Simple and Compound Interest, Stock

Question: A bank offers 12% compound interest calculated half-yearly. A customer deposits Tk. 5000 on 1st January and another Tk. 5000 on 1st July of the same year. How much interest will he earn at the end of the year?

Solution:
এখানে,
অর্ধ-বার্ষিক সুদের হার = 12% ÷ 2 = 6%
প্রথম জমা (1লা জানুয়ারী):
আসল, P₁ = 5000 টাকা
সময়, n₁ = 1 বছর = 2 অর্ধ-বছর
সুদের হার, r = 6%
চক্রবৃদ্ধি মূল (A₁) = P₁ (1 + r/100)^n₁
= 5000 × (1 + 6/100)²
= 5000 × (106/100)²
= 5000 × (106/100) × (106/100)
= 5618 টাকা

দ্বিতীয় জমা (1লা জুলাই):
আসল, P₂ = 5000 টাকা
সময়, n₂ = 6 মাস = 1 অর্ধ-বছর
সুদের হার, r = 6%
চক্রবৃদ্ধি মূল (A₂) = P₂ (1 + r/100)^n₂
= 5000 × (1 + 6/100)¹
= 5000 × (106/100)
= 5300 টাকা

মোট প্রাপ্ত মূল = A₁ + A₂ = 5618 + 5300 = 10918 টাকা
মোট জমা = 5000 + 5000 = 10000 টাকা

∴ মোট অর্জিত সুদ = মোট প্রাপ্ত মূল - মোট জমা
= 10918 - 10000 = 918 টাকা

Question: An amount of money is invested in a savings account for two years. It increased by Tk. 102.50 in two years after annual compounding at the rate of 5% per year. What is the amount invested initially?

Solution:
ধরি, আসল = P টাকা
সময়, n = 2 বছর
মুনাফার হার, r = 5% = 5/100 = 1/20
চক্রবৃদ্ধি মুনাফা = 102.50 টাকা

আমরা জানি, চক্রবৃদ্ধি মুনাফা = P(1 + r)ⁿ - P
⇒ P(1 + 1/20)² - P = 102.50
⇒ P(21/20)² - P = 102.50
⇒ P(441/400) - P = 102.50
⇒ (441P - 400P)/400 = 102.50
⇒ 41P/400 = 102.50
⇒ 41P = 102.50 × 400
⇒ 41P = 41000
⇒ P = 41000/41
∴ P = 1000

সুতরাং, শুরুতে বিনিয়োগকৃত আসলের পরিমাণ ছিল 1000 টাকা।

Question: In a certain time, a sum becomes 4 times of itself on simple interest at the rate of 10% per annum. What is the rate of interest if the same sum becomes 7 times of itself in the same duration?

Solution:
Principal = P
Time = n (same in both cases)
Rate, r =  10%
Simple Interest, SI = 4P - P = 3P

We know,
SI = Prn 
⇒ 3P = (P × 10 × n)/100
⇒ 3P = (P × n)/10
∴ n = 30 years

Again,
In the second scenario, the same sum becomes 7 times itself in the same duration (30 years).
Final Amount, A = 7P
Simple Interest, SI = A - P = 7P - P = 6P
Time, n = 30 years

SI = Prn 
⇒ 6P = (P × r × 30)/100
⇒ 6 = (r × 3)/10
⇒ r = 60/3
∴ r = 20%

So the new rate of interest is 20% per annum.

Question: How many years will it take for an investment of Tk. 15000 to earn Tk. 3600 in simple interest rate of 4%?

Solution:
Given that,
Principal, P = 15000
Simple Interest, I = 3600
Rate of interest, r = 4%
Time, n = ?

We know,
I = Pnr/100
⇒ n = (I × 100)/ (P × r)
= (3600 × 100)/(15000 × 4)
= 6

So, it will take 6 years for the investment to earn Tk. 3600 at 4% simple interest.

Question: A person deposited Tk. 2500 for 4 years and Tk. 3600 for 3 years at the same rate of simple interest in a bank. If he received a total of Tk. 1040 as interest, what was the rate of interest per annum?

Solution:
ধরি, বার্ষিক সরল মুনাফার হার = r%

আমরা জানি, মুনাফা = (আসল × সময় × হার)/100

প্রথম ক্ষেত্রে মুনাফা = (2500 × 4 × r)/100 = 100r টাকা
দ্বিতীয় ক্ষেত্রে মুনাফা = (3600 × 3 × r)/100 = 108r টাকা

প্রশ্নমতে,
100r + 108r = 1040
⇒ 208r = 1040
⇒ r = 1040/208
⇒ r = 5

∴ বার্ষিক মুনাফার হার 5%

Question: A woman purchased 500 shares of the face value of Tk. 100 each from the market at Tk. 160 per share. If a dividend of 20% is declared, find her earning percent on the investment.

Solution:
Given,
Price of 1 share (Market Value) = Tk. 160
∴ Price of 500 shares (Total Investment) = Tk. (160 × 500) = Tk. 80,000

Dividend per share = 20% of 100 = Tk. 20
∴ Total dividend income = 500 × 20 = Tk. 10,000

∴ Earning % = (Total Dividend/Total Investment) × 100
= (10,000/80,000) × 100
= (1/8) × 100
= 12.5%

Simple interest for 3 years
= 12005 - 9800
= 2205
Simple interest for 5 years
= (2205/3) × 5
= 3675
Some of money = 9800 - 3675
= 6125
R = (100 × 2205)/(6125 × 3)
= 12%

Question: In how many years Tk. 1000 will become Tk. 1331 at compound interest rate of 10% per annum?

Solution:
Given,
P = Tk. 1000
C = Tk. 1331
r = 10% p.a

We know,
P(1 + r)ⁿ = C
⇒ 1000(1 + 10%)ⁿ = 1331
⇒ (1 + 1/10)ⁿ = 1331/1000
⇒ (11/10)ⁿ = (11/10)³
∴ n = 3

∴ required time = 3 years

Question: A sum of Tk. 800 amounts to Tk. 920 in 3 years at a certain simple interest rate. If the rate of interest is increased by 3% per annum, what will be the total amount after 3 years?

Solution:
Here, 
Sum = Tk 800, 
Amount = Tk 920,

∴ Simple interest = Amount - Principal 
= 920 - 800
= Tk 120

∴ Rate = (Interest × 100)/(Principal × Time)
= (120 × 100)/(800 × 3)
= 5% per annum

If the rate is increased by 3%, 
∴ Rate = 8% per annum

∴ simple interest = (800 × 8 × 3)/(800 × 3)
= Tk 192

∴ Amount = Principal + S.I.
= (800 + 192)
= Tk 992

Question: A borrows sum of TK. 90000 for 4 years at 5% simple interest. He lends it to B at 7% for 4 years at simple interest. What is his gain?

Solution:
A's, Principal = TK. 90000
simple interest rate = 5%
time = 4 years

∴ simple interest = TK. (90000 × 4 × 5)/100
= TK. 18000

B's, Principal = TK. 90000
simple interest rate = 7%
time = 4 years

∴ simple interest = TK. (90000 × 4 × 7)/100
= TK. 25200

∴ Profit of A = TK. (25200 - 18000) = TK. 7200

Question: In how many years will Tk. 750 amount to Tk. 900 at 4% simple interest per annum?

Solution:
Simple Interest = Amount - Principal
= 900 - 750
= 150

Here,
Principal, P = 750
Interest Rate, R = 4%
SI = 150
Time, T = ?

SI = PRT/100
⇒  T = (SI × 100)/(P × R)
= (150 × 100)/(750 × 4)
= 15000/3000
= 5 years

Question: The simple interest at p% for p years will be Tk. p on a sum of?

Given,
Rate of interest, r = p%
Interest, I = p Tk.
Time, n = p years.

We know,
I = Pnr
P = I/nr
= (p × 100)/(p × p)
= 100/p Tk.

Question: A sum of money doubles itself in 5 years at a certain rate of simple interest. In how many years will it become five times itself at the same rate of interest?

Solution:
Let the Principal be P.
The amount doubles in 5 years, so Amount = 2P.
Interest (I) for 5 years = 2P - P = P.

Now, the sum needs to become five times itself, so New Amount = 5P.
Required Interest (I) = 5P - P = 4P.

We know that,
P amount of interest is earned in = 5 years.
∴ 4P amount of interest will be earned in = (5 × 4) years = 20 years.

∴ The sum of money will become five times itself in 20 years.

Question: What is the difference between simple and compound interest at 5% per annum on a sum of Tk. 8000 at the end of 2 years?

Solution:
এখানে,
মূলধন (P) = 8000 টাকা
সুদের হার (r) = 5%
সময় (n) = 2 বছর

সরল সুদ (SI):
SI = (P × r × n)/100
= (8000 × 5 × 2)/100
= 800 টাকা

চক্রবৃদ্ধি সুদ (CI):
A = P(1 + r/100)ⁿ
= 8000(1 + 5/100)²
= 8000(105/100)²
= 8000 × (105/100) × (105/100)
= 8820 টাকা

চক্রবৃদ্ধি সুদ (CI) = A - P
= 8820 - 8000
= 820 টাকা

∴ সুদের পার্থক্য = CI - SI
= 820 - 800
= 20 টাকা

Let P = 100,
A = 200,
It means I = 100

Now We want A = 300,
means I = 200
Years for I of Tk. 100 = 10
Years for I of Tk. 200 = (10/100) × 200 = 20 Years

Alternative Method:
Assuming simple interest, let the amount be P and rate of interest per annum be R%.

Given that P becomes 2P in 10 years.
=> P + PX10X R/100 = 2P
=> 1 + R/10 = 2
=> R = 10

Let P triple itself in N years:
=> P + PXNX 10/100 = 3P
=> 1 + N/10 = 3
=> N = 20 years

১ম 6 মাসে মুনাফা = 5000 × 6/12 × 7/100 = 175
২য় 6 মাসে মুনাফা = 5175 × 6/12 × 7/100 = 362.25/2 = 181.125
∴ বছর শেষে মোট টাকার পরিমান = 5000 + 175 + 181.125 = 5356.125 ≈ 5356

Question: What is the total interest on Tk 2,000 at 12.5% per annum for 9 months (in taka)?

সমাধান:
আসল, P = 2,000 টাকা
সুদের হার, r = 12.5% = 12.5/100 = 1/8
সময়, n = 9 মাস = 9/12 = 3/4 বছর

সুদ, I = Pnr
⇒ সুদ, I = 2,000 × (3/4) × (1/8)
∴ সুদ, I = 187.5 টাকা

Question: A sum of TK. 800 amounts to TK. 920 in 3 years at simple interest. What will it amount to if the rate of interest is increased by 3%?

Solution:
Principal (P) = 800 TK.
Amount after 3 years = 920 TK.
∴ Simple Interest (SI) for 3 years = 920 - 800 = 120 TK.

Now, SI = (P × n × r)/100
⇒ 120 = (800 × 3 × r)/100
⇒ 120 = 24r
⇒ r = 120/24
∴ r = 5%

New Interest Rate = 5% + 3% = 8%
New simple interest for 3 years = (800 × 3 × 8) / 100 = 192 TK.
New Amount = Principal + New Interest = 800 + 192 = 992 TK.

Alternative Shortcut Method:
Additional Interest for 3% increase in 3 years = (800 × 3 × 3)/100 = 72 TK.
∴ New Amount = 920 + 72 = 992 TK.

T.D. = (B.G.×100)/(R×T)
= Tk.(6×100 / 12×1)
= Tk.50

Question: If the simple interest on Tk. N at N% per annum for 5 years amounts to Tk. 2N, what is the value of N?

Solution:
Given,
Principal, P = N
Rate, r = N% = N/100
Time, n = 5 years
Simple Interest, I = 2N

We know,
I = (P × r × n)/100
⇒ 2N = N × (N/100) × 5
⇒ 2N = 5N²/100
⇒ 2N × 100 = 5N²
⇒ 200N = 5N²
⇒ N² = 40N
⇒ N = 40

∴ The value of N = 40

Question: How many years will it take for an investment of Tk. 10000 to earn Tk. 3000 in simple interest at a rate of 6%?

Solution:
Given that,
Principal, P = 10000
Simple Interest, I = 3000
Rate of interest, r = 6%
Time, n = ?

We know,
I = Pnr/100
⇒ 3000 = (10000 × n × 6)/100
⇒ 3000 = 100 × n × 6
⇒ 3000 = 600n
⇒ n = 3000/600
⇒ n = 5

So, it will take 5 years for the investment to earn Tk. 3000 at 6% simple interest.

Let the rate of interest be R%
Amount due in 10 months
= 9 + simple interest on Tk. 9 for ten months
= 9 + {9 × R × (10/12)}/100
= 9 + (3R/40)
With the formula mentioned,
1 = 100{9 + 3R/40)}/[(100 × 10) + {R × 10(10 - 1)/(2 × 21)}]
900 + (15R/2) = 1000 + (15R/4)
15R/4 = 100
R = 26.67.
Hence the interest rate is 26.67%.

Question: If the simple interest on Tk. X at X% per annum for 5 years is Tk. X, what is the value of X?

Solution:
Given,
P = X
r = X% = X/100
n = 5 years
I = X

We know,
I = Prn
⇒ X = X × (X/100) × 5
⇒ X = X × (X/20)
∴ X = 20

Let Principle = P.
Then, S.I = 2P and T = 20 years
∴ Rate = {(100 × 2P)/(P × 10)}
Now, principle = P, S.I. = P, R = 10%

∴ Time = {(100 × P)/(P × 10)} yrs
= 10 yrs.

Question: A 15% stock yields 12.5%. What is the market value of the stock?

Solution:
Earn Tk. 12.5 when market value = Tk. 100
Earn Tk. 1 when market value = Tk. (100 / 12.5)
Earn Tk. 15 when market value = Tk. (100 × 15)/12.5
= 1500/12.5
= 120

∴ The market value of the stock is Tk. 120.

Simple interest, I = pnr = 5000 × 5 × 10/100
= 2500 taka
Compound interest = Amount − Principal
Amount = 5000(1 + 10/100)⁵ = 8052.55
So, compound interest = 8052.55 - 5000 = 3052.55

∴ Difference = 3052.55 - 2500 = 552.55 ≅ 552 taka

Let x be the market value,
we don't know face value so consider 100.
6% of 100 = 8% of x.
(6/100) × 100 = (8/100) × x.
6 = 8x/100.
x = 600/8.
x = 75.

Alternative Solution:
For an income of Tk. 8, investment = Tk. 100.
For an income of Tk. 6, investment = Tk.(100/8 × 6) = Tk. 75
Market value of Tk. 100 stock = Tk. 75.

Question: What is the ratio of simple interest earned on a certain amount at the rate of 12% p.a. for 9 years and that for 12 years?

Solution: 
As the amount and interest rate are the same, 
Ratio = 9/12
= 3/4

Number of shares = (14400/120) = 120
Face value = (100 × 120) = 12000
Annual income = (5/100 × 12000) = 600

Question: Mr. Karim deposited a certain amount of money for a fixed period of time. On maturity, he received a total of Tk. 50,000 when the ratio of interest and investment became 1: 4. If the simple interest rate was 5%, calculate the time period for which the money was invested.

Solution:
প্রদত্ত তথ্য অনুযায়ী, আসল এবং সুদের অনুপাত = 4 : 1
মোট প্রাপ্ত টাকা = 50,000 টাকা
মোট অনুপাত = 4 + 1 = 5
সুতরাং, আসল = 50,000 টাকার (4/5) অংশ = 40,000 টাকা
এবং, সুদ = 50,000 টাকার (1/5) অংশ = 10,000 টাকা

এখানে,
I = সুদ = 10,000 টাকা
P = আসল = 40,000 টাকা
R = সুদের হার = 5%
T = সময়কাল = ?

আমরা জানি, সরল সুদের ক্ষেত্রে,
I = (P × R × T)/100
⇒ 10,000 = (40,000 × 5 × T)/100
⇒ 10,000 = 400 × 5 × T
⇒ 10,000 = 2,000 × T
⇒ T = 10,000/2,000
⇒ T = 5

সুতরাং, টাকাটি 5 বছরের জন্য বিনিয়োগ করা হয়েছিল।

Question: What will be the difference between simple and compound interest at 5% on a sum of Tk. 8000 after 2 years?

Solution:

দেওয়া আছে,
Principal, P = 8000 টাকা
Rate of interest, r = 5%
Time, n = 2 বছর

Simple Interest (SI):
SI = (P × r × n) / 100 = (8000 × 5 × 2) / 100 = 800 টাকা

Compound Interest (CI):
CI = P × (1 + r/100)ⁿ - P
= 8000 × (1 + 5/100)² - 8000
= 8000 × (1.05)² - 8000
= 8000 × 1.1025 - 8000
= 8820 - 8000 = 820 টাকা

∴ Difference between CI and SI:
= 820 - 800 = 20 টাকা

Question: If the simple interest on Tk. N at N% per annum for 5 years is Tk. N, what is the value of N?

Solution:
Given,
P = N
r = N% = N/100
n = 5 years
I = N

We know,
I = Prn
⇒ N = N × (N/100) × 5
⇒ N = (N² × 5)/100
⇒ N = N²/20
⇒ N² = 20N
⇒ N²/N = 20
∴ N = 20

সুতরাং, N এর মান হলো 20