Simple and Compound Interest, Stock

Question: At what annual compound interest rate will Tk. 5000 become Tk. 6050 in 2 years?

Solution:
Given that, 
Principal, P = Tk. 5000
Amount, A = Tk. 6050
Time, n = 2 years

We know, 
A = P(1 + r/100)ⁿ
⇒ 6050 = 5000 × (1 + r/100)²
⇒ (1 + r/100)² = 6050/5000
⇒ (1 + r/100)² = 121/100
⇒ 1 + (r/100) = √(121/100) = 11/10
⇒ r/100 = (11/10) - 1
⇒ r/100 = (11 - 10)/10
⇒ r/100 = 1/10
⇒ r = 100/10
∴ r = 10%

So the annual compound interest rate is 10%.

Assuming that face value of the first stock = Tk. 100 as it is not given in the question.
Since it is a 5% stock, we can take the dividend per stock = Tk. 5
Market Value of the first stock = Tk. 104
Investment on the first stock = Tk. 26000
Number of stocks purchases = 26000/104 = 250
His total income from all these stocks = Tk. 250 × 5 = Tk. 1250
He sells each of these stocks at Tk. 120
ie, amount he earns = Tk. 120 × 250 = Tk. 30000
He invests this Tk. 30000 in 6% stock (here also face value is not given and hence take it as Tk. 100)
His new income = Tk. (1250 + 2500) = Tk. 3750
ie, By Tk. 30000 of investment, he earns an income of Tk. 3750
To get an income of Tk. 6, the investment needed = (30000 × 6)/3750
= Tk. 48.

Let the sum be P
The sum P becomes 3P in 4 years on compound interest
3P = P{(1 + R/100)}⁴
3 = {1 + R/100)}⁴
Let the sum P becomes 81P in n years
81P = P{1 + (R/100)}ⁿ
81 = {1 + (R/100)}ⁿ
3⁴ = {1 + (R/100)}ⁿ
[{1 + (R/100)}⁴]⁴ = {1 + (R/100)}ⁿ
{1 + (R/100)}¹⁶ = {1 + (R/100)}ⁿ
n = 16.

Amount after 3 years = 40000 {(1 + 11)/100}³
= 40000 ( 111 /100 )³
= 54705.24
Compound interest = 54705.24 − 40000
= 14705.24

Question: Find the simple interest on BDT 15,000 at 8% per annum for 5 months.

Solution:
Principal, P = 15,000 Taka
Time, n = 5 months = 5/12 years
Rate of interest, r = 8% = 8/100

Simple Interest, I = P × n × r
= 15,000 × (5/12) × (8/100)
= (15,000 × 5 × 8)/(12 × 100)
= 600,000/1200
= 500

∴ The simple interest is Tk. 500.

Question: How many years will Tk. 1500 need at 4% simple interest to earn the same interest as Tk. 2000 earns in 3 years at 5% simple interest?

Solution:
Interest from Tk. 2000 in 3 years at 5%:
SI = (P × R × T)/100
= (2000 × 5 × 3)/100
= 300 Tk.

Now,
Tk. 1500 to Produce the Same Interest (Tk. 300) at 4%:
Principal, P = Tk. 1500
Interest, SI = Tk. 300
Interest Rate, R = 4%
​ Time, T = ? years

SI = PRT/100
⇒ T = (SI × 100)/(P × R)
⇒ T = (300 × 100)/(1500 × 4)
⇒ T = 30000/6000
∴ T = 5 years

Question: The difference between compound interest and simple interest on a certain sum for 2 years at 12% per annum is 36 Taka. What is the principal amount?

Solution: 
For 2 years, the difference between compound interest (CI) and simple interest (SI) is given by:

Difference = P × (r/100)t
where, 
P = Principal amount
r = interest rate = 12%
Time = t = 2 years
Difference = 36 Taka

So, we get, 
36 = P × (12/100)2
36 = P × (144/10000)
∴ P = 2500 Taka

Question: Find the initial sum that grows to Tk. 1500 in 5 years and Tk. 1620 in 7 years at simple interest.

Solution:
in two years the interest becomes = 1620 - 1500 = 120Tk.
in 5 years, interest = (120/2)/5 =300 Tk.

so , the initial sum is = 1500 - 300 = 1200 Tk.

Question: What is the difference between simple and compound interest at 10% per annum on a sum of Tk. 6000 at the end of 2 years?

Solution:
Principal (P) = Tk. 6000
Rate (r) = 10% per annum
Time (n) = 2 years

Simple Interest (SI):
SI = (P × R × T)/100
= (6000 × 10 × 2)/100
= 120000/100
= Tk. 1200

Compound Interest (CI):
Amount (A) = P × (1 + r/100)ⁿ
= 6000 × (1 + 10/100)²
= 6000 × (1.1)²
= 6000 × 1.21
= Tk. 7260

∴ CI = A - P = 7260 - 6000
= Tk. 1260

∴ Difference between CI and SI = 1260 - 1200
= Tk. 60

Number of shares sold = 7500/100
= 75

Proceeds from sale of Tk. 7500 stock
= Tk. [(105.50 - 0.5) × 75]
= Tk. 7875

Number of new shares purchased
= 7875/)124.50 + 0.50
= 7875/125
= 63

Original income
= 10% of Tk. 7500
= Tk. 750

New income
= 14% of Tk. 6300
= Tk. {(14/100) × 6300}
= Tk. 882

∴ Change in income
= Tk. (882 - 750)
= Tk. 132

Let the sum be x Assume that it will treble in n years.
Note that when the money doubles, simple interest is (2x - x), and when the money trebles, simple interest is (3x - x)
simple interest ∝ T (because here P and R are constants)
Therefore,
(2x - x) : (3x - x) = 12 : n
⇒ x : 2x = 12 : n
⇒ n = 24.

Since face value is not given, take it as Tk. 100.
As it is an 8% stock, income (dividend) per stock = Tk. 8
ie, For an income of Tk. 8, amount of stock needed = Tk. 100
For an income of Tk. 800, the amount of stock needed = (100 × 800)/8
= 10000

Let, Principal = x and interest 2x
ATQ, 2x = (x × 10 × r)/100
Or, r = 20%
Again, Let, Principal = 100
So, C = 100(1 + 20/100)²
         = 100(1 + 1/5)² = 144
∴ Interest = 144 – 100 = 44
∴ Required ratio = 100 : 44 = 25 : 11

P(4/100)² =1
⇒ P(1/25)² = 1
⇒ P/25² = 1
⇒ P = 625
Hence The sum is Tk. 625.

Question: A sum of Tk. 24,000 is invested at 5% per annum simple interest. Find the interest earned in 4 years 6 months.

Solution:
Given that,
Principal, P = Tk. 24,000
Rate of interest, r = 5%
Time, n = 4 years 6 months
= 9/2 years

We know,
I = Pnr
⇒ I = 24,000 × 9/2 × 5/100
= 24,000 × 9 × 5/(2 × 100)
= 24,000 × 45/200
= 24,000 × 9/40
= 600 × 9
= Tk. 5,400

Question: An investment becomes Tk. 12,100 in 2 years at compound interest, the rate being 10% per annum. Find the principal.

Solution:
Given,
Amount, A = 12,100 Taka
Rate, R = 10%
Time, T = 2 years

Compound Interest Formula:
A = P[1 + 100/R​]^T
⇒ 12,100 = P[1.1]²
⇒ 12,100 = 1.21P
⇒ P = 12,100/1.21
P = 10,000

∴ Principle = 10,000 taka

Principle = P
Compound Interest = Total amount - Principle
P = P(1 + R/100]ⁿ - P

Simple interest = PRT/100
R= 25% per annum; T and n = 3 years
Compound Interest - Simple Interest = Tk. 320

∴ [{P(1 + R/100]ⁿ - P} - (PRT/100)] = 320
⇒ [{P(1 + 25/100]³ - P} - {(P × 25 × 3)/100)}] = 320
⇒ P(5/4)³ - P - 3P/4 = 320
⇒ P(125/64) - (P + 3P/4) = 320
⇒ {(125P/64) - (7P/4)} = 320
∴ P = Tk. 1575.38

Let sum=Tk.x
C.I. when compounded half yearly = [x(1+10/100)⁴−x]=4641/10000
C.I. when compounded annually =[x(20/100)²−x]=11/25
4641/10000x−11/25x=482
=> x=20000

Market Value = Tk. 96.
Required Income = Tk. 650.
Here face value is not given. Take face value as Tk. 100 if it is not given in the question
To obtain Tk. 10 (ie,10% of the face value 100), investment = Tk. 96
To obtain Tk. 650, investment = {(96/10) × 650}
= Tk. 6240.

Question: A sum of Taka 100,000 is invested at 5% simple interest for the first 4 years and 10% compound interest for the next 2 years. What is the total amount after 6 years?

সমাধান:
দেওয়া আছে,
মূলধন, P = 1,00,000 টাকা
প্রথম 4 বছরের জন্য বার্ষিক সরল সুদের হার, r = 5%
পরবর্তী 2 বছরের জন্য বার্ষিক চক্রবৃদ্ধি সুদের হার, r = 10%
মোট সময়কাল = 6 বছর

প্রথম 4 বছরের সরল সুদ (Simple Interest), I = (P × n × r)/100
⇒ I = (100000 × 4 × 5)/100
⇒ I = 20,000 টাকা

4 বছর পর আসল (Principal) হবে = 1,00,000 + 20,000 = 1,20,000 টাকা
পরবর্তী 2 বছরের জন্য, এই 1,20,000 টাকা হবে নতুন মূলধন (New Principal)।
সুতরাং, P = 1,20,000 টাকা
সময়, n = 2 বছর
সুদের হার, r = 10%

চক্রবৃদ্ধি মূল (Compound Amount), C = P(1 + r/100)ⁿ
⇒ C = 120000(1 + 10/100)²
⇒ C = 120000(110/100)²
⇒ C = 120000 × (110/100) × (110/100)
⇒ C = 1,45,200 টাকা

সুতরাং, 6 বছর পর মোট পরিমাণ হবে 1,45,200 টাকা।

প্রশ্ন: Tk. 1500 is invested at 5% per year simple interest. In how many years the investment will be Tk. 1800?

সমাধান: 
Here,
P = Tk. 1500 
I = 1800 - 1500 = Tk. 300
r = 5% = 5/100 
n = ?

We know that,
∴ I = Pnr
⇒ n = I/(Pr)
= (300 × 100)/(1500 × 5)
= 4

∴ The investment will be Tk. 1800 in 4 years.

Question: A certain principal amount, invested at simple interest, grows to Tk. 950 after 3 years and Tk. 1010 after 4 years. What is the original principal amount?

Solution:
Given,
Amount after 3 years = Tk. 950
Amount after 4 years = Tk. 1010
∴ Interest for 1 year = 1010 - 950 = Tk. 60

∴ Interest for 3 years = 60 × 3 = Tk. 180

∴ Principal = 950 - 180 = Tk. 770

Question: A sum increases by 80% in 8 years at simple interest. If Tk 2500 is invested at the same rate for 2 years under compound interest, what is the final amount?
 
Solution:
let
the principal = Tk. 100
Simple Interest = Tk 80 (since 80% increase in 8 years)

We know,
Simple interest = Pnr/100
⇒ 80 = (100 × r × 8)/100
⇒ 80 = 8r
∴ r = 10%

The compound interest = 2500 {1 + (10/100)}² - 2500
= 2500 × {1 + (1/10)}² - 2500
= 2500 × (11/10)² - 2500
= (2500 × 121)/100 - 2500
= 3025 - 2500
= Tk. 525

So the compound interest on Tk 2500 after 2 years is Tk 525.

Question: A sum of Tk. 1750 is divided into two parts such that the interest on the first part at 8% simple interest per annum and that on the other part at 6% simple interest per annum are equal. The interest on each part is-

Solution: 
Principal = Tk. 1750
Let the first part = x
Hence, the second part = 1750 - x

x × (8/100) × 1 = (1750 − x) × (6/100) × 1
⇒ 4x = 5250 − 3x
⇒ 7x = 5250
⇒ x = 750

First part = Tk. 750
∴ Second part = Tk. (1750 − 750)
= Tk. 1000

∴ Required interest = 750 × (8/100) 
= Tk. 60

Given,
S.I = 3 Principal Amount
=> 3A = A x 16 x R/100
By solving, we get
=> R = 18.75%

Question: If Taka 200 is invested at 10% interest for 3 years, what is the difference between the simple and compound interest?

Solution: 
এখানে,
মূলধন, P = 200 টাকা
সুদের হার, r = 10% = 10/100 = 1/10
সময়, n = 3 বছর

আমরা জানি,
সরল মুনাফার ক্ষেত্রে,
SI = P × r × n
= 200 × (1/10) × 3
= 60 টাকা

আবার,
চক্রবৃদ্ধি মুনাফায়,
C = P(1 + r)ⁿ
= 200 × [1 + (1/10)]³
= 200 × (11/10)³
= 200 × (11/10) × (11/10) × (11/10)
= 200 × (1331/1000)
= 266200/1000
= 266.2 টাকা

∴ চক্রবৃদ্ধি মুনাফা = C - P
= (266.2 - 200) টাকা
= 66.2 টাকা

∴ সরল ও চক্রবৃদ্ধি মুনাফার পার্থক্য = 66.2 - 60 = 6.2 টাকা

Question: The difference between compound interest and simple interest on a sum of Taka 10,000 for 2 years at 10% per annum compounded annually is - 

Solution: 
Simple interest = 10000 × 2 × (10/100)
= 2000 Taka

Compound Interest, 

So, difference = 2100 - 2000 = 100 Taka

By investing Tk. 1552, income = Tk. 128
By investing Tk. 97, income
= (128/1552) × 97
= Tk 8
∴ Dividend 8%.

Question: Mr. Rahman deposited Tk. 40,000 in a scheme at 8% annual simple interest. He closed the account after 6 months and incurred an early withdrawal fee of Tk. 320. Calculate his net earnings.

Solution:
Given that,
Principal, P = Tk. 40000
Rate of interest, r = 8% per annum
Time, n = 6 months = 6/12 = 1/2​ year
Early withdrawal fee = Tk. 320 

We know, 
Simple interest = Prn/100
= (40000 × 8 × 0.5)/100
= Tk. 1600

∴ Net earnings = Interest earned - Early withdrawal fee
= 1600 - 320
= Tk. 1280

Therefore, his net earnings from the investment are Tk. 1280.

The population of the town decreases by 10% every year.
Thus, it has a new population every year. So the population for the next year is calculated on the current year population.
For the decrease, we have the formula A = P(1 - R/100)ⁿ

Therefore,
the population at the end of 5 years
= 10000(1 - 10/100)⁵
= 10000(1 - 0.1)⁵
= 10000 x 0.9⁵
= 5904.9

Suppose he buys each share for Tk. x.
Then, Tk.(25 × 9/100) = (x × 10/100)
or x = Tk. 22.50.
Cost of each share = Tk. 22.50.

P/S.I.=10/3
Let Principal = 10
S.I. for 5 year = 3
S.I. for 1 year = 0.6
Rate = S.I./Principal×100
Rate = 0.6/10×100
=6%

Two parts are = A and (40800 - A)
Time period and Simple interest are same for both

Simple Interest = (P × R × T)/100

∴ (A × 6.25 × 8)/100 = {(40800 - A) × 7 × 5}/100
∴ 50A = 35 × 40800 - 35A
∴ 85A = 35 × 40800
∴ A = 8400

Smaller amount is Tk. 16800
By direct observation, we can say that 16800 is smaller - as it is smaller than half of 40800.

Interest for the first month = p×(10000/1000) = 10p
Interest for the 2nd month will be =  q×(10000/1000) = 10q
Interest for the 3rd month will be = q×(10000/1000) = 10q

∴ Total Interest = 10p + 20q

Question: A sum of money is borrowed and paid back in two annual instalments of Tk. 882 each allowing 5% compound interest. The sum borrowed was -

Solution:
Principle,
= (P.W. of Tk. 882 due 1 year hence) + (P.W of Tk. 882 due 2 years hence)
= [{882/(1 + 5/100)} + {882/(1 + 5/100)²}]
= [{(882 × 20)/21} + {(882 × 400)/441}]
= 840 + 800
= Tk. 1640

The sum borrowed was Tk. 1640.

Question: A man invested TK 4500 in a stock at 108 to obtain an income of TK 250. What is the dividend from the stock?

Solution:
By investing TK 4500, income  = TK 250
By investing TK 108 = (108 × 250)/4500 = 6

Hence, the dividend is 6%.

Investment = Tk. 333000
Since face value is not given, we can take it as Tk. 100
and dividend per share = Tk.11/2
Market Value = 110 + 1 = 111
Number of shares purchased = 333000/111 = 3000
Total income = 3000 × 11/2 = Tk.16500

Let investment in 12% stock be Tk. x.
Then,
investment in 15% stock = Tk. (12000 - x)

∴ (12/120) × x + (15/125) × (12000 - x) = 1360
⇒ x/10 + 3/25(12000 - x) = 1360
⇒ 5x + 72000 - 6x = (1360 × 50)
⇒ x = 4000

Hence, Investment in 12% stock is Tk. 4000

Question: A sum of money amounts to Tk. 15000 in 4 years at 25% simple interest per annum. Find the sum. 

Solution:
Given, 
A = 15000 
T = 4
R = 25%

We know,
SI = P × R × T
= P × (25/100) × 4
= P

A = P + SI 
⇒ A = P + P
⇒ A = 2P
⇒ 15000 = 2P
⇒ P = 15000/2
∴ P = 7500

∴ P = I/nr = 1600/(4/100 x 4) = 10000
So, I = P(1 + r)ⁿ - P
= 10000(1 + 10/100)⁴ - 10000
= 14641 - 10000
= 4641

Question: A person pays Tk. 8000 as an amount on the sum of Tk. 6000 that he had borrowed for 3 years. What will be the rate of interest?

Solution:
Amount, A = Tk. 8000
Principal, P= Tk. 6000
Time, T = 3 years
Interest Rate, R =?

Amount = Principal  + Simple Interest
SI = A – P
= 8000 – 6000
= Tk. 2000

SI = (P × R ×T)/100
⇒ R = (SI × 100)/(P × T)
= (2000 × 100)/(6000 × 3) 
= 11.11 %
∴ The rate of interest is 11.11 %.

Question: How many years will it take for an investment of Tk. 7500 to earn Tk. 2250 in simple interest rate of 6%?

Solution:
Given that,
Principal, P = 7500
Simple Interest, SI = 2250
Rate of interest, r = 6%
Time, n = ?

We know,
SI = Pnr/100
⇒ n = (S × 100)/(P × r)
= (2250 × 100)/(7500 × 6)
= 5 years

So, it will take 5 years for the investment to earn Tk. 2250 at 6% simple interest.

Question: Compound interest on a certain sum at the rate of 25% per annum after 2 years is Tk. 5,625. Find the simple interest on that sum at the rate of 12% per annum for 5 years.

Solution:

Let the sum be, P = x.
Rate of interest, r = 25%
Period, n = 2 years

We know,
Compound Principal, C = P(1 + r)ⁿ

∴ Compound interest = P(1 + r)ⁿ - P

Hence, compound interest = x(1 + 25/100)² - x = 5,625
⇒ x(5/4)² - x = 5,625
⇒ x(25/16) - x = 5,625
⇒ x(9/16) = 5,625
⇒ x = (5,625 × 16)/9
∴ x = 10,000 Tk.

Again,
Principal, P = 10,000 Tk.
Rate of interest, r = 12%
Period, n = 5 years

We know,
Simple Interest, I = Pnr/100

Hence, simple interest = 10,000 × 5 × 12/100
= 6,000 Tk.

প্রশ্ন: At what rate of compound interest per annum will a sum of Tk. 5000 become Tk. 7200 in 2 years?

সমাধান:
দেওয়া আছে,
মূলধন (Principal), P = 5000 টাকা
চক্রবৃদ্ধি মূল (Compound Amount), C = 7200 টাকা
সময় (Time), n = 2 বছর
মুনাফার হার (Rate), r = ?

আমরা জানি,
চক্রবৃদ্ধি মূল, C = P(1 + r/100)ⁿ
⇒ 7200 = 5000(1 + r/100)²
⇒ (1 + r/100)² = 7200/5000
⇒ (1 + r/100)² = 72/50
⇒ (1 + r/100)² = 36/25​ 
⇒ (1 + r/100) = 6/5
⇒ (100 + r)/100 = 6/5
⇒ (100 + r) = 600/5
⇒ r = 120 - 100
∴ r = 20
সুতরাং, বার্ষিক চক্রবৃদ্ধি সুদের হার 20%।

Question: A certain principal amount, invested at simple interest, grows to Tk. 815 after 3 years and Tk. 854 after 4 years. What is the original principal amount?

Solution:
Given,
Amount after 3 years = Tk. 815
Amount after 4 years = Tk. 854
∴ Interest for 1 year = 854 - 815 = Tk. 39
∴ Interest for 3 years = 39 × 3 = Tk. 117
∴ Principal = 815 - 117 = Tk. 698

Question: Tk. 1,000 becomes Tk. 1,200 in 4 years at a certain rate of simple interest. If the rate of interest is increased by 2%, what amount will Tk. 1,000 become in 4 years?

Solution:
Principal, P = 1000
Interest, I = 1,200 - 1,000 = 200
Time, T = 4 years

SI = PRT/100
⇒ R = (SI × 100)/(P × n)
⇒ R = (200 × 100)/(1000 × 4)
⇒ R = 20000/4000
⇒ R = 5

∴ Interest rate, R = 5%

∴ New Interest rate = 5% + 2% = 7%

New interest at 7% for 4 years:
New interest = (1,000 × 7 × 4)/100
= (28,000)/100
= 280 Tk.

New amount = Principal + Interest
= 1,000 + 280
= Tk. 1,280

Question: There is 40% increase in an amount in 8 years at simple interest. What will be the compound interest of TK. 8,000 after 2 years at the same rate?

Solution:
ধরি, আসল P = 100 টাকা
8 বছরে বৃদ্ধি পায় 40 টাকা (সরল মুনাফা, I = 40 টাকা)

আমরা জানি, I = (P × n × r)/100
⇒ 40 = (100 × 8 × r)/100
⇒ 40 = 8r
⇒ r = 5%

এখন, 8,000 টাকার 2 বছরের চক্রবৃদ্ধি মুনাফা (একই হারে অর্থাৎ r = 5%):
আসল P = 8,000 টাকা
সময় n = 2 বছর
হার r = 5% = 5/100 = 1/20

চক্রবৃদ্ধি মুনাফা = P(1 + r)ⁿ - P
⇒ 8,000(1 + 1/20)² - 8,000
⇒ 8,000(21/20)² - 8,000
⇒ 8,000(441/400) - 8,000
⇒ (20 × 441) - 8,000
⇒ 8,820 - 8,000
∴ 820

সুতরাং, চক্রবৃদ্ধি মুনাফা হবে 820 টাকা।

Interest charged on Tk 10,000 is,
(15/100) × 10000 = Tk. 1500

Interest for two years =
Tk. 1500 × 2 = Tk. 3000

The amount payable after two years = principal + interest
= Tk. 10000 + Tk. 3000 = Tk. 13000

P = (6000×100)/(5×6) = 20000
∴ Compound interest = 20000 × (1 + 5/100)² - 20000
= 2050 

Tk. 20000 after 4 years = 20000{1 + (10/100)}⁴ =Tk. 29282
Tk. 20000 after 3 years = 20000{1 + 10/100)}³ =Tk. 26620
Tk. 20000 after 2 years = 20000{1 + 10/100)}² =Tk. 24200
Tk. 20000 after 1 years = 20000{1 + 10/100)}¹ =Tk. 22000
Total amount after 4 years = 29282 + 26620 + 24200 + 22000 = Tk. 102102