[Aptitude] Averages: More than two elements added / removed : shortcut

technique explained

In the previous article, we saw how to approach the concepts for “Averages->

Entry/removal of One element”. (click me) Now extending that concept for

Entry/removal of multiple elements.

1. Situation: Removing 2 Elements

1. Case: highest score in (40/38) innings? (SSC CGL)

2. Case: Highest marks in (10 / 8) papers (CAT-1997)

3. Case: average price of (12/10) books (SSC CGL)

2. Situation: More than Two elements

1. Approach #1: Money problem

2. Approach #2: allegation method

3. Case: Average run rate (10/40) overs (MAT)

4. Case: Avg age of boys (18/30)

1. Approach: money problem

2. Approach: alligation

3. Mock questions

Situation: Removing 2 Elements

Case: highest score in (40/38) innings? (SSC CGL)

A Batsman played 40 innings with average score of 50 runs. His highest score

exceeds his lowest score by 172 runs. If these two innings are excluded, then

score average score of his remaining innings is 48 runs. What is the highest

score of this batsman? (SSC CGL 2011)

Convert this into a Money problem

1. Old gang has 40 guys and each of them got Rs.50

2. Two guys leave the gang. One of them has Rs.172 more than the other

guy.

3. Now Average of the old gang declined by 50-48=Rs.2

Ok now let‟s think about it.

Initially everyone has Rs.50 each.

So two guys who left the game, had total Rs.50 x 2 =Rs.100.

We are left with 40-2=38 people.

Two guys left the group. If they wanted to decrease the avg. of those 38

people by Rs.2. So they must have got away with 38 x 2 =Rs.76

So the two guys had total cash of Rs.100 (theirs) + Rs.76 (they stole from

others)

=100+76=176.

But the question says difference between highest and lowest is 172.

And now we know that total is 176.

Meaning lowest was 2 and highest was 174. (because 2 + 174 =176)**

Therefore, final answer=174 was the highest score.

**in case you‟re confused:

Suppose the two observations were “a” and “b” where (a>b).

i) total of those two observations is 176 (we found this via “money problem”

approach): a + b =176

ii) difference between them is 172 (already given in question): a-b =172

Solve these two equations and you get: 2a=348=> a=174

A similar question was asked in CAT-1997

Case: Highest marks in (10 / 8) papers (CAT-1997)

The average marks of a student in 10 papers are 80. If highest and lowest scores

are not considered, then the average score is 81. If the highest score is 92, then

what is the lowest score?

Convert this into a Money problem.

1. 10 kids have 80 rupees each.

2. 2 kids run away, so 8 kids remain. (10=8-2)

3. Average of remaining 8 kids is 81. (Meaning average increased by 1 (81-

80=1))

Ok, now let‟s think about it.

10 kids had 80 rupees each.

Meaning those two kids also had 80 each

So two kids originally had 80 x 2 =160 rupees.

They wanted to raise the average of remaining eight kids by 1.

So they (two kids) gave 1 rupee to each of those eight remaining kids.

Meaning two kids gave away Rs.1 x 8kids = 8 rupees.

So now two kids are left with 160 minus 8 =152 rupees.

But the question says highest of those kids was 92.

So the other kid had 152 minus 92 =Rs.60 that‟s our final answer.

Case: average price of (12/10) books (SSC CGL)

Average price of 10 books is Rs.12. Average price of 8 books is Rs.11.75.

Of the remaining two books, if one book has price 60% higher than the other

book, find the price of both books.

Convert this into a Money problem. (although it is already a money problem but

„men‟ are not included).

1. 10 guys had 12 rupees each.

2. 2 guys left. Average goes down by 25 paisa. (12 minus 11.75=0.25)

Ok now let‟s think about it.

Originally two guys had total 2 x 12 = 24 rupees.

They wanted to bring down the average of remaining eight guys by 0.25

paisa.

So two guys stole 25 paisa each from the remaining eight people.

Thus two guys stole: 8 x 0.25 =Rs.2

Now two guys have total rupees = Rs.24 (they originally had) PLUS Rs.

2 (they stole)

=24+2=26 rupees.

But the question says, price of one book is 60% higher than the other one.

Meaning cheap book is Rs”m” then expensive book is “1.6 times m”. **

**side concept

Expensive book is 60% higher than cheap book.

Therefore, price of expensive book

= 100% (of cheap book) + 60% of cheap book

=160% of cheap book

=1.6 of cheap book. (because %=1 upon 100 so 160%=160/100=1.6)

=1.6m

Anyways back to the main process: We found that total is Rs.26

Therefore Rs. 26 = Cheap book + expensive book

=1.0m + 1.6m

=2.6m

If 26=2.6m, then m=26/2.6=Rs.10

And if m=10 then expensive book=1.6 x 10 = Rs.16.

So final answer is

Price of those two books are Rs.10 and Rs.16

I hope the concept of “Averages: Entry/Removal of Two elements” is clear by

now.

Now let‟s check “more than two elements.”

Situation: More than Two elements

Raynolds produces average 4000 pens per month for the first three months.

How many pens should this company produce on an average for remaining

months to get yearly production average of 4375?

There are two ways to solve this problem

1. Convert in Money problem

2. Wine and Water mixture (alligation method)

Approach #1: Money problem

Convert this into a money problem

12 guys have avg. Rs.4375 each (because 1 years= 12 months)

9 guys leave the gang. (because we are given avg of 3 months=12-3=9)

3 guys remain and their average is Rs.4000 each. (meaning average

declined by 4375 minus 4000=375)

Find the average money held by those 9 guys.

Ok now let‟s think about it.

12 guys have avg Rs.4375 each.

Meaning originally those 9 guys had total Rs. 4375 x 9

They want to decrease the average of remaining 3 guys by Rs.375.

So they steal Rs. 375 x 3 rupees from those three guys.

Now total money with 9 guys = 4375 x 9 (they originally had) + 375 x 3

(they stole)

But the question is asking us to find the average money held by these 9 guys

=total money with 9 guys divided by 9

=(4375×9)+(375×3)/9

=Rs.4500

Even more shortcut!

Recall what we do with “Average: errors” concept.

New average = original average plus or minus average change.

In this case original average of 9 guys =Rs.4375

And they stole Rs.375 x 3 (=increase)

So what‟s the average increase= 375 x 3 / 9 =Rs.+125 (this is plus because they

gained money)

And new average = original average (Rs.4375) + average change (+125)

=4375+125

=Rs.4500

Approach #2: allegation method

If you don‟t know the allegation (visual move) technique, then go through my

old article (click me)

Company wants average yearly production of 4375 (this is our “middle” value)

For 3 months , they had average of 4000

That means for the remaining 9 months, average production must be higher than

4375. (let‟s assume this is “m”)

Make the table.

Avg. production 4000 4375 M

Months 3

9

That‟s it. Apply the visual move.

m-4375=3….eq1

4375-4000=9…eq2

Divide eq1 by 2 and solve it

You get m=4500 that‟s our final answer.

Case: Average run rate (10/40) overs (MAT)

In a cricket match, team Pakistan makes 281 runs. Now team India starts

batting.

In the first ten overs, we have (bogus) run rate of 3.2 only. If we want to

win this match, what must be the run rate for remaining 40 overs?

I suggest you try the money problem method here on your own.

Interestingly, we can solve this problem via alligiation (wine water

mixture) as well!

How?

Since Pakistan made 281 runs, we must make 282 runs to win the game.

And we want to do this in 50 overs (40+10)

So what should be our “overall” runrate= 282/50=5.64

For the first ten overs we had run rate of only 3.2. so for remaining 40

overs, we must make run rate higher than 5.64.

Run rate 3.2 5.64 M

Weight 10

40

That‟s it, just apply the visual move!

M-5.64=10….eq1

5.64-3.2=40…eq2

Divide equation 1 by equation 2

When you solve this equation, you get m=25/4=6.25

Final answer: India must get run rate of 6.25 in the remaining 40 overs, to win

the match.

Case: Avg age of boys (18/30)

A group of 30 boys have average age of 13. Later, 18 boys, with an average of

15 years left the group. What is the average age of remaining 12 boys?

Approach: money problem

1. 30 boys had Rs.13 each

2. 18 of them left, with Rs.15 each.

Let‟s think about it.

Originally those 18 boys had Rs.13 (when they were part of the gang)

Meaning remaining 12 boys also had 12 x 13 = 156 rupees.

Now 18 boys form a new gang with Rs.15 each. So, each of these 18 boys

need Rs.2 more (15-13=2). So total they need Rs.2 x 18 = Rs.36

They steal this money from those remaining 12 boys.

So how much money is left with those 12 boys? = 156 they originally had

MINUS Rs.36 stolen= Rs.120

And since we are asked “average” of those 12 boys

So we‟ll divide 120 by 12=Rs.10 is our final answer.

Approach: alligation

I‟m directly constructing the table

Avg age M 13 15

No of boys 12

18

Apply the visual move, you get two equations.

When you solve them, you get m=10.

Final answer: average age of remaining 12 boys is 10 years.

Mock questions

1. A class has 35 people with avg weight of 62 kilos. When we donot

consider the weight of teacher and class Secretary, the average goes down

to 61.5 kilos. If the weight of class Secretary is 60 kilos, how much does

the teacher weight?

2. A batsman made average 85 runs in 50 matches. When we donot consider

his last two matches, average falls down to 82 runs. If he made 150 runs

in his second last match, how many runs did he score in his last match?

3. Gokuldham Society has 30 families with average annual income of 8 lakh

rupees. If we ignore the families of Jethalal and Bhide, the average

declines by 1.25%. If Jetalal earns 740% higher than Bhide, what is the

annual income of Bhide?

4. In a bookstore, there are 100 books with average price of Rs.300. Two of

them are removed and average declines by 10 rupees. of the two books

that were removed, if the expensive book cost Rs.1330, find the price of

cheaper book.

1.5. In a mobile store, there are 200 handsets with an average MRP of

Rs.8000. When we ignore a Nokia Lumia and a Samsung Galaxy note2,

the average of remaining phones goes down to Rs.7850. If Nokia Lumia

costs Rs.9999, find the price of that Samsung phone.

6. Answers

1. 80.5kg

2. 164 runs

3. 2 lakh rupees. 1.25% decline in average = 0.0125×8 lakhs=10,000 rupees.

Meaning Jetha+Bhide had total money of [10k x 28 +

(2x8lakh)]=18.8lakhs. And Jetha+Bhide(B)=8.4B+1B=18.8lakhs. Hence

Bhide(B)=2 lakhs.

4. Rs.250

1.5. Rs.35701

[Aptitude] Averages: Entry / Removal of 1 element => Increase / Decrease

in Average Shortcut method explained

1. Situation: Entry of Element-Increase in Avg

1. Case: Avg age of Students + Teacher

1. Approach #1: convert into money problem

2. Approach #2: Formula method

2. Case: Avg salary workers+manager

3. Case: Avg weight students+teachers

4. Case: original average not given

2. Situation: Entry of 1 element – Decrease in Avg

1. Case: younger Kid joins: average goes down

2. Case: Average Runs decline (original not given)

3. Situation: Removal of 1 element -increase in avg

4. Situation: Removal of 1 element -decrease in avg

5. Mock questions

Situation: Entry of Element-Increase in Avg

Let‟s start with really simple case. In Gokuldham society, the average age of

four kids is 14 years. If Mr. Champak is included in this group, the average age

increases by 11 years. How old is Mr.Champak?

Convert this into money problem

Four kids got Rs.14 each.

Champak makes a new gang with rule, “everyone must have Rs.25 (bcoz

14+11=25)”

But those kids got only Rs.14. so how much cash should Champak bring,

to form this new gang?

Let‟s think about it

Those kids got only Rs.14 each and want to join the new group.

But each of them would need Rs.11 extra.

So Champak must give them Rs. 11 x 4 kid = Rs.44

And Champak also wants to be a part of this gang, so he should also bring

Rs.25 for himself.

Therefore, Champak must bring Rs. 44 (to give to those kids) + 25 (for

himself) =44+25=69

Hence final answer: age of Champak is 69 years.

Observe how numbers are moving while the average changes.

Case: Avg age of Students + Teacher

Average age of 36 students is 14 years. If teacher‟s age is included, then

average increases by one. Find the age of teacher. (RBI)

We can solve it by two approaches

1. Convert into Money problem (easy, quick)

2. Formula method. (lengthy, time consuming)

Approach #1: convert into money problem

Whenever you face such question on average, try to convert it into a money

problem, then it becomes very easy to visualize.

Converting into money problem

36 kids had Rs.14 each

1 teacher comes.

New average is 15 (original 14+ increases by 1)

How much cash does teacher have?

Ok let‟s think about it.

There is a gang of 36 kids. And it has rule: everyone must have Rs.14.

Teacher wants to form a new gang. And it has rule: everyone must have

Rs.15.

But those kids have only Rs.14 each.

So teacher must give Rs.1 to each of those 36 kids.

Meaning teacher must bring Rs.1 x 36 =Rs.36

And teacher should keep Rs.15 aside for himself. (because new gang rule

is “everyone must have Rs.15)

So how much cash did teacher bring in? =Rs.36 (to give to kids) PLUS

Rs.15 (for himself)

=36+15=Rs.51 that‟s our final answer. Age of Teacher is 51 years.

Approach #2: Formula method

Formula for average: average = sum of observation divided by number of

observation

Situation 1: without teacher

36=(sum of kids)/14

Therefore, sum of kids(s)=36 x 14

S=504

Situation 2: with teacher

37 =((sum of kids)+age of teacher))/15

Therefore, S + teacher=37 x 15

Now plugin the value of s=504 from previous equation

504+teacher =555

Teacher =555-504=51 that‟s our final answer.

As you can see, the problem with this “formula” method = you‟ve to do lengthy

multiplications (37 x 15 and 36 x 14). Thus

1. it breaks your momentum, decreases your speed in the exam.

2. you might make some silly mistake in multiplications or during

“plugging” the values in equations.

So better just try to solve such questions using “money problem” method.

Case: Avg salary workers+manager

The average salary of 20 workers is Rs.1500. If manager‟s salary is added, this

average increases by Rs.100. Find the salary of Manager.(RRB)

It is already in “money problem” format.

20 guys have Rs.1500 each

Now Manager wants to increase this average by Rs.100. (new avg

=1500+100=1600)

So manager must give Rs. 100 to each of these 20 guys= manager must

bring total 100 x 20 =Rs.2000 with him.

But this new group has a rule “everybody must have Rs.1600”. Hence, If

manager wants to be part of this “new” group, he must keep Rs.1600

aside for himself.

So, how much cash should manager bring? =Rs.2000 (to give to workers)

+ Rs.1600 for himself

=2000+1600

=3600 this is the final answer.

Case: Avg weight students+teachers

There are 24 kids in a class with average weight of 35 kilograms. If Teacher‟s

weight included, the average increases by 400 grams. Find weight of teacher.

Convert this into money problem

1. 24 kids have 35 rupees each

2. Teacher wants to form new gang: “everyone must have 35.4 rupees each”

(because 400 gms =400/1000 kg =0.4 kg)

Ok let‟s think about it

Teacher must give 0.4 rupees to each of those 24 kids=24 x 0.4 =9.6

And he must bring Rs.35.4 for himself.

so how much cash should teacher bring in? =9.6 (for kids) + 35.4 (for

himself)

=9.6+35.4

=45. That‟s our final answer: weight of teacher is 45 kilos.

Case: original average not given

In his 11th

inning, a batsman scored 216 runs and his average increased by 12

runs. What‟s his new average? (Bank)

Money problem

1. 10 guys had Rs.”m‟ each

2. New guy (11th) has Rs.216 and he forms a new gang.

3. In this new gang of 11 guys, each person has Rs.(m+12) each. Find value

of m+12.

Let‟s think about it:

New guy wanted to form a new gang where everyone had Rs.m+12.

But those 10 guys from old gang, had only Rs.m

That means new guy should give each of those 10 people Rs.12

So new guy gave away Rs.12 x 10 =Rs.120

So new guy is left with Rs. 216 (he originally had) MINUS Rs.120 (that

he gave away)

216-120=Rs.96 that‟s our answer.

so far we saw “Average: Entry of 1 Element=>leading to Increase in Average”.

Now let‟s check “Average: Entry of 1 Element => leading to decrease in

average”

Situation: Entry of 1 element – Decrease in Avg

Case: younger Kid joins: average goes down

The average age of six kids is 12 years. A new kid joins the gang and average

decreased by one year. What is the age of this new kid?

Money problem

Six kids got 12 rupees each.

New kid wants to form a new gang with rule “eachone must have only

Rs.11”

Ok let‟s think about it

Since average is declining, we are sure that new kid has less money than

Rs.11.

But he wants everyone to have (12-1)=11 rupees.

He tells those 6 kids, “each of you give me 1 rupees.” = 6 x 1 = Rs.6

borrowed.

So how much money did that new kid bring? Well, Now he is part of new

gang so obviously he has Rs.11 .

but he has borrowed Rs.6 from others.

Meaning, he originally had 11-6 =Rs.5

Final answer: age of new kid is 5 years.

Let‟s try a case where one element is removed.

Case: Average Runs decline (original not given)

A cricketer made 90 runs in his 12th

inning, and his average declined by 5 runs.

Find new average.

Money problem

11 guys had average Rs.m each

12th guy has Rs.90 and he makes new gang, with rule “everyone must

have Rs.(m-5)”

Let‟s think about it

Since average is declining, we can be sure that new guy has less money

than previous average (“m”).

But he wants everyone to have (m-5) rupees (he should also have m

minus 5 rupees.)

So he tells those 11 people, give 5 rupees each to me. = 11 x 5 = Rs.55

How much cash does the new guy have now?= 90 (he originally had) +

Rs.55 (that he borrowed)=Rs.90+55=145 that‟s our new average.

Final answer: new average runs after 12 innings is 145.

Situation: Removal of 1 element -increase in avg

A batsman played 30 innings with an average of 78 runs.

If we ignore the score of his last innings, then average increases by 2 runs. How

many runs did he score in last inning?

Money problem

30 guys with Rs.78 each

1 guy (Jethalal) leaves this gang after settling his debts.

Average of remaining 29 guys is (78+2)=80 runs.

Ok let‟s proceed

Since average is increases after member leaves the gang= this member

(Jetha) had less money than others.

That means other people had lended him some money (to raise him to the

average of 78.)

When Jethalal leaves the gang, remaining guys get 2 rupees each.

That means Jetha gave them Rs. 2 to each of those 29 remaining guys.=2

x 29= Rs.58

And once upon a time, Jethalal was part of this gang, so he had Rs.78.

It means Jetha‟s original money = Rs.78 minus Rs.58 (that he

borrowed)=78-58=Rs.20

Final answer: batsman‟s score in last inning was 20 runs.

Situation: Removal of 1 element -decrease in avg

Four members of a family earn average Rs.7350. One of them, leaves and

average goes down to Rs.6500. How much did that person earn, who has now

left the family? (SSC)

Money problem in our format:

1. 4 guys with Rs.7350 each

2. One of them (known as Jethalal) steals money from others and leaves.

3. Now 3 guys left and they‟ve Rs.6500 each

Ok let‟s think about it

Jetha wants to bring down average from 7350 to 6500 =decline of 850.

So he must steal Rs.850 from each of the remaining three guys =Rs.850 x

3=2550.

And since Jetha was once part of this “gang”, he too used to earn Rs.7350

himself.

Therefore total cash with Jethalal =7350 (he originally had) + 2550 (that

he stole) =Rs.9900.

Final answer: the income of person who left family, is Rs.9900

Mock questions

1. Average age of 30 students is 11 .When teacher‟s age is added, average

increases by 1 year. Find the age of teacher.

2. Average of 24 wrestlers is 90. A new wrestler joins the gym and average

increases by 2 kilos. Find weight of new wrestler.

3. Average annual income of 25 families in a society is 1.9 lakhs. A new

family comes to live in this society and the new average annual income of

the whole society is increased by 10,000 rupees. Find the monthly income

of this new family.

4. Average salary of 21 workers+1 manager is Rs.2000. When manager‟s

income is not considered, the average declines by 25%. How much does

Manager earn?

5. A batsman makes average 50 runs in 36 matches. After 37th match, his

average increased by 1 run. How many runs did he score in last match?

6. A batsman made 102 runs in 21st inning and his average increased by 2

runs. Find total runs scored by him in first 20 matches.

7. A batsman scored 199 runs in his last match and his average increased

from 44 to 49. Find total number of matches played by him.

8. A thief steals average 2 cars per night for 13 nights continuously. But

after 14th night‟s “adventure”, his average increases to 3 cars. How many

cars did he steal in that last night?

9. Average salary of 21 workers is 2000. A new worker joins the company

and overall average declines by 2.5%. What is the salary of new worker?

10. A batsman made average 80 runs in 29 matches. After 30th match, his

average declined by 2 runs. How many runs did he score in that last

match?

11. In a dance troupe, average weight of 20 junior dancers is 60 kilos. If the

weight of lead dancer is also considered, the average declines by 500

gms. How much does the lead dancer weigh?

12. In a society, average annual income of 30 families is 9 lakhs. One new

family starts to live here and average declines to 8.8 lakhs. Find the

annual income of this new family.

Answers

1. 42

2. 140

3. 4.5 lakhs annually / 12 = 37500 monthly income.

4. Decline of Rs.500. So manager had Rs.500 x 21 + 2000= Rs.12500

5. 87 runs.

6. 20 x 60 = 1200 runs.

7. 30 matches. he kept 49 for himself and gave 5 rupees each to “m” number

of people. 199=49+5m. so m=30 matches.

8. 16 cars. (Rs.3 for himself + 1 x 13 donated=16 cars stolen last night)

9. 2.5% decline= 50 rupees on 2000. so 21 workers gave him 21 x 50

=1050. and he is part of new gang= he has 1950 minus 1050 borrowed

=His salary is 900.

10. 20 runs.

11. 49.5 kilos.

12. 2.8 lakh.