176
6*4 24
..............
200
144
.....................
56
2 years ago -2
...........
54 years c age
EG:
average of 5 numbers is 6 fine the sum of numbers 5*6 = 30
EG:
the average of 5 elements is 12 and another 10 elements is 9find new avg?
(60+90)/15 = 150/15 =10
EG:
the average marks obtained by 30 boys is 55 and the average marks obtained of 20 girls is 40 avg?
(1650 +800)/50 = 49
EG:
the average of 8 observations is 26 if the average of first 5 observations is 18.2 what is the average of remain?
8*26 = 5*18.2 +b
208 = 91.0 +b
117/3 = 39
EG:
average of 11 numbers is 40 the average of first 5 numbers is 38 and the average of last 5 numbers
is 41 find middle number?
5*38 = 190
5*41 = 395
.........
45
EG:
average of 7 results is 20 = 140
first 4 results is 25 == 100
last four results is 15 = 60
160-140 =20
EG:
T average temperature of monday,tuesday,wednesday,thursday is 48 and the average temperaturw on tuesday,wednesday,thursday,friday was 46
if the average temperature on friday was 34 find the temperature on monday?
m,t,w,th ---------48
t,w,th,f ------------46
f --- 34
t,w,th,34 ..... 46
46-34 = 12
m,12 = 48
m=48-12 = 36
EG:
the average of 4 elements is 11 if each element is increased by 7 what will be new average?
28/4 = 7
11+7 = 18
EG:
the average of 13 num 18 if each num is multipled by 3 find new average?
18 *3 = 54
3*13*18 = 702
702/13 = 54
EG:
A batsman 17 innings make a score of 85 runs and there by his average increased by 3 runs find
16 innings avg?
16 x = 16x
17 x+3 = 17(x+3)
16x+85 = 17(x+3)
x = 85 -51 = 34
EG:
The average marks obtained by 60 students in an exam is 35 if the average marks of pass students 39 and that of failed students
is 15 how many students passed the exam?
60
x 60-x
39x+ 15(60-x) = 60*35
EG:
the average age of a family of 6 members is 44 if the age of a youngest member be 14 years find the average age of the family at the birth
of the younger member?
6*44 = 264
-14*6 = -84
....................
180
180/5 = 36
EG:
The average age of a and b is 20 years if c were to replace a the average would be 19 nif c were
to replace b the average would be 21 find the ages of a,b,c?
a+b = 20*2 = 40
c+b = 19*2 = 38
a+c = 21*2 = 42
2(a+b+c) = 120
a+b+c = 60
a+b = 40
c= 20 a =22 b =18
EG:
35 students in a hostel if no of students increases by 7 the expenses of the hostel increased by 42 rs perday while the average expenditure per head decreases by 1 find the original
expenditure of the hostel?
no avgexp totalexp
35 x = 35x
42 (x-1) = 42(x-1)
35x+42 = 42(x-1)
75x = 42(x-1)
7x = 84
x =12
35x = 35*12
EG:
IN A CLASS TOTAL AVERAGE OF 20 STUDENTS IS 12 YEARS AFTER ADDING TEACHER AGE THEIR AVERAGE INCREASED BY 2 YEARS THEN FIND TEACHER AGE?
CASE 1
N = 20
TOTAL = 20*12
240
CASE 2:
N = 20+1 = 21
AVERAGE = 12+2 = 14
TOTAL = 14*12
294
TEACHER AGE = 294
240
..........
54 YEARS
METHOD 2
12 + (21*2)
12+42
54 YEARS
EG:
30 MEMBERS AVERAGE AGE IS 16 YEARS AFER ADDING TEACHER AGE TOTAL AVERAGE INCREASED BY 1 YEAR THEN
FIND TEACHER AGE?
N =30
AVERAGE = 16
30*16
480
CASE : 31
AVG = 17
31*17
TEACHER AGE = (31*17) - 480
47
EG:
15 MATCHES BATSBAN AVERAGE SCORE IS 35 RUNS THEN HOW MANY RUNS HE HAS TO SCORE IN IN HIS NEXT MATCH
THERE BY HIS AVG TO INCREASE 3 RUNS?
35 + (16 MATCHES * 3 )
35+48
83 RUNS
EG:
11 MEMBERS AVERAGE WEIGHT 80 KGS AFTER ADDING COACH AGE AVERAGE DECREASED BY 2 KGS THEN FIND COACH AGE?
80- (12*2)
80-24
56
EG:
6 YEARS BACK A,B,C,D AVERAGE AGE 44 YEARS 2 YEARS BACK C DIED NOW PRESENT AVERAGE OF A,B,D IS 48 YEARS THEN FIND C AGE AT THE
TIME OF HIS DEATH?
CASE 1
6 YEARS BACK = 44*4
176
A+B+D = 48*3
144
PRESENT C = 200-144
56 YEARS
2 YEARS BACK = 56 -2 =54
EG:
4 YEARS BACK AVERAGE AGE OF A 3 FAMILY MEMBERS IS 14 YEARS IN THESE DAYS ONE BABY BORN THEREBY NO CHANG
IN AVERAGE OF THEIR AGES THEN FIND BABY AGE?
CASEI
4 YEARS BACK:
AVERAGE : 14 YEARS
TOTAL = 14*3 = 42
CASE 2
NOW :
3 PEOPLE TOTAL = 42+12
54
PRESENT
4 MEMBERS AVG = 14*4 = 56
BABY AGE = 56-54= 2
EG:
3 YEARS BACK IN A FAMILY 5 MEMBERS AVERAGE IS 17 YEARS IN THESE DAYS ONE BABY BORN THEREBY
NO CHANG IN THEIR AVERAGE THEN FIND BABY AGE?
CASE 1:
3 YEARS BACK 5*17
85
PRESENT 85+15 100
6 PEOPLE AVE = 17
17*6 = 102 YEARS
BABY AGE = 102 - 100 = 2 YEARS
RATION AND PROPORTIONS
.......................
Ratio is comparison between two or more quantitys having same units
any ratio does not have units
the general form to represent a ratio is a : b or a/b
a : anticipate
b : consequent
NOTE: Ka : Kb
2 : 3 = 10 : 15
K = 5
Model 1 :
1. An amount of rs 500 is shared between a,b,c in the ratio 2 : 3: 5 find the share of each person?
A : B : C
2: 3 :5
2X : 3X : 5X
TOTAL = 500
2x+3x+5x = 500
10 x = 500
x = 50
2x = 100
3x = 150
5x = 250
2nd method :
A : B : C
A= 2/10(500) = 100
B = 3/10 (500) = 150
C = 5/10 (500) = 250
3rd method
2 : 3 : 5
100 150 250
10 parts = 500
1 part = 50
2 )An amount of rs 600 is shared between a,b,c in the ratio 3 : 4: 5 find the share of each person?
ans
3/12(600) = 150
4/12(600) = 200
5/12(600) = 250
3) A and b are in the ratio 3 : 5 difference between their ages is 12 years find the sum :?
5parts - 3parts = 2parts
2parts = 12 years
1 part = 6
3parts *6 = 18
5parts * 6 = 30
total = 18 +30 = 48
4) A and b are in the ratio 3 : 4 square of their ages is 225 years then find sum of their ages?
(3x)(3x) +(4x)(4x) = 225
(25x)2= 225
x= 3
5) in a school boys and girls are in the ratio 6 : 5 then find the total number of studens?
a)33
b) 35
c) 37
6x +5x =11x
so eleven multiple is 33
ans : 33
Method 2 :
in a school boys and girls are in the ratio 2 : 3 then find the total number of studens?
ans :
42 44 45 47
7) in a class there are 60 students which of the following can not be ration between boys and girls?
a: 2 :3
b) 5 :1
c) 7 : 5
d) 5 : 2
MODEL 2:
method 1
A : B = (2:3 )4 = 8 :12
B :C = (4 :5 )3 = 12 : 15
A : B :C = 8 :12 :15
METHOD 2 :
a : b = 2:3
b :c = 4 :5
ans : 2*4 + 3 *4 +3*5
8 :12 :15
..............................................
eg :
a: b = 3 :5 b : c = 4 :7 then a : b :c
12 : 20 :35
EG :
A : B = 4 :5 B : C =7 :9 THEN A :B :C
28 : 35 : 45
EG:
A : B = 2 :3 b :c =4:5 c :d = 2:1 then
a :b c : d =
2 : 3
4 :5
2 :1
a : 2 * 4 * 2
b = 3 * 4 * 2
c = 3 * 5 *2
d = 3 * 5 * 1
eg :
a : b = 2 : 5 b : c = 2 :3 c :d = 5 :4 d : e = 7 :3
then a : e?
MODEL 3
EG:
In a vessel milk and water are in the ratio 5 : 2 if 6 leters of water is added then the ratio 5 : 3
find the quality of the milk ?
5-2 = 1part
1 part = 6
5parts = 5*6 = 30
eg:
milk and water are in the ratio 7 :5
if 20 liters of water is added then ratio 1 : 1
7 : 5
1: 1
multiple (1:1) with 7 = 7 : 7
7 : 5
7 : 7
2p = 20
p = 10
7*10 = 70
eg:
In a class boys and girls are in the ratio 3 :2 if 20 girls newly joined in the class then the ratio 2 :3
find the intial no of students?
3 , 2 lcm 6
2(3 :2 ) = 6 :4
3(2 :3 ) = 6 :9
5p =20
p = 4
6 * 4 = 24
4 *4 =16
24 +16 =40
EG:
In a solution salt and water there is 75 % of salt if 5 liters of water there is 80 % of salt find the initial quality of that solution
75 %
75 : 25 80 : 20
3 : 1 4 : 1
3 , 4 lcm 12
12 :4
12 : 3
1p = 5
16 part = 16 * 5 = 80
MODEL 4 :
Eg :
a : (b+c) = 3 : 5
b : (c+a) = 1 :7
then a : b : c
1st cond = 3 +5 = 8
2nd cond = 1+7 = 8
total 8 parts
a = 3 b = 1 c = 8-4 = 4
Eg :
a : (b +c ) = 3 : 7
b : (c+a) = 1 :4
a : b : c
1st cond 3 +7 = 10
2nd cond 1+4 = 5
multiple 2nd cond with 2
3 :7
2 :8
3 : 2 : 5
Eg :
a : (b+c) = 2 :5 b : (c +a) = 1 :3 then a :b :c
7 *4
4 *7
(2 :5 ) 4 = 8 :20
(1:3)b 7 = 7 :21
a : b : c 8: 7 :13
total = 28
a+b = 15 c = 13
MODEL 5
Three equal capacity vassel contains milk and water in the ratio 3 :1 , 5 :3,7:5 if they are mixed in a baig container find the new ratio?
m w
(3 :1 )6 =18 :6
(5 :3 ) 3 = 15 :9
(7 :5 )2 = 14 :10
............
47 : 25
Eg:
the equal capacity vassel contains milk and water in the ratio 3 :2, 7 :3, 8:7
if they are mix in a big container fine new ratio?
(3 :2 )6 = 18 :12
(7 :3)3 = 21 :9
(8:7)2 = 16 :14
...............
55 :35
11 :7
Eg
two vessels contains milk and water in the ratio 3 ;1 and 5 ;3 respectivily if they mixed in the proportion 3 :4 then find new ratio?
(3 :2 ) 2 (6 :2)3 = 18 :6
5 :3 (5 : 3)4 = 20 :12
.............
38 :18
19 :9
MODEL 6 :
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