chapter 8.2

Average 179

Soln: Applying the above formula, we have

Exercise

1. I f the sum o f x number o f quantities is 162 and the aver-age is 9. Find the value o f x. a) 18 b)28 c)19 d) 17

2. Average daily income o f a rickshaw puller is Rs 45. I f after x days, rickshaw puller earns Rs 315, find the value ofx. a )8 days b) 15 days c ) 5 days d ) 7 days

3. Total temperature o f the month October is 775 C. I f the average temperature o f that month is 25 C, find o f how many days is the month o f October? a) 30 days b) 29 datys c) 31 days d) Data inadequate

4. In a coconut grove, (x + 2) trees yield 60 nuts per year, x trees yield 120 nuts per year and (x - 2) trees yield 180 nuts per year. I f the average yield per year per tree be 100, find x. [MBA 1986] a) 4 b )2 c )8 d)6

Answers 2 . d 3.c

(* + 2 ) x 6 0 + ; cx l20 + ( x - 2 ) x l 8 0 4. a; Hint: ~ 4 = 100

x+2+x+x-2

x = 4

Rule 4 Theorem: If the average of'm' boys is 'x'and the average of 'n' boys is 'b' then the average of all of them put together

mx + ny = total average) is

m + n

Illustrative Example Ex.: The average age o f students in section A o f 50 stu-

dents is 14 years and the average age o f students in section B o f 30 students is 6 years. Find the average age o f students in both sections taken together.

Soln: Following the above formula, we have

50x14 + 3 0 x 6 , . the required average = + = v e a r s -

Note: The above rule is a k ind o f discrete series. In a discrete series the values of the variables are multi-plied by their respective frequencies and the products so obtained are totalled. This total is divided by the number of items, which in a discrete series, is equal to the total of the ^equencies. The resulting quotient is a simple arithmetic jverage of the series. In the form offormula it is written as

Average = XJ\+X2fl+- + X,Jn A+f2+A+- + f

Where, f , f,, f3 f are the frequencies ie no. of items

and Xj , x2, x 3 , x - are the values of each respective item.

Ex. 1: A man bought 13 shirts o f Rs 50 each, 15 pants o f Rs 60 each and 12 pairs o f shoes at Rs 65 a pair. Find the average value o f each article.

Soln: Direct Method:

13x50 + 15x60 + 12x65

A v e m g e = 13 + 15 + 12 = R S 5 8 2 5

Ex. 2: The average score o f a cricketer in two matches is 27 and in three other matches is 32. Then find the aver-age score in all the five matches.

Soln: Direct Method: Average in 5 matches

2 x 2 7 + 3 x 3 2 54 + 96

2 + 3 :30

Exercise 1. A student bought 4 books for Rs 120 from one bool<

shop and 6 books for Rs 150 from another. The average price ( in rupees), he paid per book was _ . a)Rs27 b)Rs27.50 c ) R s l 3 5 d ) R s l 3 8

2. In a class o f 100 students, the mean marks obtained in < subject is 30 and in another class o f 50 students the mean marks obtained in the same subject is 60. The mear marks obtained by the students o f two classes takei together is . a) 30 b)50 c)40 d)45

3. A class has 20 boys and 30 girls. The average age o boys is 12 years and that o f girls is 11 years what is th< average age o f the whole class? a) 11.4 years b) 11.6 years c) 11.2 years d) 12 years

4. O f 20 men 12 gain Rs 335 each and 8 men gain Rs 24( each. What is the average gain per man? a)Rs297 b)Rs290 c)Rs279 d)Rs397

5. I f 20 chairs are bought at Rs 50 each, and 15 at Rs 4: each and 15 more at Rs 40 each. What is the averagi price o f a chair? a)Rs60 b )Rs45 c)Rs45.5 d)Rs50.5

6. The average height o f 3 0 girls out o fac las so f40 i s 16i cm and that o f the remaining girls is 156 cm. What is tfo average height o f the whole class? a) 159 cm b) 160 cm c) 159.5 cm d) 160.5 cm

7. The average expenditure o f a man for the first five month is Rs 1200 and for the next seven months is Rs 1300. F in his monthly average income i f he saves Rs 2900 durin the year. a)Rs750 b ) R s l 5 0 0 c ) R s l 7 5 0 d)Rs500

8. A man bought 13 tins at Rs 50 each, 15 tins at Rs 60 eac and 12 tins at Rs 65 each. What is the average price pai per tin? a)Rs58 b)Rs 58.50 c)Rs 58.25 d)Rs 58.75

180 PRACTICE BOOK ON QUICKER MATHS

In a certain primary school there are fifteen boys at the age o f 12, sixteen at 15, and eighteen at 14 years. Find the average age o f boys.

a) 13 years

1 c J c ) 1 y y e a r s

b) 1 7 years

3 d) 13 years

10. Out o f 24 girls 6 are 1 m 15 cm in height, 8 are 1 m 5 cm and the rest 1 m 11 cm. What is he average height o f the girls? a) 1 m b ) 2 m c ) l m l 0 c m d ) 2 m l 0 c m

11. The average daily number o f persons passing a certain point on Monday, Tuesday, Wednesday, and Thursday is 2163. The average daily number passing on Friday, Saturday and Sunday is 1960. What is the daily average for the whole weak? a) 2706 b)2067 c)2076 d)3076

12. The average age o f the boys in a class o f 20 boys is 15.6 years. What w i l l be the average age i f 5 new boys come whose average is 15.4 years? a) 15.56 years b) 13.36 years c) 15 years d) 15.50years

13. The average age o f 600 scholars o f a school is 10.75 years. By the enrolment o f 40 new scholars the average is reduced to 10.4375 years. Find the average age o f the new scholars.

4 3 3 3 a) 3 y years b) 5 years c) 4 y years d) 6 years

14. In a certain primary school, there are 60 boys o f age 12 each, 40 o f age 13 each, 50 o f age 14 each and 50 o f age 15 each. The average age ( in years) o f the boys o f the school is: [Clerical Grade Exam, 1991] a) 13.50 b) 13 c) 13.45 d) 14

15. The average height o f 30 girls out o f a class o f 40 is 160 cm and that o f the remaining girls is 156 cm. The average height o f the whole class is:

[Central Excise & I Tax Exam, 1988] a) 158 cm b) 158.5 cm c) 159 cm d) 159.5 cm

16. The average height o f 30 boys, out o f a class o f 50, is 160 cm. I f the average height o f the remaining boys is 165 cm, the average height o f the whole class ( in cm) is:

[Clerical Grade Exam, 1989] a) 161 b)162 c)163 d)164

Answers l . d 2. c 3. a 4. a 5.c 6.a 7. b; Hint: Required average monthly income

5x1200 + 7x1300 + 2900 18000

12 12

8.c 9. a 10. c 11. c; Hint: Required daily average

4x2163 + 3x1960 = = 20 lo

1 12. a 13. b; Let the average age o f the new scholars be x.

Now,

600x10.75+ 40 x x

600 + 40

or, 6450+40x = 6680

23 c 3 ' x = T = 5 7 4 4

= 10.4375

14.c 15.c 16. b

Rule 5 Theorem: If the average age of'm' boys is 'x' and the aver-age age of 'n' boys out of them (m boys) is 'y' then the

average age of the rest of the boys is mx-ny

= Rsl500

Illustrative Example Ex.: The average o f 10 quantities is 12. The average o f 6 o f

them is 8. What is the average o f remaining four num-bers.

Soln: By the above theorem, we have

1 0 x 1 2 - 6 x 8 1 0 the required average = * .

10 6 Exercise 1. A group o f 20 girls has average age o f 12 years. Average

age o f first 12 from the same group is 13 years. What is the average age o f other 8 girls in the group?

[ B S R B BhopalPO2000] a) 10 b ) l l c) 11.5 d) 10.5

2. A group o f 30 girls has average age o f 13 years. Average age o f first 18 from the same group is 15 years. What is the average age o f other 12 girls in the group? a) 12 years b) 10 years c) 16 years d) 10.5 years

3. The average age o f 30 students in a class is 12 years. The average age o f a group o f 5 o f the students is 10 years and that o f another group o f 5 students is 14 years. Find the average age o f the remaining students. a) 14 years b) 10 years c) 12 years d) Data inadequate

4. 30 horses are bought for Rs 150000. The average cost of 18 o f them is Rs 4500. What is the average cost o f oth-ers? a)Rs5750 b)Rs7550 c)Rs5760 d)Rs4750

5. The average o f 12 results is 15, and the average o f the first two is 14. What is the average o f the rest? a) 15.2 b) 13.2 c) 15 d) 16

Average 18

6. The average o f 3 numbers is 7, that o f the first two is 4, find the third number. a) 13 b) 10 c)12 d ) l l

7. The average score o f a cricketer for 10 matches is 38.9 runs. I f the average for the first 6 matches is 42, find the average for the last four matches. a)34.52 b)43.25 c)34.25 d)35

Answers l . d 2.b 3. c; Hint: Required average

3 0 x l 2 - { ( 5 x l 0 ) + ( 5 x l 4 ) } _ 240

3 0 . - ( 5 + 5) ~ 20 ~

4. a; Hint: Required average

150000(30x5000)-18x4500

3 0 - 1 8

5.a

6. a; Hint: Required number =

7.c

3 x 7 - 2 x 4

3 - 2

= Rs5750

13

Rule 6 Theorem: If the average of '' quantities is equal to 'x'. When a quantity is removed the average becomes 'y\the value of the removed quantity is [n (x -y) +yj.

Illustrative Example Ex.: The average age o f 24 boys and a class teacher o f a

class is equal to 15 years. I f class teacher left the class due to health problem the average becomes 14. Find the age o f class teacher who left the class.

Soln: Following the above theorem, we have

therequiredanswer = 2 5 ( 1 5 - 1 4 ) + 14 = 2 5 + 1 4 = 39

years.

Exercise 1. The average age o f 24 students and the class teacher is

16 years. I f the class teacher's age is excluded, the aver-age reduces by 1 year. What is the age o f the class teacher? [ B S R B Mumbai P O , 1998] a) 50 years b) 45 years c) 40 years d) Data inadequate

2. The total age o f 26 persons are 442 years. Out o f these persons one is a teacher and others are students. I f the teacher's age is excluded, the average reduces by 2 years. What is the age o f the teacher? a) 50 years b) 55 years c) 60 years d) 67 years

3. The average age o f 30 students and the class teacher is 20 years. I f the class teacher's age is excluded, the aver-age reduces by 1 year. What is the age o f the class teacher? a) 39 years b) 50 years c) 40 years d) 49 years

4. The average age o f 13 students and the class teacher i 19 years. I f the class teacher's age is excluded, the avei age reduces by 2 years. What is the age o f the cla; teacher? a) 45 years b) 40 years c) 38 years d) 39 years

5. The average age o f 15 students and the class teacher 15 years. I f the class teacher's age is excluded, the ave age reduces by 1 year. What is the age o f the cla teacher? a) 30 years b) 31 years c) 29 years d) 28 years

Answers 1. c

442 2. d; Hint: Here, n = 26, x = = 1 7 years and y

ZD

= 1 7 - 2 = 15 years. Now apply the formula and get the answer = 67 year:

3. b 4. a 5.a

Rule 7 Theorem: If the average of 'n' numbers is'm' and if 'x added to or subtracted from each given number, the av age of'n' numbers becomes (m+x) or (m-x) respectivi In the other words average value will be increased or i creased by 'x'.

Illustrative Examples Ex. 1: The average o f 11 numbers is 21 . I f 3 is added to e;

given number, what w i l l be the new average? Soln: From the above theorem, we have

new average = 21 + 3 = 24 Ex. 2: The average o f 6 numbers is 15. I f 3 is subtracted fi

each given number, what w i l l be the new average Soln: From the above theorem, we have

newaverage= 1 5 - 3 = 12

Exercise 1. The average o f 15 numbers is 25. I f 5 is added to e

given number, what w i l l be the new average? a) 20 b)30 c)25 d)Datainadeq

2. The average o f n numbers is 4n. I f n is added to t given number, what w i l l be the new average? a ) ( n + l ) 4 b ) 5 n c ) ( n + l ) 5 d ) N o n e o f t h (

3. The average o f 8 numbers is 14. I f 2 is subtracted i each given number, what w i l l be the new average? a) 12 b) 10 c)16 d ) N o n e o f t h

4. The average o f x numbers is 3x. I f x - I is subtra from each given number, what w i l l be the new aver a ) 2 x + l b ) ( x - l ) 3 C )2JC -1 d)Datainadec

Answers l . b 2 .b 3.a 4. a

L82 PRACTICE BOOK ON QUICKER MATHS

Rule 8 'heorem: If the average of'n' quantities is equal to 'x'and >hen a new quantity is added the average becomes y \ "hen the value of the new quantity is [n (y - x) + yj. In nother words it may be written as, 'alue of new entrant = No. of old members x Increase in verage + New average.

llustrative Example Ix.: The average age o f 30 boys o f a class is equal to 14

years. When the age o f the class teacher is included the average becomes 15 years. Find the age o f the class teacher.

loin: Detailed Method: Total ages o f 3 0 boys = 1 4 x 3 0 = 420 years

Total ages when class teacher is included = 15 >

Average i s :

And this increase in weight is due to the extra weight included due to the inclusion o f new person. .-. Weight o f new man = 120+ 12 = 132 kg. Quicker Method: Weight o f new person = weight o f removed person + No. o f persons * increase in aver-age = 120 + 4 x 3 = 132 kg.

Ex. 2: The average age o f 8 persons in a committee is in-creased by 2 years when two men aged 35 years and 45 years are substituted by two women. Find the av-erage age o f these two women.

Soln: By the direct formula, we have the total age o f two women = 2 x 8 + (35 + 45)

= 16+ 80 = 96 years

96

.-. Average age o f two women = = 48 years.

Exercise 1. The average weight o f 8 persons increases by 1.5 kg. I f

a person weighing 65 kg is replaced by a new person, what could be the weight o f the new persons?

[ B S R B Delhi P O 2000] a) 76 kg b ) 7 7 k g c) 76.5 kg d) Data inadequate

2. The average weight o f 10 men is increased by 1 kg

when one o f the men who weighs 68 kg is replaced by a new man. Find the weight o f the new man. a) 73 kg b ) 8 3 k g c) 82.5 kg d) Data inadequate

3. The average weight o f 15 men is increased by 2 kg when one o f the men who weighs 48 kg is replaced by a new man. Find he weight o f the new man. a) 88 kg b ) 7 8 k g c) 77.5 kg d) Data inadequate

4. The average age o f a committee o f 7 trustees is the same as it was 5 years ago, a younger man having been sub-stituted for one o f them. How much younger was he than the trustee whose place he took? a) 3 0 years b) 3 5 years c) 25 years d) Data inadequate

5. The average age o f 10 persons in a committee is increased by 1 year when two men aged 42 years and 38 years are substituted by two women. Find the average age o f these two women. a) 46 years b) 45 years c) 42 years d) 44 years

6. The average age o f 11 persons in a committee is increased by 2 years when three men aged 32 years, 33 years and 34 years are substituted by three women. Find the aver-age age o f these three women.

feS: .\ . , i a) 40 years b) 4 1 T years

c )41 years d) 40 years

7. The average weight o f the 8 oarsmen in a boat is in creased by 1 kg when one o f the crew, who weighs 60 k is replaced by a new man. What is the weight o f the ne\ man? a) 78 kg b ) 6 6 k g c ) 6 8 k g d ) 7 2 k g

8. The average weight o f 8 persons is increased by 2.5 k | when one o f them whose weight is 56 kg is replaced b a new man. The weight o f the new man is:

[Central Excise & I Tax 198* a) 66 kg b ) 7 5 k g c ) 7 6 k g d ) 8 6 k g

Answers l . b 2 .h 3.b 4. b; Hint: The average age o f committee o f 7 trustees is t l

same as it was 5 years ago. But today committee gains x 5 = 35 years. Hence, we can conclude that the youngi man who replaced the trustee 5 years ago is 35 yea younger than the trustee. In another way it can be explained as the following, the new member would have not been substituted, thf the increased age = (5 x 7) = 35 years. So the new mer ber is 35 years younger than the trustee whose place ! took.

5. b 6 .d 7.c 8.c

Rule 11 Theorem: The average age of 'n'persons is decreased 'x'years when some of them [np n2... n; where n, + n2 +


I In a class there are 36 boys whose average age is de-creased by 5 months, when 1 boy aged 30 years is re-placed by a new boy. Find the age o f the new boy. a) 16 years b) 10 years c) 15 years d) 20 years

5. In a class there are 15 boys whose average age is de-creased by 4 months, when 1 boy aged 23 years is re-placed by a new boy. Find the age o f the new boy. a) 20 years b) 18 years c) 21 years d) 18.5 years

i. In a class there are 27 students whose average age is decreased by 4 months, when 4 students aged, 16,17,18 and 19 years respectively are replaced by the same num-ber o f students. Find the average age o f the new stu-dents. a) 15.25 years b) 15 years c) 15.5 years d) 16 years

5. In a class there are 18 students whose average age is decreased by 2 months, when 3 students aged 12, 13 and 14 years respectively are replaced by the same num-ber o f students. Find the average age o f the new stu-dents. a) 18 years b) 12 years c) 16 years d) 14 years

6. In a class there are 16 students whose average age is decreased by 3 months, when 2 students aged 24 and 26 years respectively are replaced by the same number o f students. Find the average age o f the new students. a) 23 years b) 21 years c) 18 years d) None o f these

Answers l . a 2.c 3.b 4.a 5.b 6.a

Rule 12 Theorem: The average of marks obtained by 'n' candidates in a certain examination is 'T'. If the average marks of passed candidates is 'P' and that of thefailed candidates is 'F'. Then the number of candidates who passed the exami-

n(T - F)~ P-F

ie Number of passed candidates

Total candidates (Total Average - Failed Average)

Passed Average - Failed Average

Illustrative Example Ex. : The average o f marks obtained by 120 candidates in a

certain examination is 35. I f the average marks o f passed candidates is 39 and that o f the failed candi-

dates is 15, what is the number o f candidates who passed the examination?

Soln: Detai l M e t h o d : Let the number o f passed candidates hex.

Then total marks = 1 2 0 x 3 5 = 39x + ( l 2 0 - x ) x l 5

or, 4200 = 3 9 * + 1 8 0 0 - 1 5 * or, 24* = 2400

nation is

. - . x = 1 0 0

.-. number o f passed candidates = 100. Quicker Method: Apply ing the above formula, we have

no. o f passed candidates 120(35-15) _

= 100

3 9 - 5

Exercise 1. The average o f marks obtained by 90 candidates in a

certain examination is 38. I f the average marks o f passed candidates is 40 and that o f the failed candidates is 30, what is the number o f candidates who passed the exami-nation? a) 72 b)70 c)75 d)80

2. The average o f marks obtained by 80 candidates in a certain examination is 32. I f the average marks of passed candidates is 34 and that o f the failed candidates is 18, what is the number o f candidates who passed the exami-nation? a) 170 b)70 c)80 d)75

3. The average o f marks obtained by 65 candidates in a certain examination is 25. I f the average marks o f passed candidates is 27 and that o f the failed candidates is 14, what is the number o f candidates who passed the exami-nation? a) 55 b)65 c)60 d)75

4. The average o f marks obtained by 75 candidates in a certa.ii examination is 31 . I f the average marks o f passed candidates is 35 and that o f the failed candidates is 25, what is the number o f candidates who passed the exami-nation? a) 40 b)46 c)45 d)54

5. The average o f marks obtained by 77 candidates in a certain examination is 17. I f the average marks o f passed candidates is 19 and that o f the failed candidates is 8, what is the number o f candidates who passed the exami-nation? a) 36 b)63 c)40 d)70

Answers l . a 2 ,b 3.a 4.c 5.b

Rule 13 Theorem: The average of marks obtained by V candidates in a certain examination is 'T'. If the average marks of passed candidates is 'P' and that of thefailed candidates is 'F'. Then the number of candidates who failed the exami-

' n ( P - T ) l nation is P - F

ie, the number of failed candidates

_ Total candidates (Passed average - Total average)

Passed average - Failed average

Average

Illustrative Example Ex.: The average o f marks obtained by 120 candidates in a

certain examination is 35. I f the average marks o f passed candidates is 39 and that o f the failed candi-dates is 15, what is the number o f candidates who failed the examination?

Soln: Following the above formula, we have 120(39-35)

the no. o f failed candidates = r r - - 2 0 .

Exercise I . The average o f marks obtained by 108 candidates in a

certain examination is 20. I f the average marks o f passed candidates is 28 and that o f the failed candidates is 16, what is the number o f candidates who failed the exami-nation? a) 70 b)78 c)81 d)72

2 The average o f marks obtained by 110 candidates in a certain examination is 15. I f the average marks o f passed candidates is 25 and that o f the failed candidates is 14, what is the number o f candidates who failed the exami-nation? a) 100 b)90 c)105 d)95

3. The average o f marks obtained by 102 candidates in a certain examination is 18. I f the average marks o f passed candidates is 21 and that o f the failed candidates is 15, what is the number o f candidates who failed the exami-nation? a) 51 b)52 c)61 d)50

4. The average o f marks obtained by 115 candidates in a certain examination is 36. I f the average marks o f passed candidates is 40 and that o f the failed candidates is 17, what is the number o f candidates who failed the exami-nation? a) 30 b)25 c)20 d)34

5. The average o f marks obtained by 125 candidates in a certain examination is 29. I f the average marks o f passed candidates is 36 and that o f the failed candidates is 11, what is the number o f candidates who failed the exami-nation?

a) 32 b)35 c)30 d)40

Answers l . d 2. a 3. a 4.c 5.b

Rule 14

Theorem: If the average of n results (where n is an odd

number) is 'a' and the average of first I ^ I results is 'b'

and that of last

~n + \

n + l

{ 2 ) is 'c'. Then

n + l th result is

-(b + c)- na

Illustrative Example Ex.: The average o f 11 results is 50. I f the average o f fir

six results is 49 and that o f last six is 52, find the six result.


sixth result = ^ y ^ ( 4 9 + 52) - 1 1 x 50

= 6 x ( l 0 l ) - 5 5 0 = 56

Exercise 1. The average o f 17 numbers is 45. The average o f firsl

o f these numbers is 51 and the last 9 o f these numbers 36. What is the ninth number?

[BSRB Mumba iPO , 199 a) 14 b)16 c)22 d) 18

2. The average o f 19 numbers is 40. The average o f first o f these numbers is 39 and the last 10 o f these numbt is 36. What is the 10th number? a) 12 b) 14 c)18 d) 10

3. The average o f 15 numbers is 50. The average o f firs o f these numbers is 52 and the last 8 o f these numbers 39. What is the 8th number? a)32 b)22 c)31 d)30

4. The average o f 13 numbers is 30. The average o f firs o f these numbers is 32 and the last 7 o f these number: 22. Find the 7th number. a) 18 b) 12 c)16 d) 10

5. The average o f 21 numbers is 3 5. The average o f first o f these numbers is 42 and the last 11 o f these numb is 23. What is 11th number? a)20 b)22 c)18 d)23

6*. The average o f 25 results is 18; that o f first 12 is 14 a o f the last 12 is 17. Thirteenth result is:

[CBIExam,19 ( a) 78 b)85 c)28 d)72

Answers l . d 2 . d 3.a 4 .b 5.a 6.a

Rule 15 Ex.: The average o f 11 results is 30, that o f the first fiv

25 and that o f the last five is 28. Find the value o f 6th number.

Soln: Direct Formula: 6th number = Total o f 11 results - (Total o f first fiv

Total o f last five results) = H x 3 0 - ( 5 x 2 5 + 5x28)=330-265 =

Exercise 1. The average o f 11 numbers is 37.15, the average o f

first 3 is 36.41 and o f the last 7 is 37.51. The fourth ni ber is found to be wrong. Find the average o f the re a) 37.28 b)37 c)37.18 d) 37.08

2. The average o f 12 results is 36, that o f the first six is


and that o f the last five is 35. Find the value o f the 7th number. a) 65 b)60 c)64 d)56 The average o f 15 results is 28, that o f the first seven is 26 and that o f the last seven is 25. Find the value o f the 8th number. a) 36 b)66 c)65 d)63

. The average o f 13 results is 39, that o f the first five is 38 and that o f the last seven is 36. Find the value o f the 6th number.

a) 64 b)46 c)65 d)56

Answers .c;Hint: The fourth number = 11 x 37.15 - ( 3 x 36.41 + 7 x

37.51) = 36.85 Number can not be in fraction. Hence this is wrong. .-. Required average

11x37.15-36.85 371.8 , _ 1 0 = = = 3 / . 18

10 10 l.a 3 .d 4.c

Rule 16 Theorem: If a batsman in his nth innings makes a score of 'x', and thereby increases his average by 'y', then the aver-age after 'n' innings is fx y (n -1)].

Illustrative Example Ex.: A batsman in his 17th innings makes a score o f 85

and thereby increases his average by 3. What is his average after 17 innings?

Soln: Following the above formula,

8 5 - 3 ( 1 7 - l ) = 37.

Exercise 1. A batsman in his 16th innings makes a score o f 92 and

thereby increases his average by 4. What is his average after 16 innings? a) 32 b)30 c)34 d)23

2. A batsman, in his 19th innings, missed a century by 2 runs and thereby increases his average by 3. What is his average after 19 innings. a) 54 b)44 c)45 d)43

3. A batsman in his 21st innings makes a score o f 88 and thereby increases his average by 2. What is his average after 21 innings? a) 46 b)48 c)45 d)44

4. A batsman in his 20th innings makes a score o f 110 and thereby increases his average by 4. What is his average after 20 innings? a) 34 b)43 c)36 d)30

5. A batsman in his 44th innings makes a score o f 86 and thereby increases his average by 1. What is his average after 44 innings? a) 34 b)43 c)46 d)40

Answers l . a 2 .b 3.b 4.a 5.b

Rule 17 Theorem: If a cricketer has completed 'n' innings and his average is 'x' runs. The number of runs, he must make in his next innings so as to raise his average to 'y' are fn (y -x) +yj.

Illustrative Example Ex.: A cricketer has completed 10 innings and his average

is 21.5 runs. H o w many runs must he make in his next innings so as to raise his average to 24?

Soln: Following the above theorem, we have the required answer = 10 (24 - 2 1 . 5 ) + 2 4 = 2 5 + 2 4 = 4 9 .

Exercise 1. A cricketer has completed 15 innings and his average is

20 runs. How many runs must he make in his next in-nings so as to raise his average to 25? a) 75 b)50 c)100 d)85

2. A cricketer has completed 20 innings and his average is 44.5 runs. How many runs must he make in his next in-nings so as to raise his average to 45? a) 45 b)60 c)40 d)55

3. A cricketer has completed 31 innings and his average is 18 runs. How many runs must he make in his next in-nirig,3 so as to raise his average to 22? a) 124 b)146 c) 136 d) 142

4. A cricketer has completed 18 innings and his average is 26.5 runs. How many runs must he make in his next in-nings so as to raise his average to 27? V v a) 63 b)36 c)45 d)54

5. A cricketer has completed 14 innings and his average is 30 runs. How many runs must he make in his next in-nings so as to raise his average to 32? a) 60 b)55 c)65 d)50

Answers l . c 2 . d 3.b 4 .b 5.a

Rule 18 Theorem: If a person travels a distance at a speed ofxkm/ hr and the same distance at a speed of y km/hr, then the

2xy

average speed during the whole journey is given by x + y

km/hr.

or, If half of the journey is travelled at a speed of x km/hr and the next half at a speed of y km/hr, then average speed

2xy during the whole journey is ~~ km/hr.

or,

Average 187

// a man goes to a certain place at a speed of x km/hr and returns to the original place at a speed ofy km/ltr, then the

average speed during up-and-down journey is 2xy

x + y km/

Note: In all the above three cases the two parts o f the jour-ney are equal, hence the last two may be considered as a special case o f the first. That's why all the three lead to the same result.

Illustrative Example KJL : A train travels from A to B at the rate o f 20 k m per

hour and from B to A at the rate o f 30 km/hr. What is the average rate for the whole journey?

Soln: By the formula:

2 x 2 0 x 3 0 Average speed = 2 Q + 3 0 = 24 km/hr.

Ixercise A constant distance from A to B is covered by a man at 40 km/hr. The person rides back the same disance at 30 km/hr. Find his average speed during the whole journey, a) 34 km/hr b) 35.29 km/hr c) 34.29 km/hr d) 35 km/hr

1 A man goes to a certain place at a speed o f 15 km/hr and returns to the original place at a speed o f 12 km/hr, find the average speed during up-and-down journey.

a) 13 km/hr

c) 1 3 - km/hr

, 1 b) 1 3 - km/hr

d) 1 1 - km/hr

A man goes to a certain place at a speed o f 30 km/hr and returns to the original place at a speed o f 20 km/hr, find the average speed during up-and-down journey. a)24km/hr b)25km/hr c)28km/hr d)23km/hr Ram travels half o f a journey at the speed o f 24 km/hr and the next half at a speed o f 16 km/hr. What is the average speed o f Ram during the whole journey?

a) 19 km/hr 5

c) 19-j km/hr

b) 20 km/hr

d) 1 6 - km/hr 5

A person travels ha l f o f a journey at the speed o f 30 km/ hr and the next half at a speed o f 15 km/hr. What is the average speed o f the person during the whole journey? a)20km/hr b)25km/hr c)18km/hr d)24km/hr

B t w e r s I c 2. b 3. a 4. c 5. a

Rule 19 Theorem: If a person travels three equal distances at a speed of x km/ltr, y km/ltr and z km/lir respectively, then the

3xyz

average speed during the whole journey is Xy + yZ + x z

km/hr.

Illustrative Example Ex.: A person divides his total route o f journey into three

equal parts and decides to travel the three parts with speeds o f 40, 30 and 15 km/hr respectively. Find his average speed during the whole journey.

Soln: By the theorem:

Average speed 3 x 4 0 x 3 0 x 1 5

4 0 x 3 0 + 30x15 + 40x15

3 x 4 0 x 3 0 x 1 5

2250 = 24 km/hr.

Exercise 1. A person divides his total route o f journey into three

equal parts and decides to travel the three parts w i t ! speeds o f 20,15 and 10 km/hr respectively. Find his av-erage speed during the whole journey.

a) 13 km/hr

3 c) 13 km/hr

b) 1 1 km/hr

d) 1 1 km/hr

2. A person divides his total route o f journey into thre< equal parts and decides to travel the three parts w i t l speeds o f 5,10 and 15 km/hr respectively. Find his aver-age speed during the whole journey.

a) 8 km/hr

c) 8 km/hr

b) 1 1 km/hr

d) 9 km/hr

3. A person divides his total route o f journey into thre( equal parts and decides to travel the three parts wi t l speeds o f20 ,25 and 40 km/hr respectively. Find his av erage speed during the whole journey.

a) 26 km/hr

c) 2 5 km/hr

1 b) 2 6 km/hr

d) 2 5 km/hr ' 23

4. The average speed o f a cyclist who covers first, secon< and third km at 20,16 and 12 km/hr respectively ( in km hr) is . a) 16.24 km/hr b) 16 km/hr c) 15.66 km/hr d) 15.32 km/hr

1 8 8 PRACTICE BOOK ON QUICKER MATHS

Answers l . a 2.c 3.a 4 . d

Note: 1. I f instead o f Ath , Bth and Cth part we are given A % , B % and C% the expression changes to aver-

Rule 20 Theorem: If a person covers A km atx km/hr and B km aty km/hr and Ckmatz km/hr, then the average speed in cov-

ering the whole disance is A + B + C

~A B C + + x y z

km/hr.

Illustrative Example Ex.: A person covers 12 km at 3 km/hr, 18 k m at 9 km/hr

and 24 k m at 4 km/hr. Then find the average speed in covering the whole distance.

Soln: Applying the above formula, we have

12 + 18 + 24 54 the average speed - ^ j | = km/hr.

T + ~9~ + T

Exercise 1. A person covers 15 km at 5 km/hr, 12 km at 6 km/hr and 16

km at 4 km/hr. Then find the average speed in covering the whole distance.

a) 7 km/hr

c) 4 - km/hr

km/hr

d) 7 - km/hr

2. A person covers 9 km at 3 km/hr, 25 km at 5 km/hr and 30 km at 10 km/hr. Then find the average speed in covering the whole distance.

a) 5 km/hr

c) 9 km/hr

b) 1 1 - km/hr

d) 5 km/hr

3. A person covers 18 km at 6 km/hr, 16 k m at 8 km/hr and 30 km at 6 km/hr. Then find the average speed in covering the whole distance.

a) 6.5 km/hr b) 6.4 km/hr c) 6.2 km/hr d) 6 km/hr

Answers l . c 2.a 3.b

Rule 21 Theorem: If a person covers Ath part of the distance atx km/hr, Bth part of the distance aty km/ltr and the remain-ing Cth part at z km/hr, then his average speed is

1

A B C + - + x y

km/hr.

age speed :

100

A B C + +

y

km/hr.

2. Ath Bth and Cth part or A % , B % and C% together

constitute the total distance covered.

Illustrative Example 1

Ex.: A person runs the first - th o f the distance at 2 km/hr.

the next one half at 3 km/hr and the remaining dis-tance at 1 km/hr. Find his average speed.

Soln: Remaining distance = 1

Now, applying the above rule,

1 1 - + 5 2

the average

1 1

1/5 1/2 3/10 _1_ 1 _3_

2 + 3 + 1 10 6 10

3_

10

speed

_ 30 _

~ 17 " , 1 3

Exercise 1. Find the average speed o f a person when he covers firs

1 one-third o f the distance at 10 km/hr, next rd at 8 km b

and the last one-third at 6 km/hr. a) 7.66 km/hr b) 6.77 km/hr c) 6.67 km/hr d) 7.86 km/hr

2. A person runs the first th o f the distance o f 8 k m ^

3 the next th at 6 km/hr and the remaining distance at M

km/hr. Find his average speed. a) 17 km/hr b) 17.87 km/hr c) 17.78 km/hr d) None o f these

3. A man covers first 20% o f the distance at 10 km/hr, n 50% at 5 km/hr and the remaining distance at 15 km Find his average speed. a)7km/hr b)7.14km/hr c) 7.24 km/hr d) 4.17 km/hr

4. A train covers 50% o f the journey at 30 km/hr, 25% of journey at 25 km/hr and the remaining at 20 km/hr. F the average speed o f the train during the whole jour

^ 2 5 ^1 5 a) 2 5 km/hr b) 2 5 km/hr

47 23

c) 2 5 km/hr 47

47

d) None o f these

Average 15?

On a journey across Mumbai , a taxi averages 30 km/hr for 60% o f the distance, 20 km/hr for 20% o f it and 10 km/ hr for the remainder. The average speed for the whole journey (in km/hr) is a) 20 km/hr b) 22.5 km/hr c) 25 km/hr d) 24.625 km/hr

A n s w e r s (La 2.c 3.b 4. a 5. a

Rule 22 : Theorem: If the average value of all the members of a group m the average value of the first part of members is y', '^Ae average value of the remaining part of members is 'z'

the number of the first part of members is 'n', then the

n(x-y)

ber of the other part of members is

Bnstrative Example i u The average salary o f the entire staff in a office is Rs

120 per month. The average salary o f officers is Rs 460 and that o f non-officers is Rs 110. I f the number o f officers is 15, then find the number o f non-officers in the office.

in: Detail Method: Let the required number o f non-offic-ers = * Then, 110* + 460x 15 = 120(15+*) or, 120* - 1 1 0 * = 460 x 15 - 1 2 0 x 15 = 15(460 - 1 2 0 ) or, 10*= 15 x340; .-. x = 15 x 34 = 510. Quicker Method: No. o f non-officers = No. o f officers x

A v . salary o f officers - Mean average

Mean average - A v . salary o f non - officers

= 1 / 4 6 0 - 1 2 0 , 5 1 Q

U20-110

I tercise The average salary o f the entire staff in a office is Rs 130 per month. The average salary o f officers is Rs 540 and that o f non-officers is Rs 114. I f the number o f officers is 16, then find the number o f non-officers in the office, a) 140 b)410 c)510 d) 150

1 The average salary o f the entire staff in a office is Rs 220 per month. The average salary o f officers is Rs 650 and that o f non-officers is Rs 170. I f the number o f officers is 25, then find the number o f non-officers in the office. a)215 b)315 c)250 d)350 The average salary o f the entire staff in a office is Rs 166 per month. The average salary o f officers is Rs 456 and that o f non-officers is Rs 142. I f the number o f officers is 24, then find the number o f non-officers in the office.

a) 300 b)290 c)390 d)310 4. The average salary o f the entire staff in a office is Rs 200

per month. The average salary o f officers is Rs 318 and that o f non-officers is Rs 183. I f the number o f officers is 34, then find the number o f non-officers in the office, a) 118 b)240 c)246 d)236

5. The average salary o f the entire staff in a office is Rs 150 per month. The average salary o f officers is Rs 450 and that o f non-officers is Rs 80. I f the number o f officers is 14, then find the number o f non-officers in the office, a) 65 b)55 c)60 d)70

Answers l . b 2. a 3.b 4 . d 5.c

Rule 23 Theorem: If the average value of all the members of a group is x, the average value of the first part of members isy, the average value of the remaining part of members is z and the number of the remaining part of members is n, then the

number offirst part of members is n(x - z)

v-

Illustrative Example Ex.: The average age o f all the students o f a class is 18

years. The average age o f boys o f the class is 20 years and that o f the girls is 15 years. I f the number o f girls in the class is 20, then find the number o f boys in the class.


20(18-15) the required answer :

2 0 - 1 8 = 30

Exercise 1. The average age o f al l the students o f a class is 16 years.

The average age o f boys o f the class is 21 years and that o f the girls is 12 years. I f the number o f girls in the class is 10, then find the number o f boys in the class. a) 4 b )8 c)12 d) 10

2. The average age o f all the students o f a class is 24 years. The average age o f boys o f the class is 29 years and that o f the girls is 20 years. I f the number o f girls in the class is 25, then find the number o f boys in the class. a)30 b) 15 c)24 d)20

3. The average age o f all the students o f a class is 22 years. The average age o f boys o f the class is 26 years and that o f the girls is 19 years. I f the number o f girls in the class is 16, then find the number o f boys in the class. a) 12 b) 10 c)6 d)8

4. The average age o f all the students o f a class is 25 years. The average age o f boys o f the class is 27 years and that o f the girls is 23 years. I f the number c f girls in the class is 32, then find the number o f boys in the class.

190 PRACTICE BOOK ON QUICKER MATH ;

a)24 b) 16 c)32 d)28 5. The average age o f al l the students o f a class is 22 years.

The average age o f boys o f the class is 24 years and that o f the girls is 20 years. I f the number o f girls in the class is 30, then find the total number o f students in the class, a) 60 b)30 c)45 d)50

Answers l . b 2 .d 3.a 4.c 5.b

Ex. :

Soln:

Rule 24 There were 35 students in a hostel. I f the number o f students increases by 7, the expenses o f the mess increase by Rs 42 per day while the average expendi-ture per head diminishes by Re 1. Find the original expenditure o f the mess. Detail Method: Suppose the average expenditure was Rs *. Then total expenditure = 35*. When 7 more students j o i n the mess, total expendi-ture = 3 5 * + 4 2

Now, the average expenditure : 3 5 * + 42 35*+ 42

35 + 7 42

Now, we have 3 5 * + 42

42 = *- !

or, 35* + 42 = 4 2 * - 4 2

or, 7* = 84 . \ = 12

Thus the original expenditure o f the mess = 3 5 x 1 2 = Rs420 Direct Formula: I f decrease in average = * increase in expenditure = v increase in no. o f students = z and number o f students (originally) = N , then

x(N + z)+ y

z the original expenditure = N

In this case, 35 l(35 + 7 ) + 4 2

7 = 35 x !2 = Rs420

Exercise 1. There were 40 students in a hostel. I f the number o f

students increases by 8, the expenses o f the mess in-crease by Rs 48 per day while the average expenditure per head diminishes by Rs 2. Find the original expendi-ture o f the mess. a)Rs620 b)Rs720 c)Rs750 d)Rs820

2. There were 45 students in a hostel. I f the number o f students increases by 9, the expenses o f the mess in -crease by Rs 25 per day while the average expenditure per head diminishes by Re 1. Find the original expendi-ture o f the mess.

4.

a)Rs390 b)Rs295 c)Rs395 d)Rs400 There were 42 students in a hostel. I f the number students increases by 7, the expenses o f the mess i crease by Rs 32.5 per day while the average expenditu per head diminishes by Rs 1.5. Find the original expenc ture o f the mess. a)Rs636 b)Rs536 c)Rs630 d)Rs656 There were 36 students in a hostel. I f the number students increases by 4, the expenses o f the mess crease by Rs 32 per day while the average expend' per head diminishes by Re 1. Find the original expe -ture o f the mess. a)Rs640 b)Rs648 c)Rs650 d)Rs658

Answers l . b 2.c 3.a 4 .b

Rule 25 n+l

The average offirst 'n' natural numbers is

Illustrative Example Ex.: Find the average o f first 61 natural numbers. Soln: Apply ing the above rule, we have

the required average = 61 + 1

3 1 .

Exercise 1. Find the average o f first 62 natural numbers.

a)31 b)31.5 c)31.2 d)32 2. Find the average o f first 31 natural numbers,

a) 15 b)14 c)16 d) 17 3. Find the average o f first 91 natural numbers,

a) 45 b)47 c)45.5 d)46 4. Find the average o f first 101 natural numbers.

a) 50.5 b)52 c)51 d) None of

Answers l . b 2.c 3 .d 4.c

Rule 26 Ex.: The average weight o f 50 balls is 5 gm. I f the

o f the bag be included the average weight incr by 0.05 gm. What is the weight o f the bag?

Soln: Direct Formula: Weight o f bag = Old average + Increase in averJ Total no. o f objects

= 5 + 0.05 x 51 = 5 + 2.55 = 7.55 g J

Exercise 1. The average weight o f 30 balls is 3 gm. I f the weuj l

the bag be included the average weight increassJ 0.03 gm. What is the weight o f the bag? a)3.93gm b ) 3 g m c ) 4 g m d ) 4 . 3 9 g j

2. The average weight o f 40 balls is 4 gm. I f the we J

%verage 191

the bag be included the average weight increases by 0.04 gm. What is the weight o f the bag? a) 4.64 gm b) 6.64 gm c) 5.64 gm d) None o f these The average weight o f 20 balls is 2 gm. I f the weight o f pe bag be included the average weight increases by 0.02 gm. What is the weight o f the bag? a) 2.04 gm b)2.42gm c)3.42gm d)3.04gm

~ e r s 2.c 3.b

Rule 27 average ofn (where n is an odd number) consecutive

i is always the middle number. The numbers may be merely consecutive numbers eg, 1,2,3,4, 5. Average value of these 5 consecutive numbers will be me middle number le 3

: Average = 1 + 2 + 3 + 4 + 5 15

= 3 5 5

consecutive odd numbers eg 1, 3, 5, 7, 9. The average = middle number = 5

Average = 1 + 3 + 5 + 7 + 9 25

5

tcmsecutive even numbers eg, 2,4, 6, 8,10. The average = middle number = 6

Proof: Average 2 + 4 + 6 + 8 + 10 30

= 6.

rcise Rnd the average o f these 9 consecutive numbers. 5.6,7,8,9,10,11,12,13

Find the average o f the consecutive numbers given be-

. 26,27,28,29,30,31

Find the average o f the consecutive odd numbers given below.

35,37,39,41,43,45 Fmd the average o f the consecutive odd numbers given below. | "3.75,77,79,81,83,85,87,89,91 Fmd the average o f the consecutive even numbers given below. R52,54,56 ,58 ,60 ,62 ,64 ,66 ,68 ,70 ,72 ,74

ers 2.28 3.39 4.81 5.62

Rule 28 m:Theaverageof'n'(wheren=even number) con- numbers (whether merely consecutive, consecu-

or consecutive even) is the average of the middle ers.

(a) merely consecutive numbers eg, 1, 2, 3, 4, 5, 6. Average average of the middle two numbers

\ = 3.5. - ^ 2 2

1 + 2 + 3 + 4 + 5 + 6 21 , . Proof: Average = = - 3.5 ,

6 6

(b) consecutive odd numbers eg, 1, 3, 5, 7, 9, 11

5 + 7 Average = = 6 .

1 + 3 + 5 + 7 + 9 + 11 36 Proof: Average = - 6 .

6 6

(c) consecutive even numbers eg 2, 4, 6, 8, 10, 12.

6 + 8 Average

Proof: Average =

= 7.

2 + 4 + 6 + 8 + 10 + 12 72 = 7

6 6 Note: In all the above series, there are two middle terms.

Hene the required average can be calculated by the fol lowing methods.

I . In the case of'consecutive numbers': Average = Smaller middle term + 0.5 or Greater middle

t e rm-0 .5 I I . In the case o f 'consecutive odd' and 'consecutive

even': Average = Smaller middle term + 1 or Greater middle

term - 1 .

Exercise 1. Find the average value o f the fo l lowing consecutive

numbers. (i) 5 ,6 ,7 ,8 ,9 ,10,11,12,13,14 (ii) 22,23,24,25,26,27,28,29,30,31,32,33 (iii) 71,72,73,74,75,76

2. Find the average value o f the fol lowing consecutive odd numbers. (i) 9,11,13,15,17,19 (ii) 35,37,39,41,43,45,47,49 (iii) 81,83,85,87,89,91,93,95

3. Find the average value o f the following consecutive even numbers. (i) 82,84,86,88,90,92 (ii) 50,52,54,56,58,60,62,64

Answers 1. (i)9.5 (ii) 27.5

(iii) 73.5 2. CO 14 0042

(iii) 88 3. (i)87 (ii)57

192 PRACTICE BOOK ON QUICKER MATH

Rule 29 The average of odd numbers from 1 to n is

[Last odd number + 1J , where n = natural odd number.

2

Illustrative Example Ex.: What is the average o f odd numbers from 1 to 35? Soln: Applying the above rule, we have

35 + 1 1 0

the required answer = - = 1 .

Exercise 1. What is the average o f odd numbers from 1 to 39?

a)20 b)19 c)18 d)21 2. What is the average o f odd numbers from 1 to 79?

a) 30 b)25 c)35 d)40 3. What is the average o f odd numbers from 1 to 103?

a)53 b)51 c)52 d)50 4. What is the average o f odd numbers from 1 to 51?

a) 27 b)26 c)25 d)28 Answers l . a 2 .d 3.c 4 .b

Rule 30 The average of even numbers from 1 to n is

Last even number + 2

; where n = natural even number.

Illustrative Example Ex.: What is the average o f even numbers from 1 to 50? Soln: Applying the above formula, we have

50 + 2

2

the required answer : = 2 6 .

Exercise 1. What is the average o f even numbers from 1 to 30?

a) 16 b ) I 5 c ) I S d)17 2. What is the average o f even numbers from 1 to 80?

a)40 b)41 c)42 d)44 3. What is the average o f even numbers from 1 to 42?

a)20 b)24 c)22 d) 18

4. What is the average o f even numbers from 1 to 92?

a) 45 b)44 c)46 d)47

Answers l . a 2 .b 3.c 4 . d

Rule 31 The average of square of natural numbers till n is

(n + \)(2n + \y

Illustrative Example Ex.: Find the average o f squares o f natural numbers t i l l Soln: Following the above formula, we have

(7 + lX2x7 + l ) _ 8x15 J the required answer = - - 21

6 o

Exercise What is the average o f square o f the natural num from 1 to 10. a) 39 b )40 c)38.5 d)37.5 What is the average o f square o f the natural num from 1 to 23. a) 188 b)182 c)180 d) 183 What is the average o f square o f the natural num from 1 to 35. a) 436 b)426 c)416 d)446 What is the average o f square o f the natural num from 1 t o 4 1 .

a) 580 b)571 c)851 d)581

Answers l . c 2.a 3.b 4 . d

Rule 32 The average of cubes of natural numbers till

n{n + \f

4

Illustrative Example Ex.: Find the average o f cubes o f natural numbers t i l l Soln: Apply ing the above formula, we have

. , 7(7 + l ) 2 7 x 8 x 8 the required answer = =

M 4 4 Exercise 1. Find the average o f cubes o f natural numbers f n r

16. a) 1156 b) 1516 c ) U 5 5 d) 1165

2. Find the average o f cubes o f natural numbers from 15. a) 690 b)960 c)890 d)980

3. Find the average o f cubes o f natural numbers fro 24. a) 7350 b)3570 c)3750 these

4. Find the average o f cubes o f natural numbers fr 27. a) 756 b)9252 c)5922 d)5292

5. Find the average o f cubes o f natural numbers fr 8. a) 162 b) 172 c) 153 d) 163

Answers l . a 2 .b 3.c 4 d 5. a

erage 193

Rule 33 Mme nerage of first n consecutive even numbers is (n +1).

strative Example fa.- Find the average o f first 50 consecutive even num-

bers. Soln: Following the above rule, we have

the required answer = 5 0 + 1 = 5 1 . E x e r c i s e ,

Find the average o f first 52 consecutive even numbers. a)52 b)53 c)51 d)50

1 Find the average o f first 60 consecutive even numbers. a)61 b)59 c)62 d)58

Find the average o f first 14 consecutive even numbers. a) 14 b) 13 c)16 d) 15

4. Find the average o f first 18 consecutive even numbers. a) 16 b) 17 c)19 d) 18

Answers l> 2. a 3 .d 4. c

Rule 34 e average of first n consecutive odd numbers is 'n'.

Illustrative Example Find the average o f first 16 consecutive odd num-bers.

fata: Applying the above rule, we have

the required answer = 1 6 .

E x e r c i s e Find the average o f first 17 consecutive odd numbers, a) 16 b)17 c)18 d) 15

1 Find the average o f first 28 consecutive odd numbers, a) 28 b)27 c)29 d)26 Find the average o f first 64 consecutive odd numbers, a) 64 b)63 c)65 d)66 Find the average o f first 55 consecutive odd numbers, a) 54 b)56 c)55.5 d)55

wers M> 2. a 3. a 4 . d

Rule 35 ie average of squares of first n consecutive even numbers

~2(w + lX2 + l )~ 3

lustrative Example L : Find the average o f squares o f first 10 consecutive

even numbers.

Soln: Fol lowing the above rule, we have the required answer

2(lO + lX20 + l ) _ 2 x 1 1 x 2 1

3 3

Exercise 1. Find the average o f squares o f first 11 consecutive even

numbers. a) 184 b)148 c)186 d)174

2. Find the average o f squares o f first 14 consecutive even numbers. a) 280 b)270 c)290 d)295

3. Find the average o f squares o f first 17 consecutive even numbers. a) 450 b)420 c)430 d)410

4. Find the average o f squares o f first 23 consecutive even numbers.

a) 750 b)754 c)725 d)752

Answers l . a 2.c 3.b 4 . d

Rule 36 The average of squares of consecutive even numbers till n

~(n + \\n + 2)~ is ^

Ex.: Find the average o f squares o f consecutive even num-bers t i l l 10.

Soln: Apply ing the above rule, we have

1 1 x 1 2 , .

the required answer = - - 4 4 .

Exercise 1. Find the average o f squares o f consecutive even num-

bers from 1 to 11. a)65 b)52 c)44 d)51

2. Find the average o f squares o f consecutive even num-bers from 1 to 22. a) 184 b)174 c)182 d) 186

3. Find the average o f squares o f consecutive even num-bers from 1 to 26. a) 243 b)236 c)252 d)235

4. Find the average o f squares o f consecutive even num-bers from 1 to 35. a) 484 b)445 c)408 d)444

5. Find the average o f squares o f consecutive even num-bers from 1 to 44.

a) 680 b)960 c)690 d)860

Answers l . c 2.a 3.c 4.c 5.c


Rule 37 The average of squares of consecutive odd numbers till n is

~n(n + 2)~ . 3 _'

Illustrative Example Ex.: Find the average o f squares o f consecutive odd num-

bers t i l l 15. Soln: Following the above formula, we have

. , 15(15 + 2) the required answer = >.

Exercise 1. Find the average o f squares o f consecutive odd number

from 1 to 14. a) 70 b)65 c)75 d)66

2. Find the average o f squares o f consecutive Odd number from 1 to 20. a) 142 b) 136 c)133 d)144

3. Find the average o f squares o f consecutive odd number from 1 to 21 . a) 164 b) 161 c)144 d)184

4. Find the average o f squares o f consecutive odd number from 1 to 37. a) 404 b)464 c)481 d)444

5. Find the average o f squares o f consecutive odd number from 1 to 44.

a) 645 b)702 c)802 d)502

Answers l . b 2.c 3.b 4.c 5.a

Rule 38 Theorem: If the average of n members is 'A' and on re-checking it is noticed that some of the numbers (ie

* i , x 2 *3> *) are wrongly taken as

(x\, x'2, x ' 3 , x') then their correct average is

Previous average

Sum of correct numbers - Sum of wrong numbers

n '

or, Correct average

. . {xi+X2+Xi+...X)-(x'i+X2+X3+... + X) = A-i

n

Illustrative Example Ex.: In a calculation Mohan found that the average o f 4

numbers is 25 and on rechecking Sohan noticed that a number 15 is wrongly taken as 51 . Find the correct average.

Soln: Applying the above formula,

Correct average = 25 + - = 25 - 9 = 16.

Exercise 1. The average weight o f a group o f 20 boys was calcu-

lated to be 89.4 kg and it was later discovered that one weight was misread as 78 kg instead o f the correct one o f 87 kg. The correct average weight is a) 88.95 kg b) 89.25 kg c) 89.55 kg d) 89.85 kg

2. The average weight o f a group o f 15 boys was calcu-lated to be 60 kg and it was later discovered that one weight was misread as 24 kg instead o f the correct one o f 42 kg. The correct average weight is a) 60.2 kg b) 61.2 kg c ) 6 2 k g d ) 6 1 k g

3. In a calculation Ram found that the average o f 10 num-bers is 45 and on rechecking Shyam noticed that the some numbers 18,34,63 is wrongly taken as 81,43 and 36. Find the correct average. a) 39.5 b)40.5 c)45.5 d) 50.50

4. The average weight o f a group o f 9 boys was calculated to be 74.5 kg and i t was later discovered that one weight was misread as 56 kg instead o f the correct one o f 65 kg. The correct average weight is a) 75 kg b) 73.5 kg c) 75.5 kg d ) 7 6 k g

5. The average weight o f 15 students was calculated to be 52 kg and it was later discovered that one weight was misread as 21 kg instead o f the correct one o f 12 kg. The correct average weight is

a) 51.4 kg b) 50.6 kg c) 52.4 kg d) 51.6 kg

Answers l . d 2 .b 3.b 4.c 5.a

Rule 39 Geometric Mean: Geometric mean is useful in calculating averages of ratios such as average population growth rate, average percentage increase etc.

Geometric mean of x\, x2, J c 3 , x is denoted by

GM= ^]xx x x 2 x * 3 x...xxn

Illustrative Example Ex.: Find the geometric mean o f 4 ,8 ,16 .

Soln: G M = ^ 4 x 8 x 1 6 = \Ul2 = 8

Exercise 1. Find the geometric mean o f 2 ,4 and 8.

a) 2 b ) 4 c)3 d)5 2. Find the geometric mean o f 3,6 and 12.

a) 4 b )8 c ) 6 d) None o f these 3. Find the geometric mean o f 4 ,10 and 25.

a) 5 b) 10 c)20 d) 15 4. Find the geometric mean o f 9,12 and 16.

a) 8 b)12 c)14 d)22

Average

Answers l . b 2.c 3.b 4 .b

Rule 40 Harmonic Mean: Harmonic mean is useful forfinding out average speed of a vehicle, average production per day, etc.

Harmonic mean of x , , x2, x3,..., xn is denoted by

HM 1 1 1 1

+ + + ... + xn x2 x3 xn

Illustrative Example Ex.: Find the harmonic mean o f 2 , 3 , 4 and 5.

Soln: H M = 1

1 1 1 1 r + + +

4 _2 3 4 5_

30 + 20 + 15 + 12 4 x 6 0 _ 240

77 ~ ~ T T 60

Exercise 1. Find the harmonic mean o f 5 ,6 ,7 and 8. 2. Find the harmonic mean o f 4 ,6 and 8. 3. Find the harmonic mean o f 12,15,18 and 21 .

Answers

1. 3360

533 2.

72

13

5040

299

Rule 41 Average of a series having common difference 2 is

first term + last term

2 ~'

Illustrative Example Ex.: Find the average o f the fol lowing series

25,27,29,31,33 Soln: Applying the above formula, we have

25 + 33 the required average = - = 29 .

Exercise 1. Find the average o f the fol lowing series

(i) 22,24,26,28,30,32,36,38,40 a)32 b)31 c)29 d)33

(ii) 13,15,17,19,21,23,25,27,29,31,33

a) 23 b)32 c)24 d)22 (iii) 85,87,89,91,93,95

a) 91 b)89 c)90 d)92 (iv) 70,72,74,76,78,80,82,84,86,88,90

a) 80 b)82 c)78 d)84 (v) 35,37,39,41,43,45,47,49,51,53,55

a) 47 b)49 c)43 d)45 Note: Try to solve the above questions by using the ru

27 and 28.

Answers i . b i i . a in. c iv. a v . d

Rule 42 If the average of thefirst and the second of three numbers 'x' more or less than the average of the second and t third of these numbers, then the differene between theft, and the third of these three numbers is given by '2x'. Note: Here only 2 numbers (ie first and second or seco

and third) are involved in calculating average, thei fore, we m u l t i p l y * by 2. I f ' n ' numbers are involve for getting answer, we mult iply x by n.

Illustrative Example Ex.: The average o f the first and the second o f three nu

bers is 10 more than the average o f the second a the third o f these numbers. What is the differen between the first and the third o f these three nu; bers?

Soln: Detail Method: Average o f the first and the second numbers

First + Second = - and

2

Average o f the second and the third numbers

Second + Third

According to the question,

First + Second Second + Third = 10

2 2 .-. F i r s t - T h i r d = 20 Quicker Method: Apply ing the above rule, we c get

the required answer=2 x 10 = 20.

Exercise 1. The average o f the first and the second o f three nu

bers is 15 more than the average o f the second and t third o f these numbers. What is the difference betwe the first and the third o f these three numbers?

[ S B I P O E x a m , 2 0 ( a) 15 b)45 c)60 d)30

PRACTICE BOOK ON QUICKER MATHS

The average o f the first and the second o f three num-bers is 12 more than the average o f the second and the third o f these numbers. What is the difference between the first and the third o f these three numbers? a) 24 b)10 c)12 d) Data inadequate The average o f the first i n d the second o f three num-bers is 16 more than the average o f the second and the third o f these numbers. What is the difference between the first and the third o f these three numbers? a) 32 b)48 c)61 d) 16 The average o f the first and the second o f three num-bers is 13 more than the average o f the second and the third o f these numbers. What is the difference between the first and the third o f these three numbers? a) 25 b)24 c)26 d) 19 The average o f Suresh's marks in English and History is 55. His average o f marks in English and Science is 65. What is the difference between the marks which he ob-tained in History and Science?

[Bank of Baroda P O 1999] a) 40 b)60 c)20 d) Data inadequate The average marks scored by Ganesh in English, Sci-ence, Mathematics and History is less than 15 from that scored by h im in English, History, Geography and Math-ematics. What is the difference o f marks in Science and Geography scored by him? [ B S R B Chennai P O 2000] a) 40 b)50 c)60 d) Data inadequate The average temperature for Monday, Tuesday and Wednesday was 4 0 C. The average for Tuesday, Wednesday and Thursday was 41 C. That for Thurs-day being 42 C, what was the temperature on Monday? a ) 3 9 C b ) 4 5 C c ) 4 4 C d ) 4 0 C The average temperature for Monday, Tuesday and Wednesday was 4 0 C. The average for Tuesday, Wednesday and Thursday was 41 G and that o f Thurs-day being 45 C. What was the temperature on M o n -day? a ) 4 8 C b ) 4 1 C c )46 d ) 4 2 C The mean temperature o f Monday to Wednesday was 37 C and that o f Tuesday to Thursday was 34 C. I f the

temperature on Thursday was th that o f Monday,

what was the temperature on Thursday? a) 34 b ) 3 5 . 5 C c ) 3 6 C d ) 3 6 . 5 C

swers 2. a 3. a 4. c

;; Hint: Here, we can say that the average o f Suresh's marks in English and History is 10 less than the average marks in English and Science. (65 - 55 = 10) and now apply the above rule.

; Hint: Here four subjects are involved ie English, Sci-

ence, Mathematics and History in first group and En-glish History, Geography and Mathematics in second group. We observe that English, Mathematics and His-tory are common to both the groups. Hence the differ-ence o f marks in Science and Geography is given by 4 x 15 = 60. [Also see note.]

7. a; Hint: Average temperature for Monday, Tuesday and Wednesday was 1 C less (41 C - 40 C) than the aver-age temperature o f Tuesday, Wednesday and Thurs-day. Therefore, difference o f temperature between Thurs-day and Monday is given by 3 * 1 (By the rule) = 3 C. Temperature o f Thursday is 42 C given. Hence, tem-perature o f Monday is (42 - 3 =) 39 C.

8. d 9. c; Hint: Monday to Wednesday = Monday, Tuesday,

Wednesday Tuesday to Thursday = Tuesday, Wednesday, Thurs-day Difference o f temperature between Monday and Thurs-day = 3 x (37 - 34) = 9 C. According to the question,

4x n x - 9 [where, x = temperature o f Monday]

.'. x = 45

.-. Temperature o f Monday = 45 C and

4 temperature o f Thursday = x45 = 36 C

Rule 43 If average of 'n' consecutive odd numbers is 'x', then the difference between the smallest and the largest numbers is given by2(n-l). Note: We see that the above formula is independent o f x. That means, this formula always holds good irrespective o f the value of* .

Illustrative Example E x : I f average o f 7 consecutive numbers is 2 1 , what is the

difference between the smallest and the largest num-bers?

Soln: App ly ing the above rule, we have

the required answer = 2 (7 1) = 12.

Exercise 1. I f average o f 6 consecutive numbers is 48, what is the

difference between the smallest and the largest num-bers? [NABARD,1999] a) 10 b) 12 c)9 d) Data inadequate

2. I f average o f 8 consecutive numbers is 64, what is the difference between the smallest and the largest num-bers? a) 12 b)16 c)14 d) 18

3. I f average o f 15 consecutive numbers is 32, what is the

Average

difference between the smallest and the largest num-bers? a)28 b) 18 c)26 d)24

4. I f average o f 18 consecutive numbers is 45, what is the difference between the smallest and the largest num-bers?

a) 36 b)34 c)35 d)37

Answers l . a 2.c 3.a 4 .b

Miscellaneous 1. The average attendance o f a college for the first three

days o f a week is 325, and for first four days it is 320. How many were present on the fourth day? a) 305 b)350 c)530 d)503

2. A car runs for t\s at vj km/hr, t2 hours at v 2 km/

hr. What is the average speed o f the car for the entire journey?

a)

c)

+h V,fj + v2t2

v , f 2 + v 2 r ,

v l + v 2

km/hr

km/hr

V]?i + V2t2 b) ~ ~ km/hr

V! + V 2 d) T - " ~~~ km/hr

' V;?] + v-,t-2'2

3. A car runs x km at an average speed o f v, km/hr and y

km at an average speed o f v 2 km/hr. What is the average

speed o f the car for the entire journey?

V ! V 2 ( * + y ) a) , km/hr

x y ( v | + v 2 ) c) , .. km/hr

xv2 + yvl b) 77777T777\r

XV] + yv2 d) y . . . .. \r xvx + yv2 _ / xy(v} + v 2 )

4. A n aeroplane covers the four sides o f square field at speeds of200,400,600 and 800 km/hr. Then the average speed o f the plane in the entire journey is a) 600 km/hr b) 400 km/hr c) 500 km/hr d) 384 km/hr

5. The average age o f the three boys is 15 years. Their ages are in the ratio 3:5:7. Then the age o f the oldest is

[SBIPO Exam, 1987] a) 7 years b) 14 years c) 20 years d) 21 years

6. The population o f a town increased by 20% during the first year, by 25% during the next year and by 44% dur-ing the third year. Find the average rate o f increase dur-ing 3 years. a) 36.87% b) 37.68% c) 38.67% d) None o f these

1 7. A n investor earns 3% return on th o f his capital, 5%

on rd and 1 1 % on the remainder. What is the average

rate o f return he earns on his total capital? a) 5% b) 10% c)5.5% d) 10.5/

8. Out o f three given numbers, the first one is twi second and three times the third. I f the average o numbers is 88, then the difference between first an is . a) 48 b)72 c)96 d)32

9. The average o f 8 readings is 24.3, out o f which t h age o f first two is 18.5 and that o f next three is 21.2 sixth reading is 3 less than seventh and 8 les eighth, what is the sixth reading? a) 24.8 b)26.5 c)27.6 d)29.4

10. The average age o f a family o f 6 members is 22 y< the age o f the youngest member be 7 years, the a age o f the family at the birth o f the youngest m was [Railway; a) 15 years b) 17 years c) 17.5 years d) 18 ye

11. The average age o f a husband and wife was 2 : when they were married 5 years ago. The average the husband, the wife and a child who was born the interval, is 20 years now. How old is the chile a) 9 months b) 1 year c) 3 years d) 4 yes

12. 5 years ago, the average age o f A , B, C and D v With E jo in ing them now, the average age o f all t is 49 years. How old is E? a) 25 years b) 40 years c) 45 years d) 64 ye

13. 5 years ago, the average o f Ram and Shyam's ag 20 years. Now, the average age o f Ram, Shya Mohan is 30 years. What w i l l be Mohan's age I I hence? [LIC a) 45 years b) 50 years c) 49 years d) 60 yt

14. The average height o f 40 students is 163 cm. Or ticular day, three students A , B , C were absent i average o f the remaining 37 students was foun< 162 cm. I f A , B have equal heights and the height i 2 cm less than that o f A , find the height o f A .

[LI< a) 176cm b) 166cm c) 180cm d)186c

15. Out o f three numbers, the first is twice the seconc half o f the third. I f the average o f the three num 56, the three numbers in order are:

[Central Excise & I. Ta: a) 48,96,24 b) 48,24,96 c) 96,24,48 d)96,4!

16. O f the three numbers, second is twice the first also thrice the third. I f the average o f the three ni is 44, the largest number is: [Railway! a)24 b)36 c)72 d) 108

17. The sum o f three numbers is 98. I f the ratio betwe and second be 2 : 3 and that between second an be 5 : 8, then the second number is: [SSC E x a n a) 30 b)20 c)58 d)48

18. The average weight o f 3 men A , B and C is 84 \ other man D joins the group and the average n

PRACTICE BOOK ON QUICKER MATHS

jmes 80 kg. I f another man E, whose weight is 3 kg lore than that o f D , replaces A , then average weight o f , C, D and E becomes 79 kg. The weight o f A is:

[Bank P O 1989] 170 kg b ) 7 2 k g c ) 7 5 k g d ) 8 0 k g he average age o f A , B , C, D 5 years ago was 45 years, y including X , the present average o f all the five is 49 :ars. The present age o f X is: [Bank P O , 1988] 164 years b) 48 years c) 45 years d) 40 years he average age o f A and B is 20 years. I f C were to :place A , the average would be 19 and i f C were to place B , the average would be 2 1 . What are the ages o f , B a n d C ? [MBA 1982] 122,18,20 b) 18,22,20 c) 22,20,18 d) 18,20,22

/ers Required answer = 320 * 4 - 325 x 3 = 305.

Distance covered in f, hours = /,v, km.

Distance covered in t2 hours = t2v2 km

Total distance = txvx +t2v2

Total time = tx + 1 2

: . Average speed = km/hr

6.c;

.-. Total o f their ages = 3x + 5x + 7x = 3 * 15 or, 15x = 45 => x = 3 .-. The age o f the oldest = 7x = 21 years. Let the init ial population be 100 Population after the first year = 100 Ki .20 = 120 Population after the second year = 120 * 1.25 = 150 Population after the third year = 150 x 1.44 = 216 Net increase = 216 - 100 = 116 Net per cent increase during 3 years

116

100 -x 100 = 116%

116, Net per cent increase per year = % = 38.67%

7. a; Remainder capital

Total return

U 3 = i 3 + 8 - l 1 1 - 1

12 12 ~ 12

= 3x + 5 x + l l x 4 3 12

9 + 40 + 11 _ 60 _ 5 ~12 "72 ~

.-. Average per cent return = 5%

Time taken in the first journey = hours 8. c; Let first x. Then, second = and third 2

x

3

y Time taken in the second journey = hours

:.x ++ = 3 x 8 8 2 3

Total distance = (x + y) k m

Total time =

( \ x x + hours

11* or, = 264

or,x= 2 6 4 x 6

11 144

.-. Average speed = Distance

Time

x + y

\l V 2 ;

km/hr x v 2 + y v t

Let one side o f the square be x km. Then, the total distance=4x km.

Total t i m e - ^ + ^ + ^ + ^ = f6 hours

4 x x 9 6 .-. Average speed = = 384 km/hr

Let their ages be 3x, 5x and 7x

F i r s t - T h i r d = 1 4 4 - - x l 4 4 ] = 96

9. c; Let 6th reading=x. Then, 7th = (x + 3) and 8th = (x + 8) .-. 2 x 18.5 + 3 x21.2 + x + (x + 3) + ( x + 8 ) = 8x24.3 or,37 + 63.6 + 3 x + 1 1 = 194.4 or,x = 27.6

10. d; Total present age o f the family = (6 x 22) = 132 years

Total age o f the family 7 years ago = (132 - 7 * 6) = 90 years A t that time, the number o f members = 5.

.. Average age at that time = 90 ^ j years = 18 years.

11. d; Present total age o f husband and wife

= ( 2 x 2 3 + 2 x 5 ) = 56 years.

Average

Present total age o f husband, wife and child = 3 x 20 = 60 years.

Present age o f chi ld = (60 - 56) = 4 years.

12. c; 5 years ago, ( A + B + C + D ) = (45 x 4) years = 180 years.

Now, ( A + B + C + D) = (180+4 x 5) years=200 years. Now, (A + B + C + D + E ) = ( 5 x 49) years = 245 years. .-. Age o f E now = ( 2 4 5 - 2 0 0 ) = 45 years.

13. b; Total age o f Ram and Shyam 5 years ago = (2 x 20) = 40 years

.-. Total age o f Ram and Shyam now = (40 + 5 + 5) = 50 years.

Total age o f Ram, Shyam and Mohan now = (3 x 30) = 90 years.

Mohan's age now = (90 - 50) years = 40 years. Mohan's age 10 years hence = (40 + 10) years

= 50 years. 14. a; Let the heights o f A , B , and C be x cm, x cm and (x - 2)

cm Then, x + x + (x - 2) = (163 x 40 - 1 6 2 x 37). .-. x = 176cm

15. b; Let the numbers be 2x, x and 4x.

2x + x + 4x _ Ix Average = r

3 x 5 6 .-. x = = 24

Hence, the numbers in order are 48,24 and 96.

16. c; Let the numbers be x , 2 x and y * .

= 56

Average = x + 2x + x , .

3 1 Ix -44

17. a;

18. c;

19. c;

20. a;

4 4 x 9 x = = 36

11

So, the numbers are 36,72 and 24. Hence, the largest one is 72. Let the numbers be x, y and z. Then,

x + y + z = 98,

2v x = and z-

y 5 3 a n d 7 " 8

5

S o , f + v + f = 98 or, 49v 15 98 o r y = 3 Weight o f D = (80 x 4 - 84 x 3) kg = 68 kg Weight o f E = (68 + 3) kg = 71 kg (B + C + D + E) 's weight = (79 x 4) kg = 316 kg .-. (B + C)'s weight = [316 - (68 + 71)] kg = 1T, Hence, A's weight = [(84 x 3) - 177] kg = 75 kj Total age o f A , B , C, D 5 years ago = (45 x 4 ) ;

= 180yeai Total present age o f A , B , C, D and X = (49 x 5

= 245 ye Present age o f A , B , C and D = (180 + 5 x 4) y

= 200 years. .-. Present age o f X = 45 years. Say, a, b, c are the ages o f A , B , and C

a + b = 2 x 2 0 = 4 0 + b + c = 2 x 1 9 = 38 + c + a = 2 x 2 1 = 4 2

_+ a + b + c = 60 [Adding all the 3 eqi

b + c = 38 a = 22 a + b b

40 18

and c = .-. Age o f A = 22 years

Age o f B = 18 years Age o f C = 20 years

20