Problems on Average - Solved Examples

Problems on Average - Solved Examples

1. In the first 10 overs of a cricket game, the run rate was only 3.2. What should be the run rate in the remaining 40 overs to reach the target of 282 runs?
A. 6.25	B. 5.5
C. 7.4	D. 5

answer with explanation

Answer: Option A
Explanation:
Runs scored in the first 10 overs = 10 × 3.2 = 32
Total runs = 282

Remaining runs to be scored = 282 - 32 = 250
Remaining overs = 40

Run rate needed = $\frac{250}{40}=6.25$

2. A grocer has a sale of Rs. 6435, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. How much sale must he have in the sixth month so that he gets an average sale of Rs. 6500?
A. 4800	B. 4991
C. 5004	D. 5000

answer with explanation

Answer: Option B
Explanation:
Let the sale in the sixth month $=x$

Then $\frac{6435+6927+6855+7230+6562+x}{6}$ $=6500$

=> $6435+6927+6855+7230+6562+x$ $=6\times 6500$

=> $34009+x=39000$
=> $x=39000-34009=4991$

3. The average of 20 numbers is zero. Of them, How many of them may be greater than zero, at the most?
A. 1	B. 20
C. 0	D. 19

answer with explanation

Answer: Option D
Explanation:
Average of 20 numbers = 0

=> $\frac{\text{Sum of 20 numbers}}{20}=0$

=> Sum of 20 numbers = 0

Hence at the most, there can be 19 positive numbers.
(Such that if the sum of these 19 positive numbers is x, 20th number will be -x)

4. The captain of a cricket team of 11 members is 26 years old and the wicket keeper is 3 years older. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team. Find out the average age of the team.
A. 23 years	B. 20 years
C. 24 years	D. 21 years

answer with explanation

Answer: Option A
Explanation:
Number of members in the team = 11

Let the average age of of the team = $x$

=> $\frac{\text{Sum of ages of all 11 members}}{11}=x$

=> Sum of the ages of all 11 members = $11x$

Age of the captain = 26
Age of the wicket keeper = 26 + 3 = 29

Sum of the ages of 9 members of the team excluding captain and wicket keeper
$=11x-26-29=11x-55$

Average age of 9 members of the team excluding captain and wicket keeper
$=\frac{11x-55}{9}$

Given that $\frac{11x-55}{9}=(x-1)$

$\Rightarrow 11x-55=9(x-1)\Rightarrow 11x-55=9x-9\Rightarrow 2x=46\Rightarrow x=\frac{46}{2}=23\text{years}$

5. The average monthly income of A and B is Rs. 5050. The average monthly income of B and C is Rs. 6250 and the average monthly income of A and C is Rs. 5200. What is the monthly income of A?
A. 2000	B. 3000
C. 4000	D. 5000

answer with explanation

Answer: Option C
Explanation:
Let monthly income of A = a
monthly income of B = b
monthly income of C = c

a + b = 2 × 5050 .... (Equation 1)
b + c = 2 × 6250 .... (Equation 2)
a + c = 2 × 5200 .... (Equation 3)

(Equation 1) + (Equation 3) - (Equation 2)
=> a + b + a + c - (b + c) = (2 × 5050) + (2 × 5200) - (2 × 6250)
=> 2a = 2(5050 + 5200 - 6250)
=> a = 4000

i.e., Monthly income of A = 4000

6. A car owner buys diesel at Rs.7.50, Rs. 8 and Rs. 8.50 per litre for three successive years. What approximately is the average cost per litre of diesel if he spends Rs. 4000 each year?
A. Rs. 8	B. Rs. 7.98
C. Rs. 6.2	D. Rs. 8.1

answer with explanation

Answer: Option B
Explanation:
Total Cost = 4000 × 3

Total diesel used = $\frac{4000}{7.5}+\frac{4000}{8}+\frac{4000}{8.5}$

Average cost per litre of diesel
$=\frac{4000\times 3}{(\frac{4000}{7.5}+\frac{4000}{8}+\frac{4000}{8.5})}=\frac{3}{(\frac{1}{7.5}+\frac{1}{8}+\frac{1}{8.5})}$

It is important how you proceed from this stage. Remember time is very important and if we solve this in the normal method, it may take lot of time. Instead, we can find out the approximate value easily and select the right answer from the given choices.

In this case, answer
$=\frac{3}{(\frac{1}{7.5}+\frac{1}{8}+\frac{1}{8.5})}\approx \frac{3}{(\frac{1}{8}+\frac{1}{8}+\frac{1}{8})}\approx \frac{3}{\left(\frac{3}{8}\right)}\approx 8$

We got that answer as approximately equal to 8. From the given choices, the answer can be 8 or 7.98 or 8.1 . But which one from these?

It is easy to figure out. We approximated the denominator,$(\frac{1}{7.5}+\frac{1}{8}+\frac{1}{8.5})$ to $\frac{3}{8}$. However
$\frac{1}{7.5}+\frac{1}{8.5}=\frac{1}{8-0.5}+\frac{1}{8+0.5}=\frac{8+0.5+8-0.5}{(8-0.5)(8+0.5)}=\frac{16}{(8^2-{0.5}^2)}[\because (a-b\left)\right(a+b)=a^2-b^2]=\frac{16}{64-0.25}$

i.e., $\frac{1}{7.5}+\frac{1}{8.5}=\frac{16}{(64-0.25)}$
We know that $\frac{1}{8}+\frac{1}{8}=\frac{1}{4}=\frac{16}{64}$

=> $\frac{1}{7.5}+\frac{1}{8.5}>\frac{1}{8}+\frac{1}{8}$

Early we had approximated the denominator to $\frac{3}{8}$. However, from the above mentioned equations, now we know that actual denominator is slightly greater than $\frac{3}{8}$. It means answer is slightly lesser than 8. Hence we can pick the choice 7.98 as the answer

Try to remember such relations between numbers which can save lot of time in calculations.

7. In Kiran's opinion, his weight is greater than 65 kg but less than 72 kg. His brother does not agree with Kiran and he thinks that Kiran's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Kiran?
A. 70 kg	B. 69 kg
C. 61 kg	D. 67 kg

answer with explanation

Answer: Option D
Explanation:
Let Kiran's weight = x. Then

According to Kiran, 65 < x < 72 ----(equation 1)

According to brother, 60 < x < 70 ----(equation 2)

According to mother, x $\le$ 68 ----(equation 3)

Given that equation 1,equation 2 and equation 3 are correct. By combining these equations,we can write as

$65<x\le 68$

i.e., x = 66 or 67 or 68

Average of different probable weights of Kiran = $\frac{66+67+68}{3}=67$

8. The average weight of 16 boys in a class is 50.25 kg and that of the remaining 8 boys is 45.15 kg. Find the average weights of all the boys in the class.
A. 48.55	B. 42.25
C. 50	D. 51.25

answer with explanation

Answer: Option A
Explanation:
Average weight of 16 boys = 50.25
Total weight of 16 boys = 50.25 × 16

Average weight of remaining 8 boys = 45.15
Total weight of remaining 8 boys = 45.15 × 8

Total weight of all boys in the class = (50.25 × 16)+ (45.15 × 8)
Total boys = 16 + 8 = 24

Average weight of all the boys = $\frac{(50.25\times 16)+(45.15\times 8)}{24}$
$=\frac{(50.25\times 2)+(45.15\times 1)}{3}=(16.75\times 2)+15.05=33.5+15.05=48.55$

9. A library has an average of 510 visitors on Sundays and 240 on other days. What is the average number of visitors per day in a month of 30 days beginning with a Sunday?
A. 290	B. 304
C. 285	D. 270

answer with explanation

Answer: Option C
Explanation:
In a month of 30 days beginning with a Sunday, there will be 4 complete weeks and another two days which will be Sunday and Monday.

Hence there will be 5 Sundays and 25 other days in a month of 30 days beginning with a Sunday

Average visitors on Sundays = 510
Total visitors of 5 Sundays = 510 × 5

Average visitors on other days = 240
Total visitors of other 25 days = 240 × 25

Total visitors = (510 × 5) + (240 × 25)
Total days = 30

Average number of visitors per day
$=\frac{(510\times 5)+(240\times 25)}{30}=\frac{(51\times 5)+(24\times 25)}{3}=(17\times 5)+(8\times 25)=85+200=285$

10. A student's mark was wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by $\frac{1}{2}$. What is the number of students in the class?
A. 45	B. 40
C. 35	D. 30

answer with explanation

Answer: Option B
Explanation:
Let the total number of students = $x$

The average marks increased by $\frac{1}{2}$ due to an increase of 83 - 63 = 20 marks.

But total increase in the marks = $\frac{1}{2}\times x=\frac{x}{2}$

Hence we can write as
$\frac{x}{2}=20\Rightarrow x=20\times 2=40$

Important Formulas - Average

Important Formulas - Average

1. Average = $\frac{\text{Sum of observations}}{\text{Number of observations}}$

2. Average Speed

If a car covers a certain distance at $x$ kmph and an equal distance at $y$ kmph. Then, the average speed of the whole journey = $\frac{2xy}{x+y}$ kmph

QUIZ-12

Q1. Name the Indian-origin actor who will be felicitated with the Asia Society 2017 Asia Game Changers Award in New York.

(a) Kal Penn

(b) Dev Patel

(c) Aziz Ansari

(d) Padma Lakshmi

(e) M. Night Shyamalan

S1. Ans.(b)

Sol. Indian-origin actor Dev Patel is to be honoured by a leading educational organization for making a “transformative and positive” difference for the future of Asia and the world. London-born Dev Patel, who became popular with his debut film Slumdog Millionaire, will be felicitated with the Asia Society 2017 Asia Game Changers Award at a ceremony to be held at the United Nations, New York, in November 2017.

Q2. The Legislative Assembly of which state has recently passed a bill which reclassifies coconut as a `tree', ensuring legal protection for it?

(a) Kerala

(b) Karnataka

(c) Maharashtra

(d) Goa

(e) Andhra Pradesh

S2. Ans.(d)
Sol. The Goa Legislative Assembly passed a bill which reclassifies coconut as a `tree', ensuring legal protection for it. Goa Forward Party, a partner in the Manohar Parrikar-led government, had promised to reclassify coconut.

Q3. Name the Bollywood Celebrity who has been signed in as a brand ambassador by a health and wellness startup Cure.fit in a deal worth Rs 100 crore.

(a) Amir Khan

(b) Imran Khan

(c) Hrithik Roshan

(d) Abhishek Bachchan

(e) Ajay Devgan

S3. Ans.(c)
Sol. Hrithik Roshan has been signed as a brand ambassador by a health and wellness startup Cure.fit in a deal worth Rs 100 crore. The actor will be associated with the brand for five years.

Q4. The Nagasaki Day is observed across the world on _____________.

(a) 9 August

(b) 5 August

(c) 6 August

(d) 10 August

(e) 11 August

S4. Ans.(a)
Sol. On August 9, 1945, the second bomb—“Fat Man”—was dropped on Nagasaki (Japan), incinerating the city and its unexpecting inhabitants.

Q5. Name the e-commerce company that has launched 'Budding Star Program', an exclusive program aimed at its employees to help them achieve their dreams by providing them financial assistance and paid leave.

(a) Amazon

(b) Snapdeal

(c) Bechdo.com

(d) Yatra.com

(e) Flipkart

S5. Ans.(e)
Sol. Flipkart has launched 'Budding Star Programme', an exclusive programme aimed at Flipkart employees to help them achieve their dreams by providing them financial assistance and paid leave.

Q6. Name the Indian-American who has been appointed to a key position of Deputy Comptroller for Minority Affairs in New York's Nassau County.

(a) Latar Mangesh

(b) Dilip Chauhan

(c) Mehek Gulati

(d) Rahul Gupta

(e) Sameer Roman

S6. Ans.(b)
Sol. Indian American Dilip Chauhan has been appointed to a key position of Deputy Comptroller for Minority Affairs in New York's Nassau County.

Q7. India has recently celebrated the 75th anniversary of the Quit India Movement. What is the theme of this year?

(a) Sankalp Se Vriddhi

(b) Sankalp Se Parinam

(c) Sankalp Se Siddhi

(d) Sankalp Se Vikas

(e) Sankalp Se Kamaal

S7. Ans.(c)
Sol. India has celebrated the 75th anniversary of the Quit India Movement (on 9 August). This year’s theme is “SANKALP SE SIDDHI"- the Attainment through Resolve. Prime Minister Narendra Modi in his recent Mann Ki Baat program had called for launching the mega campaign Sankalp Se Siddhi.

Q8. Name the lender who has recently launched instant credit cards enabling its savings account customers to get a credit card instantly, in a completely digital and paperless manner.

(a) SBI

(b) HDFC Bank

(c) IDBI Bank

(d) ICICI Bank

(e) Axis Bank

S8. Ans.(d)
Sol. The country's largest private sector lender, ICICI Bank, launched instant credit cards enabling its savings account customers to get a credit card instantly, in a completely digital and paperless manner. The bank has opened up this offering to a few lakh of pre-qualified customers who can apply for the card online and generate card details and start shopping online instantly.

Q9. With which of the following digital payments company Private sector lender IDFC Bank has partnered to launch ‘IDFC Bank Benefits’?

(a) Zeta

(b) Beta

(c) Paytm

(d) MobiKwik

(e) FreeCharge

S9. Ans.(a)
Sol. Private sector lender IDFC Bank has partnered with digital payments company Zeta, to launch ‘IDFC Bank Benefits’. It is a payment solution for corporates that digitizes employee spends and claims, making the process simple, real-time and paperless.

Q10. Syndicate Bank has pared its marginal cost of funds based lending rate (MCLR) by 5-10 basis points in certain maturity buckets. Who is the Managing Director & CEO of Syndicate Bank?

(a) Rakesh Sethi

(b) Melwyn Rego

(c) Mahesh Kumar Jain

(d) Aditya Puri

(e) R S Pandey

S10. Ans.(b)
Sol. Syndicate Bank has pared its marginal cost of funds based lending rate (MCLR) by 5-10 basis points in certain maturity buckets. Mr Melwyn Rego is the Managing Director & CEO of Syndicate Bank.

Q11. Sri Lankan Navy has recently commissioned an India-built state-of-the-art AOPV to boost its search and rescue capabilities. What does 'P' stands for in AOPV?

(a) Petrol

(b) Premium

(c) Patrol

(d) Population

(e) Play

S11. Ans.(c)
Sol. Sri Lankan Navy has commissioned an India-built state-of-the-art Advanced Offshore Patrol Vessel (AOPV), the first-ever purchase from a foreign ship builder, to boost its search and rescue capabilities.

Q12. Private sector lender Axis Bank reduced the interest rate on savings bank accounts by 50 basis points to 3.5 per cent for deposits up to Rs 50 lakh. Who is the Present CEO of Axis Bank?

(a) Shikha Sharma

(b) Amit Gaba

(c) Rahul Saxena

(d) Chanda Kocchar

(e) Aditya Suri

S12. Ans.(a)
Sol. Private sector lender Axis Bank reduced the interest rate on savings bank accounts by 50 basis points to 3.5 per cent for deposits up to Rs 50 lakh. Shikha Sharma is the Present CEO of Axis Bank.

Q13. Sri Lankan President and Armed Forces commander-in-chief Maithripala Sirisena has recently commissioned the new patrol vessel in Sri Lanka. The Ship has been built at __________ in India.

(a) Mazagon Dock Ltd

(b) Goa Shipyard Ltd

(c) Cochin Shipyard Ltd

(d) Bharati Shipyard Ltd

(e) Pipavav Shipyard

S13. Ans.(b)
Sol. Sri Lankan President and Armed Forces commander-in-chief Maithripala Sirisena commissioned the new patrol vessel “SLNS Sayurala”. It is the largest vessel built by an Indian ship builder for a foreign navy to date. Built at the Goa Shipyard Ltd (GSL) in India, SLNS Sayurala is the first-ever brand new platform purchased from a foreign ship builder.

Q14. Name the Indo-Canadian entrepreneur who has been recently appointed Canada's consul general in San Francisco.

(a) Mahendra Kumar

(b) Yousuf Gilani

(c) Rana Sarkar

(d) Samvid Kumar

(e) Gajendra Verma

S14. Ans.(c)
Sol. Rana Sarkar, an Indo-Canadian entrepreneur, has been appointed Canada's consul general in San Francisco as well as a member of the Canadian official team tasked with renegotiating the North American Free Trade Agreement (NAFTA) with the US and Mexico.

Q15. Sri Lankan President and Armed Forces commander-in-chief Maithripala Sirisena has recently commissioned the new patrol vessel named ______________.

(a) SLNS Saryu

(b) SLNS Dvora

(c) SLNS Wave Rider 

(d) SLNS Sayurala

(e) SLNS Ranavijaya

S15. Ans.(d)
Sol. Sri Lankan President and Armed Forces commander-in-chief Maithripala Sirisena commissioned the new patrol vessel “SLNS Sayurala”. It is the largest vessel built by an Indian ship builder for a foreign navy to date.