## [Aptitude] Compound Interest Rate, Population Growth without Formulas

- Introduction
- Case: City’s population: Growth
- Case: City’s Population: Decline
- Case: Bank’s compound interest rate(CI)
- Case: CI →Finding Nemo Principal
- Case: Compounding Twice a Year
- Case: Compounding thrice a year
- Case: Wine Replacement (Adulteration)
- Approach 1: Alligiation loop
- Approach #2: Compound Interest

- Case: Adulteration→Finding Original volume
- Case: Simple interest rate (SI)
- Case: SI→ Finding Principal
- Practice Questions for CI/SI
- Sidenote on IBPS (Bank PO)
- 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

% 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% 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

# 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%