Simple and Compound Interest, Stock

Question: The simple interest on a certain sum at 10% per annum for 2 years is Tk. 2,000. Find the compound interest on the same sum at the same rate for 2 years.

Solution:
We know,
I = Pnr

Given,
I = 2,000 Tk.
r = 10%
n = 2 years

∴ P = I/nr
= (2,000 × 100)/(2 × 10)
= 10000

Now,
Compound Principal = 10,000 × (1 + 10/100)²
= 10,000 × (110/100)²
= 10,000 × (11/10)²
= 10,000 × (121/100)
= 12,100

So, Compound interest = 12,100 - 10,000
= 2,100 Tk.

Let the interest rate be r%
We know that,
S.I = PTR/100
=> (1540 x 5 x r)/100  +  (1800 x 4 x r)/100 = 1788
=> r = 178800/14900 = 12%

Question: The simple interest on a certain sum at the rate of 5% per annum for 3 years and 7 years differs by Tk. 400. What is the principal amount?

Solution:
Let the principal amount be Tk. P
Then, ATQ
400 = {(P × 5 × 7)​/100} - {(P × 5 × 3​)/100}
⇒ 400 = (35P - 15P)/100
⇒ 40000 = 20P
⇒ P = 40000/20
∴ P = 2000

So the principal amount is Tk. 2000

Question: At what rate of compound interest per annum will a sum of Tk. 4000 becomes Tk. 4840 in 2 years?

​Solution:
Principal, P = Tk. 4000
Compound Amount, C = Tk. 4840
Time, n = 2 years
Rate, r = ?

We know,
C = P × (1 + r/100)ⁿ
⇒ 4840 = 4000 × (1 + r/100)²
⇒ (1 + r/100)² = 4840/4000 
⇒ (1 + r/100)² = 484/400
⇒ 1 + r/100 = 22/20 [উভয়পাশে বর্গমূল করে]
⇒ r/100 = (11/10) - 1
⇒ r/100 = (11 - 10)/10
⇒ r/100 = 1/10
⇒ r = (1 × 100)/10
∴ r = 10

∴ Interest Rate = 10%

Question: What yearly income can be earned by investing Tk. 18000 in 12% stock at Tk. 90?

Solution:
12% stock at Tk. 90 বলতে বুঝায় স্টকটির ফেস ভ্যালু 100 টাকায় 12 টাকা লাভ হয় এবং স্টকটির বাজার মূল্য 90 টাকা।

এখন, 
90 টাকা বিনিয়োগ করে আয় হয় 12 টাকা
∴ 1 টাকা বিনিয়োগ করে আয় হয় 12/90 টাকা
∴ 18000 টাকা বিনিয়োগ করে আয় হয় (18000 × 12)/90 টাকা
= 216000/90 টাকা = 2400 টাকা

Question: What is the compound amount of Tk. 3200 for 2 years at a rate of interest 5% per annum?

Solution:
Given,
Principal, P = 3200
Rate, r = 5% = 5/100 = 1/20
Time, n = 2 years

We know,
A = P(1 + r)ⁿ
  = 3200 × (1 + 1/20)²
  = 3200 × (21/20)²
  = (3200 × 21 × 21) / (20 × 20)
  = (3200 × 441) / 400
  = 1411200 / 400
  = 3528

∴ The compound amount is Tk. 3528.

Let the sum be Tk. P.
Then,[p(1+10/100)²-p]=525
Sum =Tk.2500
S.I.= Tk.(2500×5×4)/100
= Tk. 500

Question: At what rate percent per annum will a sum of Tk. 64,000 amount to Tk. 85,184 in 3 years at compound interest?

Solution:
Given that,
Principal, P = Tk. 64,000
Amount, C = Tk. 85,184
Time, n = 3 years

We know,
C = P(1 + r)ⁿ
⇒ 85,184 = 64,000 × {1 + (r/100)}³
⇒ 85,184/64,000 = {1 + (r/100)}³
⇒ 1.331 = {1 + (r/100)}³
⇒ (1.1)³ = {1 + (r/100)}³

∴ 1 + (r/100) = 1.1
⇒ r/100 = 0.1
∴ r = 10%

Interest compounded half yearly = 5000(1 + 2/100)²
= 5000(102/100)²
= 5000(1.02)² .........(i)

As we know, A = P(1 + R/100)^t
Interest compound yearly = 5000(1 + 4/100)¹
= 5000(104/100)
= 5000(1.04) ............(ii)

From (i) and (ii)
Required difference = 5000(1.02)² - 5000(1.04)¹
= 5000(1.0404) - 5000(1.04)
= 5202 - 5200
= 2.

Given that, principal = P = Tk. 33600 and
R = 25/4 %
Time duration = From September 2019 to May 2020
= 9 months
= 9/12 year
= 3/4 year
S.I = (Principle × Interest × Time)/100
= PRT/100
= Tk. (33600 × 25/4 × 3/4) × 1/100
= 21 × 25 × 3
= Tk. 1575.
Hence, the answer is Tk. 1575.

Suppose 100 be the principal.
In 4 years it doubles and becomes 200.
In next 4 years 200 double to 400.
In next 4 years 400 doubles to 800.
Hence the period for 8 times is 12 years.

Question: A sum of TK. 600 amounts to TK. 720 in 4 years at simple interest. What will it amount to if the rate of interest is increased by 2%?

Solution: 
Given that,
Principal, P = Tk. 600 
Amount = Tk. 720
Time, n = 4 years
∴ Simple Interest, SI = Amount - Principal = 720 - 600 = Tk. 120

We know, 
SI = (P × r × n)/100
⇒ 120 = (600 × r × 4)/100
⇒ 120 = (2400 × r)/100
⇒ 120 = 24r
⇒ r = 120/24
∴ r = 5%

Again, 
New rate = original rate + 2% = 5% + 2% = 7%

∴ New SI = (P × r × n)/100
= (600 × 7 × 4)/100
= 16800/100
= Tk. 168

∴ New Amount = Principal + New Interest
= 600 + 168
= Tk. 768

∴ If the rate of interest is increased by 2%, the sum will amount to Tk. 768 in 4 years.

R = {(20 × 100)/1000)}
= 2% [percentage is calculated for twenty per thousand]
Population after 2 years
= P(1 - R/100)^T
= 40000{1 - (2/100)}²
= 40000{1 - (1/50}²
= 40000(49/50)²
= (40000 × 49 × 49)/(50 × 50)
= 16 × 49 × 49
= 38416.

Given, P = 4000
Interest rate per year = 10%, so per half year = 5%
Time = 3/2 years or 3 half year

C = 4000 × 105/100 × 105/100 × 105/100
= 4630.50

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.

Compound interest for 1 year.
2{3%(P)} + 3%{3%(P)}
= 6%(P) + 0.09%(P)
= 6.09%(P)
That is an effective interest rate for 1 year = 6.09%.

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) = Tk. 15

Number of shares of company Y which he purchases = (4000 × 30)/15
= Tk. 8000.

Question: A certain sum of money becomes 2.5 times itself in 10 years at simple interest. What is the rate of interest per annum? 

Solution: 
Let the principal amount be P.
The sum becomes 2.5 times itself in 10 years.

So, A = 2.5P
∴ Simple Interest, I = 2.5P - P = 1.5P

We know, I = Pnr/100
1.5P = P × 10 × r/100
10r = 150
∴ r = 15%

Question: For what percentage of profit per annum by which Tk. 12000 will be Tk. 17760 as profit-principal in 6 years?

Solution: 
Principal, P = Tk. 12000
Total Amount, A = Tk. 17760
Time, n = 6 years
Simple Interest SI = A - P = 17760 - 12000 = Tk. 5760

We know, 
SI = Prn/100
⇒ 5760 = (12000 × r × 6)/100
⇒ r × 720 = 5760
⇒ r = 5760/720
∴ r = 8

So the profit percentage of per annum is 8%.

Let the sum be P
then,
P(1 + r/100)³ = 6690 .......(i)
and
P(1 + r/100)⁶ = 10035 .......(ii)
On dividing (ii) by )i),
(1 + r/100)³ = 10025/6690 = 3/2
Substituting this value in (i) we find, P×(3/2) = 6690
⇒ P = (6690×2/3) = 4460

The difference in interest rate = 6(1/4)% - 4%
= (25/4)% - 4%
= (9/4)%
Gain per year
= simple interest on Tk. 5000 at = (9/4)% for 1 year.
= {5000 × (9/4) × 1}/100
= 225/2
= Tk. 112.5

Question: In how many years will Tk. 15,000 amount to Tk. 18,600 at 12% per annum simple interest?

Solution:
Principal, P = Tk. 15,000
Amount, A = Tk. 18,600

∴ Simple interest, I = A - P
= 18,600 - 15,000
= Tk. 3600

Rate of interest, r = 12%

We know,
I = Pnr
⇒ 3,600 = 15,000 × n × (12/100)
⇒ 3,600 = 15,000 × n × (3/25)
⇒ 3,600 = 1,800n
⇒ n = 3,600/1,800
∴ n = 2 years

ধরি, সুদের হার r %
প্রশ্নমতে, (400 × 2 × r)/100 + (550 × 4 × r)/100 + (1200 × 6 × r)/100 = 1020
⇒ 8r + 22r + 72r = 1020
⇒ 102r = 1020
⇒ r = 10%

Question: Find the compound interest on Tk. 2000 at the rate of 20% per annum for 1.5 years. When interest is compounded half-yearly.

Solution: 
Principal P = 2000
Rate, R = 20%
Time, T = 1.5 years

Now,
Compound interest for half-yearly:
A = P{1 + R/(2×100)}^2T
= 2000{1 + (20/200)}^2×1.5
= 2000 × {1 + (1/10)}³
= 2000 × (11/10)³ 
= 2000 × (1331/1000)
= 2662

Compound Interest = A – P
= 2662 – 2000 = 662 Tk

Question: A sum of 12,000 Taka is lent out in two parts. One part is lent at 8% simple interest and the other at 10% simple interest. If the total annual interest is 1,080 Taka, how much is lent at 8%?

Solution:
Let, 
Amount lent at 8% simple interest = x
Amount lent at 10% simple interest = (12000 - x)

We know,
I = Pnr
I = Pr [Since the time is 1 year]

Interest from the amount lent at 8%, 
I = x(8/100) = 0.08x

Interest from the amount lent at 10%, 
I = (12000 - x) × (10/100) = 0.10(12,000−x)

ATQ,
0.08x + 0.10(12,000−x) = 1080
⇒ 0.02x = 120
∴ x = 6000 Taka

So, Amount lent at 8% simple interest = 6000 Taka

Number of shares = (4455/8.25)
= 540
Face value = Tk(540 × 10)
= Tk. 5400
Tk (12/100 × 5400) = Tk. 648

Face value of each share = Tk.20
Market value of each share = Tk.25
Number of shares = 12500

The amount required to purchase the shares
= 12500 × 25
= 312500

Question: Find the amount of compound interest on Tk. 2000 for 1.5 years at a 20% annual rate, with interest added semi-annually.

Solution: 
Principal P = 2000
Rate, R = 20%
Time, T = 1.5 years

Now,
Compound interest for half-yearly:
A = P{1 + R/(2×100)}^2T
= 2000{1 + (20/200)}^2×1.5
= 2000 × {1 + (1/10)}³
= 2000 × (11/10)³ 
= 2000 × (1331/1000)
= 2662

Compound Interest = A – P
= 2662 – 2000 = 662 Tk

Interest for 1 year is the same whether it's simple interest or compound interest.
Now interest of third-year = 8575 - 7350 = 1225;
means principal for this interest is 7350 if compound interest is taken 7350 is the principal interest = 1225

if 100 is the principal interest = (1225/7350) × 100 = 50/3%
When a thing increases for two successive times the overall increase on initial amount = a + b + (a × b)/100

Therefore overall interest for two years = 50/3 + 50/3 + [(50/3) × (50/3)]/100 = 325/9 %
Therefore amount after 2 years = 100 + 325/9 = 1225/9

If 1225/9 is the amount principal =100
if 7350 is the amount principal =(900/1225) × 7350 =5400

So sum = 5400

Question: Find the amount if Tk. 2000 is invested at 10% compound interest p.a. for 3 years.

Solution: 
Given,
P = Tk. 2000
r = 10%
n = 3 years

We know,
C = P(1 + r)ⁿ
= 2000{1 + (10/100)}³
= 2000(1 + 1/10)³
= 2000(11/10)3
= 2000 × 1.1 × 1.1× 1.1
= 2662

For an income of Tk. 10, investment = Tk. 96
For an income of 12, investment
= Tk. (96/10) × 12
= Tk. 115.20

Hence, He must purchase a stock worth of Tk. 115.20

Question: Find the difference between the simple interest and the compound interest at 10% per annum for 2 years on a principal of Tk 3,000.

Solution:
We know, Simple Interest, I = pnr
and Compound Principal, C = p(1 + r)ⁿ

Simple Interest = 3,000 × 2 × (10/100) = 600 Tk

Compound Principal = 3,000 × (1 + 10/100)²
= 3,000 × (110/100)²
= (3,000 × 110 × 110)/(100 × 100)
= 3,630

So, Compound interest = 3,630 - 3,000 = 630 Tk

So, difference = 630 - 600 = 30 Tk.

Question: A man borrowed some money for 90 days. He paid Tk. 270 as interest at the rate of 8% per annum. What was the amount he borrowed?

Solution:
এখানে,
সময়, n = 90 দিন
= 3 মাস = 3/12 বছর
= 1/4 বছর

মুনাফা, I = 270 টাকা
মুনাফার হার, r = 8% = 8/100 = 2/25
আসল, P = ?

আমরা জানি,
I = P × n × r
⇒ P × n × r = I
⇒ P = I/(n × r)
= 270/{(1/4) × (2/25)}
= 270/(1/50)
= 270 × 50
= 13500 টাকা

1852.20 = 1600(1 + r/100)3
Or, 1852.2/1600 = (1 + r/100)3
Or, 18522/16000 = (1 + r/100)3
Or, (21/20)3 = (1 + r/100)3
Or, (1 + r/100) = 21/20
Or, r/100 = 1/20
So, r = 5

Question: Find the compound interest on TK. 30000 at 7% interest compounded annually for two years.

Solution: 
Principal, P = Rs 30000
Rate, R = 7%
Time, T = 2 year

By formula:
A = P(1 + R/100)^T 
= 30000 (1 + 7/100)²
= 30000 (107/100)²
= 30000 × (11449/10000)
= 34347

Compound Interest = A - P = 34347 - 30000
= 4347

Question: The compound interest on Tk. 30,000 at an annual rate of 7% amounts to Tk. 4,347. What is the time period (in years) for which the money was invested? 

Solution:
Amount = 30000 + 4347 = 34347 tk

Let, the time be n years,
Then,
30000{1 + (7/100)}ⁿ = 34347
⇒ (107/100)ⁿ = 34347/30000
⇒ (107/100)ⁿ = 11449/10000
⇒ (107/100)ⁿ = (107/100)²
∴ n = 2 years