# 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:

 % Form Fraction form % Form Fraction Form 10% 1/10 90% 9/10 20% 1/5 80% 4/5 25% ¼ 75% ¾ 30% 3/10 70% 7/10 33.33% 1/3 66.66% 2/3 40% 2/5 60% 3/5 50% ½ 50% 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%+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 # 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 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 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) 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]

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 # 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 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. Recall that ‘visual-move’
100-M=10ml….(1)
M-0=90ml…….(2)
Divide (1) with (2) 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  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 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 ## 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?
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

# 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

Books: UPSC Prelims Books: UPSC Mains Books: Maths /Reasoning

## 88 Comments on “[Aptitude] Compound Interest Rate, Population Growth without Formulas”

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

2. 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.