[Aptitude] Compound Interest Rate, Population Growth without Formulas

Aptitude81 Comments

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  1. Introduction
  2. Case: City’s population: Growth
  3. Case: City’s Population: Decline
  4. Case: Bank’s compound interest rate(CI)
  5. Case: CI →Finding Nemo Principal
  6. Case: Compounding Twice a Year
  7. Case: Compounding thrice a year
  8. Case: Wine Replacement (Adulteration)
  9. Case: Adulteration→Finding Original volume
  10. Case: Simple interest rate (SI)
  11. Case: SI→ Finding Principal
  12. Practice Questions for CI/SI
  13. Sidenote on IBPS (Bank PO)
  14. Verbal Sidenote:

 

Introduction

CI/Population growth is not a separate theory by itself but mere an extension of Percentage calculation theory. Hence following three questions can be solved with one and same approach, without having to mugup three separate formulas

  • A city has 10,000 residents. Its population grows at the rate of 10% per annum, what’ll be its total population after 5 years?
  • A bank offers 10% interest rate compounded annually, if you deposit Rs. 10,000 today, what’ll be the total amount in your savings account after 5 years?
  • A butler steals 10 ml of whiskey from 100 ml bottle and replaces it with water. He repeats this process 5 more times, how much % whisky is left in the bottle?

First, master the “Fraction” table method of % calculation:

The ready reference Table

% FormFraction form% FormFraction Form
10%1/1090%9/10
20%1/580%4/5
25%¼75%¾
30%3/1070%7/10
33.33%1/366.66%2/3
40%2/560%3/5
50%½50%½ haha

Case: City’s population: Growth

A city has 10,000 residents. Its population grows at the rate of 10% per annum, what’ll be its total population after 5 years?

10% increase
=100% we have already + 10% new is added
=100%+10%
=110%
But if we talk in fraction form: 100% = 1 and 10%=1/10
Hence
10% increase
=1+1/10
=11/10

After first year

The new population after 1 year, will be 11/10 times the original
=(11/10)*original ; we know that originally there are 10,000 resident. But no need to calculate that right now.
This is our new original:

After second year

The new population will be 11/10 times the original population at the end of first year
=11/10*[(11/10)*original]

After third year

The new population will be 11/10 times the original
=11/10 [11/10 [(11/10)*original]]

Continuing like this, what we get after 5 years is
aptitude CSAT CMAT CAT IBPS

CASE: City’s Population: Decline

A city has 10,000 residents. Its population declines at the rate of 10% per annum, what’ll be its total population after 5 years?
Decline = decrease
=100%-10%
=1-(1/10)
=9/10

Population after 5 years

aptitude CSAT CMAT CAT IBPS
Answer. After 5 years, city’s population will be 5904.

Case: Bank’s compound interest rate(CI)

A bank offers 10% interest rate compounded annually, what’ll be the total amount in your savings account after 5 years?
It is simple: use the same trick used in population case
100%+10%
=1+(1/10)
=11/10

Money in your account after 5 years

aptitude CSAT CMAT CAT IBPS
Rs.16105 is the total amount in your bank account.

How much interest did you earn?

= 16105 minus 10000 =Rs. 6105 earned in interest.

Case: CI →Finding Nemo Principal

A man had deposited some money in SBI compounded 10% annually. After 3 years he got Rs.3310 in interest. How much money did he deposit initially?

Principal = The money you deposit initially.  Suppose he deposited Rs.M
After 3 years he got (M+3310)
10% compound interest rate for three years means
(See the calculation  in following picture)
aptitude CSAT CMAT CAT IBPS
Final Answer: Principal was Rs. 10000

Case: Compounding Twice a Year

You deposited Rs.10,000 @10% annual compound interest rate in SBI. If the interest rate is compounded after each 6 months, how much money will be there in your account after 3 years?

Important: When interest is compounded half yearly, the interest rate will be half of the annual interest rate. (NCERT class 8 Math textbook)
So the effective interest rate
= half of 10%
=5%

Thus percentage increase
=100%+5%
=1+(1/20)
=21/20

After 6 months

=21/20 times the original amount
This becomes our “new original”

After another 6 months

=21/20 times the “new original”
=21/20  x [(21/20) x original]

Now think about it
1 year has two blocks of 6 months [6+6=12 months] 3 years has 3 x2=6 blocks of 6 months
Therefore we’ve to do this interest rate calculation 6 times because there are 6 blocks.
Amount in your bank account after 3 years
aptitude CSAT CMAT CAT IBPS

Case: Compounding thrice a year

Rs.10,000, 10% annual interest rate, compounded thrice a year.

Since it is compounded thrice a year, hence effective interest rate is one third of 10%= (10/3)%
Percentage increase
=100%+ (10/3)%
=1+(10/300)
=1+(1/30)
=31/30

1 year has 3 blocks of 4 months [3 x 4 =12 months] 3 years have 9 blocks of 4 months [3 x 4 x 3 =36 months = 3 years]

Amount after 3 years

aptitude CSAT CMAT CAT IBPS

CASE: Wine Replacement (Adulteration)

A butler steals 10 ml of whiskey from 100 ml bottle and replaces it with water. He repeats this process 5 more times, how much % whisky is left in the bottle?

This is again a compound interest rate problem. But the amount doesn’t increase like in Bank’s savings account. Because Butler makes sure that volume remains 100ml after every replacement.

Approach 1: Alligiation loop

Proceed with the technique you learned in Mixture Alligiation article.

aptitude CSAT CMAT CAT IBPS

Recall that ‘visual-move’
100-M=10ml….(1)
M-0=90ml…….(2)
Divide (1) with (2)
aptitude CSAT CMAT CAT IBPS

Therefore M=90
Means new mixture contains 90% wine.
This is 1st time stealing. Butler is going to do this 4 times more.
Situation: Now bottle contains 100 ml mixture containing 90% wine, Butler steal 10 ml from it and adds 10 ml fresh water.
Repeat the same alligiation
aptitude CSAT CMAT CAT IBPS

aptitude CSAT CMAT CAT IBPS
Final answer: after five repetitions, the concentration of wine left in bottle (quality) = 59.05%
Since it contains 100 ml mixture, so ‘absolute value’ (Quantity) is 100 x 59.05%=59.05ml
This was quite lengthy, tiresome and time consuming approach, wasn’t it?
Better try second approach using compound interest concept.

Approach #2: Compound Interest

A butler steals 10 ml of whiskey from 100 ml bottle and replaces it with water. He repeats this process 5 more times
Butler is stealing 10ml out of 100 ml everytime. So percentage wise he is stealing 10/100=10% of content everytime.
What is left:
100%-10%
=90%
=9/10

Run this loop 5 times

aptitude CSAT CMAT CAT IBPS

Case: Adulteration→Finding Original volume

A tanker is full of milk, 25% of the liquid is stolen and replaced with water. If this process is repeated 4 times and ultimate mixture contains 810 litres of milk, what is the total capacity of this tanker?

25% is stolen so what is left?
100%-25%
=75%
=3/4
Loop this 4 times
aptitude CSAT CMAT CAT IBPS

milk left in the tanker

 

What is the final concentration of milk in this tanker?

Total capacity of tanker is 2560 lt.
And we know that 810 lt. of milk is left.
So % wise (810/2560)*100=31.64% milk is left in the tanker.

CASE: Simple interest rate (SI)

Rs.10,000 @10% annual simple  interest rate, How much interest do you earn after 3 years?

No brainer! In simple interest rate, you get interest only on the initial Principal (Rs.10000 in our case)
Three years = you earn interest three times
How much interest do you earn in 1 year?
=10% of 10000
=(1/10) x 10000
=1000
Repeat this process three times, what do you get?: 1000 x 3 = Rs.3000

Case: SI→ Finding Principal

You deposited some money in bank, after three years you get back Rs.13,000 @10% simple interest rate. What was the Principal?
Suppose your Principal is Rs.M
Each year you get 10% of M in interest
=10% M
=(1/10) x M
In Three years you get = 3 x (1 /10) x M in interest
Total amount you receive after three years
=Principal + interest for three years
=M + 3 x (1 /10) x M
=M+ (3/10)M
=(13/10)M
But it is given that you received Rs.13000 so,
(13/10)M=13000
M=13000 x (10 /13)=  10000
Answer. Your Principal =Rs.10000

Verbal Sidenote:

It is “Principal” and Not “Principle”
Principle = rule /maxim / siddhant/ Niyam/usool (उसूल)
Principal =Main/ chief/money deposited initially.
In the movie Agnipath, Hrithik Roshan’s father Dinanath Master was a schoolteacher and not a ‘principal’, but he did not compromise with his ‘principles’ so Sanjay Dutt killed him.

Mrunal recommends

  1. (free) NCERT, NIOS, TN-Books
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  3. Indian Polity M.Laxmikanth (Hindi | English)
  4. Art & Culture by Nitin Singhania (Hindi | English)
  5. Spectrum: Modern History (Hindi | English)
  6. Bipin Chandra: Post Independence
  7. Fast-track to Arithmetic Rajesh Verma
  8. MK Pandey’s Analytical Reasoning
  9. Disha’s Topicwise Paperset (Hindi | English)
  10. School Atlas
  11. Mains: Language papers
  1. (free) NCERT, NIOS, TN-Books 4 History,Geo,Sci
  2. Indian Polity M.Laxmikanth (Hindi | English)
  3. Spectrum: Modern History (Hindi | English)
  4. Maths: Quantam CAT Sarvesh Kumar
  5. Objective General English SP Bakshi
  6. Word Power made Easy -Norman Lowe
  7. Topic wise Solved Paperset by Disha


So far 81 Comments posted

  1. sai sujith

    mrunal sir we are not able to view images plz rectify the technical problem.The population after 5 years will be,the amount after 5 yrs will be after this images are not able to view.

  2. sharmi

    Not able to see the picture kindly resolve the issue.Thanks in advance !!

  3. Aspirant

    Unable to see the picture…hence unable to follow up. Kindly fix the glitch. Its the same problem in all of the aptitude section

  4. Bhavik

    After 5 years city population is 16105..( growth)
    After 5 years city population is
    5904.9 (decrease)

  5. BBB

    Unable to see any pics. Please fix it..

  6. Maheswaran S

    Reinstate the pictures please

  7. emperor

    Sir ji image please..ya fir 8th class vala
    Method h

  8. nidhi

    not able to see the images..plz fix the glitch

  9. Pratyush

    Hi Mrunal,
    Images are no longer visible in this article. Please fix this problem for people like us who may not be able to follow regular coaching.

  10. aaruk

    pls update the image links

  11. sandeep

    sir not abel to see the image plz sir see to the problem as fast as possible…….

  12. Elvis

    Looks like the webmaster is not keen on rectifying the image display problem. I’m using Way Back Machine to view the image. Placing a link below for everyone who is interested.

  13. paaru

    Sir
    Not able to see the images.. ! sir please fix the issue…:(

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