[Aptitude] Averages: Entry / Removal of 1 element => Increase / Decrease in Average Shortcut method explained

Support StaffAptitude133 Comments

Mrunal Patel is providing free updates to the Economy current affairs, budget & Economic survey for the upcoming UPSC Prelims and Mains under this Win21 Series. In Pillar4C, he'll cover new developments related to GDP, Inflation, Unemployment including latest budget and economic survey!
  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.
Avg Age problem Concept

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.

133 Comments on “[Aptitude] Averages: Entry / Removal of 1 element => Increase / Decrease in Average Shortcut method explained”

  1. I have 1 question
    The average of 6 observations is 45.5. If one observation is added in them then new average will be 47. What will be the new observation?
    Solution – As per above method
    7th person should give Rs. 1.5 to 6 members i.e. 1.5 * 6 = 9
    & he should have Rs. 45.5.
    so total = 45.5 + 9 = 54.5 is answer.

    but actual answer is 56 for this..

    1. No,as new average is 47..
      This observation should be 47. and
      what he distributes is 1.5*6=9.
      9+47=56… this is the solution

  2. Sir what in case of replacement problems.e,g A member weighing 40 Kilos in a group of 12 is replaced by a member and average weight increase/decreases by 1/4th Kilo.Whats the age of the new member (the substitute)?

    1. You are right Pooja…answer is 31.
      Up to 30 matches, his average was 44.
      And when he played 31st match and scored 199 runs then his average was increased from 44 to 49.
      So answer to 31 hi banta hai.

  3. Hey Mrunal,

    Thanks for building our interest in maths.
    I request you to please help me out with circle, lines, triangle portion. Thanks.

  4. murnal sir how to apply ur trick here
    average age of 10 students is 15 when 5 more students join the group the average age rose by 1 year. The average age in years of the new students is ?

    1. Solve it as letting 5 boys = single new boy….
      10 students had Rs 15 each..
      1 new boy comes.
      New average is 16 (original 15+increases by 1)
      How much cash does new boy have?
      ———————————————————————-
      So new boy must give Rs. 1 to each of those 10 students….1*10=10
      And he should keep Rs. 16 aside for himself.
      So how much cash did new boy bring in? = 26 Rs. (10+16)
      Now as u know this new boy equals to “5 more students”
      so 26/5= 5.2 years
      so Average ages of 5 students is 5.2 years.

      1. KARAN beta book likhle…….”Quicker math with KARAN”…..(ans=18 yr)…..And he should keep Rs. 16 aside for “himself”….

        HERE himself means 5 new boys and each has avg 16..

        So how much age did new boys bring in? = 10+(16*5)=90…..so avg of new 5 boys=(90/5)=18

        (or) directly 16+(10/2)=18 yr

  5. hello murnal sir i think answer of question no. 10 shuld be 22 according to ur method or funda
    29– (m)
    30– (m-2)
    2*29 =58 borrowed from others as average is decrease so it shuld be less than previous average therefore 80-58=22 answer
    plz correct me if I am wrong anyone……

  6. answer no. Of 10th question shuld be 720 coz here runs are asked not average so it shuld be 29*80 = 2320
    30* (80-2) = 3040
    runs score in last match 3340 – 2320 = 720

  7. 10th question shuld be 720 coz here runs are asked not average so it shuld be 29*80 = 2320
    30* (80-2) = 3040
    runs score in last match 3340 – 2320 = 720

  8. pls tell me how to solve this sum… batsman has a certain average of runs for his 8 matches. in the 9th match he scores 100 runs and thereby increases his average by 9runs. his new average of runs is.. i m not able to solve this sum..

  9. The average score of sachin in 15 IPL matches is 70 runs and the average score in Border gavaskar trophy T-20 is 45 runs in 7 matches. If he had played 10 more international T-20 matches and his overall average score in all the T-20 matches was 73 runs.what was his total score in 10 international T-20 matches? please kindly solve,thanx.

  10. The average age of a family of 5 members is 20 years. If the age of the youngest member be 10 years then what was the average age of the family at the time of the birth of the youngest member?
    13.5
    14
    15
    12.5

  11. Sir I am from Tamilnadu. I want to get bank job within six months.Sir please guide me.I completed ME.

  12. Thanx Mrunal Sir :)
    I have always had problem in understanding these type of problems. I like the money concept.. very easy and practical to grab :)
    Thanx again!!

  13. Thanks for your method.
    People may forget maths and equations…
    But, logic always stands aside and alive…

    Hats off Sir ..

  14. 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. sir this model is not covered in our concept sir plzz help me

  15. never thought that these average questions can be solved so easily . thanks man for the great effort .

Leave a Reply

Your email address will not be published. Required fields are marked *