Simple and Compound Interest, Stock

S.I. for 1 year = 854 - 815
= 39
S.I. for 3 years = 39 × 3
= 117
∴ Required Sum = 815 - 117
= Tk. 698.

Question: Tk. 6000 becomes Tk. 7200 in 4 years at a certain rate of simple interest. If the rate becomes 1.5 times of itself, the amount of same principal in 5 years will be -

Solution: 
৬০০০ টাকা ৪ বছরে ৭২০০ টাকা হয়। 
৪ বছরে সুদ = ৭২০০ - ৬০০০ টাকা 
= ১২০০ টাকা 
১ বছরে সুদ = ১২০০/৪ = ৩০০ টাকা 

১.৫ গুণ বৃদ্ধিতে ১ বছরে সুদ = ৩০০ × ১.৫ টাকা 
= ৪৫০ টাকা 

৫ বছরে সুদ = ৪৫০ × ৫ টাকা 
= ২২৫০ টাকা 

∴ ৫ বছর পর সুদাসলে হবে = ৬০০০ + ২২৫০ টাকা 
= ৮২৫০ টাকা 
 

For an income of Tk. 1 in 9% stock at 96,
investment = Tk. (96/9)
= Tk. 32/3

For an income of Tk. 1 in 12% stock at 120,
investment = Tk. (120/12)
= Tk. 10

Ratio of investments = (32/3) : 10
= 32 : 30
= 16 : 15.

Question: A sum of Tk. 2500 amounts to Tk. 2809 in 2 years at compound interest. Find the rate of interest per annum.

Solution:
Here,
Principal, P = 2500 Tk.
Final amount, A = 2809 Tk.
Time, n = 2 years
Interest rate, r = ?

প্রশ্নমতে,
A = P × (1 + r/100)ⁿ
⇒ 2809 = 2500 × (1 + r/100)²
⇒ (1 + r/100)² = 2809/2500
⇒ (1 + r/100) = √(2809/2500)
⇒ 1 + r/100 = 53/50
⇒ r/100 = (53/50) - 1
⇒ r/100 = 3/50
⇒ r = (3/50) × 100
⇒ r = 6

∴ The annual rate of interest is 6%.

Question: Karim borrowed Tk. 10000 at a certain rate of simple interest for 4 years. If he paid Tk. 2500 as interest, find the rate of interest per annum.

Solution:
Given that, 
Principal, P = Tk. 10,000
Simple Interest, SI = Tk. 2,500
Time, n = 4 years

We know,
SI = (Principal × Rate × Time)/100
⇒ 2500 = (10,000 × r × 4)/100
⇒ 2500 = (40,000 × r)/100
⇒ 2500 = 400 × r
⇒ r = 2500/400
∴ r = 6.25%

Therefore, the rate of interest per annum is 6.25%.

Question: A person invests Tk. 3100 at a rate of 4% per annum under simple interest. After how many years will the total interest earned be Tk. 372?

Solution:
Given that,
Principal, P = Tk. 3100
Rate, r = 4%
Simple Interest, SI = Tk. 372

We know,
SI = (P × R × T) / 100
⇒ 372 = (3100 × 4 × n)/100
⇒ 372 = 124 × n
⇒ n = 372 ÷ 124
∴ n = 3 years

∴ Number of years = 3

Question: If the simple interest on 6,000 Taka for 3 years is 1,800 Taka, what is the rate of interest per annum?

Solution:
Given, I = 1,800 Taka
P = 6,000 Taka
n = 3 years

We know, 
I = Pnr
⇒ r = I/Pn
⇒ r = 1800/(6000×3)
⇒ r = 0.1
∴ r = 10%

Question: How many years will it take for an investment of Tk.10000 to earn Tk. 1200 in simple interest rate of 6%?
(Officer Cash 2022 অনুযায়ী)

Solution: 
Given that,
Principal, P = 10000
Simple Interest, SI = 1200
Rate of interest, r = 6%
Time, n = ?

We know,
n = I/Pr
= 1200/(10000 × 6%)
= (1200 × 100)/(10000 × 6)
= 2

So, it will take 2 years for the investment to earn Tk. 1200 at 6% simple interest.

SI =  10000 × 2 × 5/100 = 1000
CI = 10000(1 + 5/100)² – 10000 = 1025
∴ Difference =  1025 - 1000 =  25

Question: The population of a town is 20,000, and it increases by 5% each year. What will the population be after 3 years?

Solution:
We can use the compound interest formula for population growth:

Population after n years = P × [1 + (r/100)]ⁿ

Here,
P = 20,000, r = 5%, n = 3

∴ Population after 3 years = 20,000 × [1 + (5/100)]³
= 20,000 × (105/100)³
= 20,000 × 1.157625
= 23,152.5

∴ The population of the town after 3 years will be approximately 23,153.

Question: An amount of Tk. 12,000 yields a simple interest of Tk. 2,160 in 4 years. What is the annual rate of interest?

Solution:
Given, Principal, P = 12000
Simple Interest, SI = 2160
Time, n = 4 years
Rate of interest, r = ?

We know, I = Pnr/100
⇒ r = (I × 100)/(P × n)
⇒ r = (2160 × 100)/(12000 × 4)
⇒ r = 216000/48000
⇒ r = 4.5%

So, the annual rate of interest is 4.5%.

Question: A sum of Taka 50,000 is invested at 6% simple interest for the first 3 years and 8% compound interest for the next 2 years. What is the total amount after 5 years?

Solution: 
Given,
Tk. 50,000 is invested.
6% simple interest for the first 3 years.
8% compound interest for the next 2 years.

Simple Interest, I = Pnr
= 50000 × 3 × (6/100)
= 9000

New Principal = 50,000 + 9,000 = 59,000

Compound Principal = P × (1 + r)ⁿ
= 59000 × (1 + (8/100))²
= 59000 × (1.08)²
= 68,817.60

Question: The difference in Taka between simple and compound interest at 4% annually on a sum of Tk. 8,000 after 2 years is:

Solution:
Given that, Principal, P = Tk. 8,000
Rate, r = 4%
Time, n = 2 years

We know that,
Simple Interest = Prn/100
= (8000 × 4 × 2)/100
= 640

And Compound Interest = P(1 + r/100)ⁿ - P
= 8000{1 + (4/100)}² - 8000 
= 8000(104/100)² - 8000
= {8000 × (104/100) × (104/100)} - 8000 
= 8652.8 - 8000
= 652.8

∴ Difference = 652.8 - 640 = 12.8

The difference between compound and simple interest is Tk. 12.8

Dividend on Tk.20 = Tk. (9/100 × 20) = Tk. 9/5
Tk.12 is income of Tk.100.
∴Tk. 9/5 is an income of = Tk (100/12 × 9/5) = Tk.15

Let the sum be x
Then,
Simple interest = x/4
T = 3(1/8)
= 25/8 years
R = {100 × (x/4)}/{x × (25/8)}
= 8
Hence Required interest rate = 8%

Question: If the simple interest on a sum of money is 40% of the principal over 8 years, what is the annual interest rate?

Solution: 
We know, Simple Interest, I = Pnr

Given, 
Simple Interest = 0.4 × Principal (since 40% of the principal)
= 0.4 × P

Where, Principal = P
Rate = r
Time, n = 8 years

So, 0.4 × P = P × 8 × r
⇒ r = 0.4/8
⇒ r = 0.05 × 100%
∴ r = 5%

Question: A sum of money at simple interest doubles in 10 years. In how many years, at the same rate, will it be tripled?

Solution:
Given that,
Principal = P
Simple interest doubles the money in 10 years
 ∴ SI = P in 10 years.

We know,
SI = Prn/100
⇒ P = Prn/100
⇒ r × 10 = 100
⇒ r = 100/10
∴ r = 10% per annum 

Again,
To triple, total amount = 3P,  SI = 2P
2P = (P × 10 × n)/100
⇒ 2 = n/10
∴ n = 20

∴ Time taken to triple the amount is 20 years.

By investing Tk. 1500, the man obtained an income of Tk. 130.
So, by investing Tk. 95, the income will be = (130/1500) × 96
= Tk 8.32
So, the divided = 8.32%

By investing tk. 1552, income = tk. 128.
By investing tk. 97, income = tk.128/1552 x 97= tk. 8.
Dividend = 8%

Question: Find the difference between the compound profit and the simple profit on Tk. 12,000 for 2 years at 25% per annum.

Solution:
Given that,
P = 12000 Tk
r = 25%
= 25/100
= 1/4

t = 2 years

We know,
The compound profit = P (1 + r)ⁿ - P
= 12000 (1 + 1/4)²- 12000
= (12000 × 25/16)- 12000
= 18750 - 12000
= 6750 Tk

and
Simple profit I = Prn
= 12000 × 1/4 × 2
= 6000

∴ The different between compound profit and simple profit = (6750 - 6000) Tk
= Tk 750

Question: The difference between simple interest for 4 years and 6 years at 5.5% per annum is BDT 220. Find the principal amount.

Solution:
Given that,
Rate of interest, r = 5.5%
Difference in simple interest for 6 years and 4 years = 220
Time difference, n = 6 - 4 = 2 years

We know that,
I = (P × r × n)/100
⇒ 220 = (P × 5.5 × 2)/100
⇒ 11 × P = 22000
⇒ P = 22000/11
∴ P = 2000
∴ The sum is Tk. 2000.

Given amount = Tk. 1348.32
Principle = Tk. 1200
And time = 2 years

According to the law,
A = P(1 + R/100)ⁿ
1348.32 = 1200(1 + R/100)²
1348.32/1200 = (1 + R/100)²
11236/10000 = (1 + R/100)²
(106/100)² = (1 + R/100)²
(1 + 6/100)² = (1 + R/100)²
R = 6% per annum.

Question: The compound interest on Tk. 30000 at 7% per annum is Tk. 4347. The period (in years) is-

Solution:
Given that,
Principal, P = Tk. 30000
Compound Interest, CI = Tk. 4347
Rate, r = 7% per annum
And, Amount, A = P + CI = 30000 + 4347 = 34347

We know,
A = P(1 + r/100)ⁿ
⇒ 34347 = 30000(1 + 7/100)ⁿ
⇒ (107/100)ⁿ = 34347/30000
⇒ (107/100)ⁿ = 11449/10000
⇒ (107/100)ⁿ = (107/100)²
∴ n = 2 

Hence, the period = 2 years.

Just too much information is given in the question to confuse. This is a straight and simple question
Market Value of Company X (his selling price) = Tk. 30
Total shares sold = 4000
The amount he gets = Tk. (4000 × 30)
He invests this amount in ordinary shares of Company Y
Market Value of Company Y(His purchasing price) = 15
Number of shares of company Y which he purchases = (4000 × 30)/15
= Tk. 8000.

Question: Find the simple interest on BDT 12000 at 8% per annum for 6 months.

Solution:
Principal, P = 12000 Taka
Time, n = 6 months = 6/12 = 1/2 years
Rate of interest, r = 8% = 8/100

Simple Interest, I = P × n × r
= 12000 × (1/2) × (8/100)
= (12000 × 1 × 8)/(2 × 100)
= 96000/200
= 480

∴ The simple interest is Tk. 480.

The population grew from 3600 to 4800 in 3 years.
That is a growth of 1200 on 3600 during a three year span.
Therefore, the rate of growth for three years has been constant.

The rate of growth during the next three years will also be the same.
Therefore, the population will grow from 4800 by 4800 × 1/3 = 1600
Hence, the population three years from now will be 4800 + 1600 = 6400

S.I. for 1 year=Tk.(854−815)=Tk.39
S.I. for 3 year=Tk.(39×3)=Tk. 117
∴Principal=Tk.(854−117)=Tk. 698

Let,
The amount invested at 12% be Tk. x and that invested at 10% be Tk. y
Then, 12% x + 10% of y = 130
⇒ 12x + 10y = 13000
⇒ 6x + 5y = 6500 .............(i)

And, 10% x + 12% of y = 134
⇒ 10x + 12y = 13400
⇒ 5x + 6y = 6700 .......(ii)

Adding (i) and (ii) we get,
11(x + y) = 13200
⇒ x + y = 1200 ..........(iii)

Subtracting (i) from (ii) we get,
x + y = 200 .........(iv)

Adding (iii) and (iv) we get,
2y = 1400
⇒ y = 700

∴ x = 1200 - 700 = 500

So,
The amount invested at 12% is TK 500
And the amount invested at 10% is TK 700.

Question: A certain principal amount, invested at simple interest, grows to Tk. 920 after 2 years and Tk. 1010 after 5 years. What is the original principal amount?

Solution:
Given,
Amount after 2 years = Tk. 920
Amount after 5 years = Tk. 1010

∴ Interest for (5 - 2) = 3 years = 1010 - 920 
= Tk. 90
∴ Interest for 1 year = 90/3 = Tk. 30
∴ Interest for 2 years = 30 × 2 = Tk. 60

∴ Principal = 920 - 60 = Tk. 860

Question: On a certain sum of money, the simple interest for 2 years is Tk. 200 at the rate of 5% per annum. If it was invested at compound interest at the same rate for the same duration as before, how much more interest would be earned?

Solution:
Simple Interest (I) = (P × r × n)/100
⇒ 200 = (P × 2 × 5)/100
∴ P = 2000

∴ Compound Interest = P(1 + r)ⁿ - P
= 2000(1 + 0.05)² - 2000
= 2000 × 1.1025 - 2000
= 2205 - 2000
= 205

∴ Difference = 205 - 200 = 5 Tk

Question: A sum of money amounts to Tk. 1800 in 5 years and Tk. 2400 in 8 years at simple interest. Find the annual rate of interest.

Solution:
দেওয়া আছে,
5 বছরের সুদ-আসল = 1800 টাকা
8 বছরের সুদ-আসল = 2400 টাকা
^{——————————————————}
∴ 3 বছরের সুদ = (2400 - 1800) = 600 টাকা
∴ 1 বছরের সুদ = 600/3 = 200 টাকা

এখন,
আসল = 5 বছরের সুদ-আসল - 5 বছরের সুদ
⇒ আসল = 1800 - (5 × 200)
⇒ আসল = 1800 - 1000
⇒ আসল = 800 টাকা

এখানে,
মোট সুদ (I) = 1000 টাকা
আসল (P) = 800 টাকা
সময় (n) = 5 বছর
আমরা জানি, সুদের হার, r = (I × 100)/(P × n)
 ⇒ r = (1000 × 100)/(800 × 5)
⇒ r = 100000/4000
⇒ r = 25
অতএব, বার্ষিক সুদের হার 25%।

Question: A sum of money doubles itself in 8 years at a certain rate of simple interest. In how many years will it become four times itself at the same rate of interest?

Solution: 
Given that, 
The sum doubles itself in 8 years.
Amount after 8 years = 2P
Simple Interest for 8 years = P

We know,
SI = (P × r × n)/100
⇒ P = (P × r × 8)/100
⇒ 8r = 100
⇒ r = 100/8
∴ r = 12.5% per year

Now, we want to find in how many years the sum becomes four times itself.
Amount = 4P
Interest needed = 4P - P = 3P

We know,
SI = (P × r × n)/100 
⇒ 3P = (P × 12.5 × n)/100.  ; [r = 12.5%]
⇒ 3 = (12.5 × n)/100
⇒ n = (3 × 100)/12.5
∴ n = 24 years

∴ The sum will become four times itself in 24 years.

Question: A person who pays income tax at the rate of 4 paise per tk finds that a fall in the interest rate from 4% to 3.75% diminishes his net yearly income by tk 48. What is his capital?

Solution:
If the capital after tax deduction be p, then
p × (4 - 3.75) % = 48
⇒ (p × 0.25)/100 = 48
⇒ (p × 25)/10000 = 48
⇒ p/400 = 48
⇒ p = 48 × 400 = tk 19200

∴ Required capital = (19200 × 100)/96
= 20000 tk

Question: What is the compound amount of Tk. 4000 for 2 years at a rate of interest 5% per annum?

Solution:
Given,
Principal, P = 4000
Rate, r = 5% = 5/100 = 1/20
Time, n = 2 years

We know,
A = P(1 + r)ⁿ
= 4000 × (1 + 1/20)²
= 4000 × (21/20)²
= (4000 × 21 × 21)/(20 × 20)
= (4000 × 441)/400
= 4410

∴ The compound amount is Tk. 4410.

Question: Mohan lent some amount of money at 8% simple interest and an equal amount of money at 10% simple interest each for two years. If his total interest was TK. 720, what amount was lent in each case ?

Solution:
Let the amount invested = TK. P
According to the questions,
⇔ (P × 8 × 2)/100 + (P × 10 × 2)/100 = 720
⇔ (16P + 20P)/100 = 720
⇔ 36P = 72000
⇔ P = 2000

Question: Find the simple interest on BDT 8,000 at 6% per annum for 9 months.

Solution: 
Given, P = 8000 Taka
n = 9 months = 0.75 years
r = 6% 

Simple Interest, I = Pnr
= 8000 × 0.75 × 6/100
= 80 × 4.5
= 360 Taka

Question: If principal M becomes N in 2 years when interest R% is compounded half-yearly. And if the same principal M becomes N in 2 years when interest S% is compound annually, then which of the following is true?

Solution:
ধরি, আসল = M এবং সবৃদ্ধি মূলধন = N
সময় n = 2 বছর

অর্ধবার্ষিক চক্রবৃদ্ধির ক্ষেত্রে (মুনাফার হার R%):
N = M(1 + (R/2)/100)^{2 × 2}
= M(1 + R/200)⁴

বার্ষিক চক্রবৃদ্ধির ক্ষেত্রে (মুনাফার হার S%):
N = M(1 + S/100)²

যেহেতু উভয় ক্ষেত্রে আসল (M), সময় (2 বছর) এবং সবৃদ্ধি মূলধন (N) একই,
সেহেতু যে পদ্ধতিতে বছরে বেশিবার মুনাফা গণনা করা হয় (Half-yearly), সেখানে কাঙ্ক্ষিত মুনাফা পেতে তুলনামূলক কম সুদের হার প্রয়োজন।
∴ R < S

Let the rate of interest be R%
Amount due in 6 months
= 10 + simple interest on Tk. 10 for six months.
= {10 + 10 × R × (1/2)}/100
= 10 + (R/20)
With the formula mentioned,
3 = 100(10 + 9R/20)/{(100 × 6) + R × 6(6 - 1)}/(2 × 12)
⇒ 3 = (1000 + 5R)/{600 + (5R/4)}
⇒ 1800 + 15R/4 = 1000 + 5R
⇒ 5R/4 = 800
⇒ R = 640.
Hence interest rate is 640%

Question: How many years will it take for an investment of Tk.1000 to earn Tk. 200 in simple interest rate of 5%?

Solution: 
Given that,
Principal, P = 1000
Simple Interest, SI = 200
Rate of interest, r = 5%
Time, n = ?

We know,
n = I/Pr
= 200/(1000 × 5%)
= (200 × 100)/(1000 × 5)
= 4

So, it will take 4 years for the investment to earn Tk. 200 at 5% simple interest.

Question: A man invests Tk. 8,100 partly in 14% stock at 294 and partly in 12% stock at 288. If his income from both is the same, find his investment in the 14% stock.

Solution:
Let he invests x at 14% stock.
x Investment at 12% stock = 8100 - x
​
As income is same.
x × (14/100) × (1/294) = (8100 - x) × (12/100) × (1/288)
⇒ x/2100 = (8100 - x)/2400
⇒ x = {(8100 - x)/2400} × 2100
⇒ 24x = 170100 - 21x
⇒ 45x = 170100
∴ x = 3780 Tk

Question: A man borrowed some money for six months. He paid Tk. 500 at an interest rate of 10% per annum. What was the amount he borrowed?

Solution:
এখানে,
সরল সুদ (SI) = Tk. 500
সুদের হার, r = 10%
সময়, n = 6 মাস = 6/12 = 1/2 বছর
আসল (Principal), P = ?

আমরা জানি,
I = Pnr/100
⇒ 500 = (P × 1/2 ×10)/100
⇒ 500 = 5P/100
⇒ 500 = P/20
⇒ P = 20 × 500
∴ P = 10000

অতএব, তিনি মোট Tk. 10,000 ধার নেন।

Question: An investment doubles in 8 years at simple interest. In how many years will it become four times?

Solution:
Given:
In 8 years, interest earned = P (because, P + interest = 2P)

To become 4 times (Total amount = 4P):
Interest needed = 4P - P = 3P

Since interest is earned linearly with time at simple interest:

P interest in = 8 years
3P interest in = 8 × 3 = 24 years

∴ 24 years

Question: What will be the total amount after 3 years if Tk. 1200 is invested at a simple interest rate of 5% annually?

Solution:
Here,
Principal, P = Tk. 1200
Rate of Interest, r = 5% = 5/100
Time, n = 3 years

We know,
Simple Interest, I = P × n × r
I = 1200 × 3 × (5/100)
I = 1200 × 3 × 5/100
I = 3600 × 5/100
I = 180 Tk.

Now, the Amount after 3 years, A = P + I
A = 1200 + 180
A = 1380 Tk.

∴ The amount after 3 years will be Tk. 1380.