### AVERAGE ARTHMETIC MATERIAL RATION AND PROPORTIONS

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 : 