Average

1.The average of marks obtained by 120 candidates in a certain examination is 35. If the average marks of passed candidates is 39 and that of the failed candidates is 15, what is the number of candidates who passed examination?

(a) 100

(b) 200

(c) 300

(d) 400

Solution: (a)
Let the number of passed candidates be x.
Then total marks =120×25=39x+(120-x)×15 
or, 4200=39x+1800-15x
or, 24x=2400
∴ x=100

∴ number of passed candidates = 100

2.A constant distance from A to B is covered by a man at 40 km/hr. The person rides back the same distance 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

Solution: (c) 
By the formula:
2xy/(x+y) km/hr=2*40*30/40+30


=34.29 

3.A person divides his total route of journey into three equal parts and decides to travel the three parts with speeds of 40, 30 and 15 km/hr respectively. Find his average speed during the whole journey. 

(a)  22

(b)  24

(c)  34

(d)  44

Solution: (b)
By  Formula:-
3xyz/(xy+yz+xz) 
Average speed =(3×40×30×15)/(40×30+30×15+40×15)
=(3×40×30×15)/2250=24 km/hr. 

4.There were 35 students in a hostel. If the number of students increases by 7, the expenses of the mess increase by Rs. 42 per day while the average expenditure per head diminishes by Re 1. Find the original expenditure of the mess. 

(a) 110 

(b) 220

(c) 320

(d) 420

Solution: (d)
Suppose the average expenditure was Rs. x. Then total expenditure =35x.
When 7 more students join the mess, total expenditure =35x+42 
Now, the average expenditure =(35x+42)/42=x-1
or, 35x+42=42x-42  
or, 7x=84∴ x=12

Thus the original expenditure of the mess=35×12= Rs. 420 

5.There were 40 students in a hostel. If the number of students increases by 8, the expenses of the mess increase by Rs. 48 per day while the average expenditure per head diminishes by Rs. 2. Find the original expenditure of the mess. 

(a) Rs. 620  

(b) Rs. 720 

(c) Rs. 750 

(d) Rs. 820 

Solution: (b)
Suppose the average expenditure was Rs. x. Then total expenditure =40x.
When 7 more students join the mess, total expenditure =40x+48 
Now, the average expenditure =(40x+48)/48=x-2
or, 40x+48=48x-96  
or, 8x=144∴ x=18

Thus the original expenditure of the mess=40×18= Rs. 720

6.The average weight of a group of 15 boys was calculated to be 60 kg and it was later discovered that one weight was misread as 24 kg instead of the correct one of 42 kg. The correct average weight is?

(a) 60.2 kg

(b) 61.2 kg

(c) 62 kg

(d) 61 kg
Solution: (b)
=15*60-24+42/15= (900+18)/15

=61.2 

7.The average of Suresh’s marks in English and History is 55. His average of marks in English and Science is 65. What is the difference between the marks which he obtained in History and Science?

(a) 40 

(b) 60

(c) 20

(d) Data inadequate 

Solution : (c) 
E+H= 55*2=110
E+S= 65*2= 130

= 130-110 = 20

8.The population of a town increased by 20% during the first year, by 25% during the next year and by 44% during the third year. Find the average rate of increase during 3 years. 

(a) 36.87% 

(b) 37.68%

(c) 38.67%

(d) None of these

Solution: (c) 
Let the initial population be 100
Population after the first year =100×1.20=120 
Population after the second year =120×1.25=150
Population after the third year =150×1.44=216
Net increase =216-100=116
Net per cent increase during 3 years =116/100×100=116%
Net per cent increase per year =116/100×100=116%

Net increase= 116%/3 = 38.67%

9.The average age of a husband and wife was 23 years when they were married 5 years ago. The average age of the husband, the wife and a child who was born during the interval, is 20 years now. How old is the child now?

(a) 9 months 

(b) 1 year

(c) 3 years

(d) 4 years

Solution: (d)
Present total age of husband and wife =(2×23+2×5)=56 years
Present total age of husband, wife and child =3×20=60 years.

Present age of child =(60-56)=4 years.

10.The average height of 40 students is 163 cm. On a particular day, three students A, B, C were absent and the average of the remaining 37 students was found to be 162 cm. If A, B have equal heights and the height of C be 2 cm less than that of A, find the height of A.

(a) 176 cm 

(b) 166 cm

(c) 180 cm

(d) 186 cm

Solution: (a)
Let the height of A, B and C be x cm, x cm and (x-2) cm 
Then, x+x+(x-2)=(163×40-162×37). 

∴x=176 cm

Problems on Average - Solved Examples

21. The average age of boys in a class is 16 years and that of the girls is 15 years. What is the average age for the whole class?
A. 15	B. 16
C. 15.5	D. Insufficient Data

answer with explanation

Answer: Option D
Explanation:
We do not have the number of boys and girls. Hence we cannot find out the answer.

22. The average age of 36 students in a group is 14 years. When teacher's age is included to it, the average increases by one. Find out the teacher's age in years?
A. 51 years	B. 49 years
C. 53 years	D. 50 years

answer with explanation

Answer: Option A
Explanation:
average age of 36 students in a group is 14
Sum of the ages of 36 students = 36 × 14

When teacher's age is included to it, the average increases by one
=> average = 15
Sum of the ages of 36 students and the teacher = 37 × 15

Hence teachers age
= 37 × 15 - 36 × 14
= 37 × 15 - 14(37 - 1)
= 37 × 15 - 37 × 14 + 14
= 37(15 - 14) + 14
= 37 + 14
= 51

23. The average of five numbers id 27. If one number is excluded, the average becomes 25. What is the excluded number?
A. 30	B. 40
C. 32.5	D. 35

answer with explanation

Answer: Option D
Explanation:
Sum of 5 numbers = 5 × 27
Sum of 4 numbers after excluding one number = 4 × 25

Excluded number
= 5 × 27 - 4 × 25
= 135 - 100 = 35

24. The batting average for 40 innings of a cricket player is 50 runs. His highest score exceeds his lowest score by 172 runs. If these two innings are excluded, the average of the remaining 38 innings is 48 runs. Find out the highest score of the player.
A. 150	B. 174
C. 180	D. 166

answer with explanation

Answer: Option B
Explanation:
Total runs scored by the player in 40 innings = 40 × 50
Total runs scored by the player in 38 innings after excluding two innings = 38 × 48
Sum of the scores of the excluded innings = 40 × 50 - 38 × 48 = 2000 - 1824 = 176

Given that the scores of the excluded innings differ by 172. Hence let's take
the highest score as x + 172 and lowest score as x

Now x + 172 + x = 176
=> 2x = 4
=> x $=\frac{4}{2}$ = 2

Highest score = x + 172 = 2 + 172 = 174

25. The average score of a cricketer for ten matches is 38.9 runs. If the average for the first six matches is 42, what is the average for the last four matches?
A. 34.25	B. 36.4
C. 40.2	D. 32.25

answer with explanation

Answer: Option A
Explanation:
Total runs scored in 10 matches = 10 × 38.9

Total runs scored in first 6 matches = 6 × 42

Total runs scored in the last 4 matches = 10 × 38.9 - 6 × 42

Average of the runs scored in the last 4 matches = $\frac{10\times 38.9-6\times 42}{4}$
$=\frac{389-252}{4}=\frac{137}{4}=34.25$

26. The average of six numbers is x and the average of three of these is y. If the average of the remaining three is z, then
A. None of these	B. x = y + z
C. 2x = y + z	D. x = 2y + 2z

answer with explanation

Answer: Option C
Explanation:
Average of 6 numbers = x
=> Sum of 6 numbers = 6x

Average of the 3 numbers = y
=> Sum of these 3 numbers = 3y

Average of the remaining 3 numbers = z
=> Sum of the remaining 3 numbers = 3z

Now we know that 6x = 3y + 3z
=> 2x = y + z

27. Suresh drives his car to a place 150 km away at an average speed of 50 km/hr and returns at 30 km/hr. What is his average speed for the whole journey ?
A. 32.5 km/hr.	B. 35 km/hr.
C. 37.5 km/hr	D. 40 km/hr

answer with explanation

Answer: Option C
Explanation:
-------------------------------------------
Solution 1
--------------------------------------------
If a car covers a certain distance at x kmph and an equal distance at y kmph. Then,
average speed of the whole journey = $\frac{2xy}{x+y}$ kmph.

Therefore, average speed
$=\frac{2\times 50\times 30}{50+30}=\frac{2\times 50\times 30}{80}=\frac{2\times 50\times 3}{8}=\frac{50\times 3}{4}=\frac{25\times 3}{2}=\frac{75}{2}=37.5$
-------------------------------------------
Solution 2
--------------------------------------------
Though it is a good idea to solve the problems quickly using formulas, you should know the fundamentals too. Let's see how we can solve this problems using basics.

Total time taken for traveling one side = $\frac{\text{distance}}{\text{speed}}=\frac{150}{50}$
Total time taken for return journey = $\frac{\text{distance}}{\text{speed}}=\frac{150}{30}$
Total time taken = $\frac{150}{50}+\frac{150}{30}$

Total distance travelled $=150+150=2\times 150$

Average speed = $\frac{\text{Total distance traveled}}{\text{Total time taken}}$

$=\frac{2\times 150}{\frac{150}{50}+\frac{150}{30}}=\frac{2}{\frac{1}{50}+\frac{1}{30}}=\frac{2\times 50\times 30}{30+50}=\frac{2\times 50\times 30}{80}=\frac{2\times 50\times 3}{8}=\frac{50\times 3}{4}=\frac{25\times 3}{2}=\frac{75}{2}=37.5$

28. The average age of a husband and his wife was 23 years at the time of their marriage. After five years they have a one year old child. What is the average age of the family ?
A. 21 years	B. 20 years
C. 18 years	D. 19 years

answer with explanation

Answer: Option D
Explanation:
Total age of husband and wife (at the time of their marriage) = 2 × 23 = 46

Total age of husband and wife after 5 years + Age of the 1 year old child
= 46 + 5 + 5 + 1 = 57

Average age of the family = $\frac{57}{3}$ = 19

29. In an examination, a student's average marks were 63. If he had obtained 20 more marks for his Geography and 2 more marks for his history, his average would have been 65. How many subjects were there in the examination?
A. 12	B. 11
C. 13	D. 14

answer with explanation

Answer: Option B
Explanation:
Let the number of subjects = x
Then, total marks he scored for all subjects = 63x

If he had obtained 20 more marks for his Geography and 2 more marks for his history, his average would have been 65
=> Total marks he would have scored for all subjects = 65x

Now we can form the equation as 65x - 63x = additional marks of the student = 20 + 2 = 22
=> 2x = 22
=> x = $\frac{22}{2}$ = 11

30. The average salary of all the workers in a workshop is Rs.8000. The average salary of 7 technicians is Rs.12000 and the average salary of the rest is Rs.6000. How many workers are there in the workshop?
A. 21	B. 22
C. 23	D. 24

answer with explanation

Answer: Option A
Explanation:
Let the number of workers = x

Given that average salary of all the workers = Rs.8000
Then, total salary of all workers = 8000x

Given that average salary of 7 technicians is Rs.12000
=> Total salary of 7 technicians = 7 × 12000 = 84000

Count of the rest of the employees = (x - 7)
Average salary of the rest of the employees = Rs.6000
Total salary of the rest of the employees = (x - 7)(6000)

8000x = 84000 + (x - 7)(6000)
=> 8x = 84 + (x - 7)(6)
=> 8x = 84 + 6x - 42
=> 2x = 42
=> x = $\frac{42}{2}$ = 2
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