Simple and Compound Interest, Stock

Question: A man buys Tk. 20 shares paying 9% dividend. The man wants to have an interest of 12% on his money. The market value of each share is-

Solution:
Dividend on Tk. 20 = Tk. (9/100) × 20 = Tk. 9/5

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

Question: There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Tk. 12000 after 3 years at the same rate?

Solution:
Given that,
Increase in amount after 6 years = 60%

We know,
A = P(1 + r/100)ⁿ
simple Interest = (principal × rate × time)/100

Now,
Let the amount at 1^st year be 100x
∴ Increased in amount = 60% of 100x = 60x
⇒ 60x = (100x × 6 × r)/100
⇒ 6r = 60
⇒ r = 60/6 = 10
∴ r = 10% 

∴ Compound Interest after 3 years = P(1 + r/100)ⁿ - P
= 12000(1 + 10/100)³ - 12000
= 12000 × (11/10)³ - 12000
= 12000 × (11/10) × (11/10) × (11/10) - 12000
= 15972 - 12000
= Tk. 3972

∴ The required answer is Tk. 3972

Question: The difference between simple and compound interests compounded annually on a certain sum of money for 2 years at 5% per annum is Tk. 3. The sum is-

Solution:
Let, Sum = x
Here, r = 5% = 5/100 and n = 2 

Now, S.I. = (x × 5 × 2)/100
= x/10

And, C.I. = [x(1 + (5/100))2 - x]
= (441x - 400x)/400
= 41x/400

ATQ,
(41x/400) - (x/10) = 3
⇒ (41x - 40x)/400 = 3
⇒ x/400 = 3
∴ x = 1200

Number of shares purchased = 5508/102
= 54.

Income from each share = 4% of Tk. 100
= Tk. 4

∴ Original income = Tk. (54 × 4) = Tk. 216

Money incurred from sale of share = Tk. (105 × 54)
= Tk. 5670

Number of new shares purchased = 5670/126 = 45

New income = Tk. (45 × 5)
= Tk. 225

∴ Change in income = Tk. (225 - 216)
= Tk. 9.

By investing Tk. 1552, income = Tk. 128
By investing Tk. 97, income = (128 × 97)/1552
= 8
Hence, the dividend is 8%.

6% হার সুদে 360 টাকার 5 বছরের সুদ = (360 × 5 × 6)/100 = 108 টাকা
∴ 360 টাকার 4 বছরের সুদ = 540 - (360 + 108) = 72 টাকা
∴ প্রথম 4 বছরে সুদের হার ছিল = (100 × 72)/(360 × 4) = 5% [এখানে, r = (100 × 1)/(p × n) সূত্র ব্যবহার করে]

Question: Kamal invested Tk. 500 in EBL for 2 years and Tk. 300 in SIBL for 4 years. At the end of the period, he received a total of Tk. 220 as simple interest from both banks. What was the rate of interest per annum?

Solution:
Let the common rate of interest be r % per annum.

We know,
SI = (P × r × t)/100

Interest from EBL,
(500 × r × 2)/100 = (1000r)/100 = 10r

And, 
Interest from SIBL,
(300 × r × 4)/100 = (1200r)/100 = 12r

∴ Total interest received = Interest from EBL + Interest from SIBL
⇒ 220 = 10r + 12r 
⇒ 220 = 22r
⇒ r = 220/22
∴ r = 10

So the common rate of interest is 10% per annum.

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}]
= Tk. 1640.

Question: The list price of a commodity is the price after a 20% discount on the retail price. The festival discount price on the commodity is the price after a 30% discount on the list price. Customers purchase commodities from stores at a festival discount price. What is the effective discount offered by the stores on the commodity on its retail price?

Solution: 
Let r be the retail price.
The list price is the price after a 20% discount on the retail price.
Hence, the list price = r[1 - (20/100)] 
= r[1 - (1/5)]
= 4r/5

The festival discount price is the price after a 30% discount on the list price.
Hence, the festival discount price = (4r/5) [1 - (30/100)]
= (4r/5) × [1 - (3/10)] 
= (4r/5) × (7/10)
= 14r/25

Hence, the total discount offered is [(Original Price - Price after discount) × 100]/Original Price
= [r - (14r/25) × 100]/r 
= (11r × 100)/25r
= 44%

Total cost of the computer = Tk. 39000
Down payment = Tk. 17000
Balance = Tk. (39000 - 17000) = Tk. 22000.
Let the rate of interest be R% p.a.
Amount of Tk. 22000 for 5 months
= {22000 + 22000 × (5/12) × R/100}
= {22000 + (275R/3)}

The customer pays the shopkeeper Tk. 4800 after 1 month,
Tk. 4800 after 2 months, ...... and Tk. 4800 after 5 months.

Thus, the shopkeeper keeps Tk. 4800 for 4 months, Tk. 4800 for 3 months, Tk. 4800 for 2 months, Tk. 4800 for 1 months and Tk. 4800 at the end.

∴ sum of the amounts of these installments
= (Tk. 4800 + S.I. on Tk. 4800 for 4 months) + (Tk. 4800 + S.I. on Tk. 4800 for 3 months) + ...... + (Tk. 4800 + S.I. on Tk. 4800 for 1 month) + Tk. 4800
= Tk. (4800 × 5) + S.I. on Tk. 4800 for (4 + 3 + 2 + 1) months
= Tk. 24000 + S.I. on Tk. 4800 for 10 months
= 24000 × 4800 × R × (10/12) × (1/100)
= (24000 + 40 R)

∴ {22000 + (275R/3)} = (24000 + 40 R)
⇒ 155R/3 = 2000
⇒ R = (2000 × 3)/155
= 38.71

Question: A sum of Tk. 1600 is invested at 15% simple interest per annum for 10 months. What will be the interest?

Solution:
দেওয়া আছে,
আসল, P = Tk. 1600
সুদের হার, r = 15%
সময়, n = 10 months = 10/12 year = 5/6 year

আমরা জানি,
সুদ (I) = Pnr/100
= 1600 × (5/6) × 15/100
= (16 × 5 × 15)/6
= 1200/6
= 200 Taka

∴ 10 মাসে মোট মুনাফা হবে 200 টাকা।

Question: If a person invests Tk. 900 at 12% simple interest and Tk. 700 at 8% simple interest, how much interest will he receive after four years?  

Solution:
এখানে,
I₁ = p₁n₁r₁
= (900 × 4 × 12)/100 
= 432 টাকা 

আবার,
I₂ = p₂n₂r₂
= (700 × 4 × 8)/100 
= 224 টাকা

∴ মোট সুদ = (432 + 224) টাকা
= 656 টাকা

Face value of each share = Tk. 20
Dividend per share = 9% of 20 = (9 × 20)/100
= 9/5.
He needs to have an interest of 12% on his money
Money Paid for a share = (9/5) × (12/100)
Money Paid for a share = (9/5) × (100/12)
= 15.
ie, Market Value of the share = Tk. 15.

Principle = P, Simple interest = P (since both are same), R = 25/4, time = n
Interest = pnr/100
Or, p = pn(25/4)/100
Or, n = p × 100 × 4/25 × p
Or, n = 16

Question: A person lends Tk. 10,000 at 10% per annum simple interest and Tk. 5000 at 20% per annum simple interest. Find the total interest earned in 2 years.

Solution:
First Loan,
Principle₁ = 10,000 Taka
Rate₁ = 10% Per Annum
Time = 2 Year

Simple Interest:
SI = (P × R × T) / 100
SI₁ = (10,000 × 10 × 2) / 100 = 2,000 Taka

Second Loan,
Principle₂ = 5,000 Taka
Rate₂ = 20% Per Annum
Time = 2 Year

Simple Interest:
SI₂ = (5,000 × 20 × 2) / 100 = 2,000 Taka

Total Interest:
SI₁ + SI₂ = 2,000 + 2,000 = 4,000 Taka

∴ Total earned interest = 4,000 Taka

Question: A certain sum amounts to Tk. 12,100 in 2 years at 10% per annum compound interest. Find the principal.

Solution:

Principal, P = Tk. P
Amount, C = Tk. 12,100
Rate, r = 10%
Time, n = 2 years

We know,
C = P(1 + r)ⁿ
⇒ 12,100 = P × (1 + 10/100)²
⇒ 12,100 = P × (110/100)²
⇒ 12,100 = P × (11/10)²
⇒ 12,100 = P × (121/100)

∴ P = (12,100 × 100)/121
= 10,000

Hence, Principal = Tk. 10,000

Question: What is the difference between the simple interest on a principal of Tk. 1000 being calculated at 7% per annum for 2 years and 6% per annum for 3 years?

Solution:
Given that,
Principal, P = Tk. 1000
Rate, r₁ = 7% and Time, n₁ = 2 years
Rate, r₂ = 6% and Time, n₁ = 3 years

We know,
I = Prn/100
1st case:
I₁ = Prn/100 = (1000 × 7 × 2)/100 = 140

2nd case:
I₂ = Prn/100 = (1000 × 6 × 3)/100 = 180

∴ Difference = 180 - 140 = Tk. 40

Question: The least number of complete years in which a sum of money put out at 21% compound interest will be more than doubled is:

Solution:
ধরি, আসল (Principal) = P
বার্ষিক সুদের হার (Rate of Interest), r = 21%
সময় (Time in years) = t
শর্ত অনুযায়ী, চক্রবৃদ্ধি মূলধন আসলের দ্বিগুণের বেশি হবে।
অর্থাৎ, A > 2P

আমরা জানি, চক্রবৃদ্ধি সুদের সূত্র হলো,
A = P(1 + r/100)ⁿ
প্রশ্নমতে,
P(1 + 21/100)ⁿ > 2P
⇒ (1 + 0.21)ⁿ > 2
⇒ (1.21)ⁿ > 2
এখন, আমরা n-এর বিভিন্ন পূর্ণসংখ্যা মান বসিয়ে পরীক্ষা করে দেখি:
যদি n = 1 হয়, (1.21)¹ = 1.21 (যা 2-এর থেকে ছোট)
যদি n = 2 হয়, (1.21)² = 1.4641 (যা 2-এর থেকে ছোট)
যদি n = 3 হয়, (1.21)³ = 1.772 (যা 2-এর থেকে ছোট)
যদি n = 4 হয়, (1.21)⁴ = 2.144 (যা 2-এর থেকে বড়)

∴ সর্বনিম্ন 4 বছরে মূলধন দ্বিগুণের চেয়ে বেশি হবে।

Question: For three years, Tk 40,000 is invested at a 10% yearly interest rate. What is the difference between the Compound Interest (CI) and the Simple Interest (SI)?

Solution:
Given:
Principal (P) = Tk. 40,000
Rate of Interest (r) = 10% per annum
Time (n) = 3 years

Simple Interest, I = Pnr
= 40000 × 3 × (10/100)
= 12000

Compound Principal = P × (1 + r)ⁿ
= 40000 × (1 + (10/100))³
= 40,000×(1.1)³
= 53240

Compound Interest, C = 53240−40000 = 13240

So, Difference = 13240 - 12000 = 1240 Taka

Given, P = 5000
Interest rate per year = 10%, so per half year = 5%
Time = 3/2 years or 3 half year

C = 5000 × 105/100 × 105/100 × 105/100
=  5788.125

Question: The simple interest on a certain sum of money at 4% per annum for 3 years is 1,200 Taka. The compound interest on the same sum for the same period and at the same rate is - 

Solution: 
Given, Simple Interest, I = 1,200 Taka
r = 4% = 0.04
n = 3 years

We know,
I = Pnr
P = I/nr
P = 1200/(3×0.04)
P = 10,000 Taka

For Compound Interest, 

So, Compound Interest = 11248.64 - 10000 
= 1248.64 Taka
= 1249 Taka

Question: The compound interest on a sum of money for 2 years at 10% per annum is Taka 525. The simple interest on the same sum for the same period and rate is - 

Solution: 
Compound Interest, 

For Simple Interest, 
I = Pnr
= 2500 × 2 × (10/100)
= 500 Taka

Question: What is the difference between simple and compound interest at 8% per annum on a sum of Tk. 12,500 at the end of 2 years?

Solution:
Here,
Principal, P = 12500 Tk.
Interest Rate, R = 8%
Time, T = 2 years

We know,
Simple Interest, SI = PRT/100
= (12500 × 8 × 2)/100
= 200000/100
= 2000 Tk.

For Compound Interest,
Amount, A = P{1 + (R/100)}^T
= 12500{1 + (8/100)}²
= 12500(1.08)²
= 12500 × 1.1664
= 14580 Tk.

∴ CI = A - P
= 14580 - 12500
= 2080 Tk.

∴ Difference between compound interest and simple interest,
= 2080 - 2000
= 80 Tk.

Question: The compound interest on Tk. 10,000 at 10% per annum for a certain period is Tk. 2,100. The time period (in years) is -

Solution: 
Principal amount, P = 10,000 taka
Compound Interest, I = 2100 taka

So, Total amount, A = 12100 Taka
r = 10% = 0.1

Now, (1.1)² = 1.21
So, n = 2 (Time period)

Question: Mamun invested 77500 tk in CT bank. In two years how much compound interest will he get, if the first year rate of interest was 10% and second year had 2% more than first year?

Solution:
Rate of Interest for Year 1 = 10% 
Rate of Interest for Year 2 = 10 + 2 = 12%

For compound Interest, Total Amount = P {1 + (R/100)}ⁿ
∴ Compound Interest = Amount - Principal = P {1 + (R/100)}ⁿ - P
= 77500{1 + (10/100)}¹ × {1 + (12/100)}¹ - 77500   [1st year 10% and 2nd year 12%]
= 77500[{1 + (10/100)}¹ × {1 + (10/100)}1 - 1]
= 77500 {(11/10) (112/100) - 1}
= 77500{(1232/1000) - 1}
= 77500(232/1000)
= 775(232/10)
= 17980 tk

Question: On a certain sum of money, the simple interest for 2 years is Tk. 300 at the rate of 6% per annum. If the same sum is invested at compound interest at the same rate for the same period, how much more interest would be earned?

Solution:
We know,
SI = (P × r × n)/100
⇒ 300 = (P × 6 × 2)/100
⇒ 300 = 12P/100
⇒ P = (300 × 100)/12
∴ P = Tk. 2500

Compound Interest = P(1 + r/100)ⁿ - P
= 2500(1 + 6/100)² - 2500
= 2500 × (1.06)² - 2500
= 2500 × 1.1236 - 2500
= 2809 - 2500
= Tk. 309

∴ Extra interest earned = 309 - 300 = Tk. 9

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

Total price of TV = Tk. 16000
Initial payment = Tk. 4000
Remaining amount = Tk. 12000
Simple interest in 15 months for Tk. 12000

S.I. = (P × R × T)/100
= ( 12000 × 12 × 15)/(100 × 12)
= Tk. 1800

⇒ With S.I. total amount to be paid for principal amount Tk. 12000
= Tk. (12000 + 1800)
= Tk. 13800

Therefore,
total amount he pays for the TV is
= 4000 + 13800
= Tk. 17800

Face Value of a share = Tk. 60
He bought each share at Tk. 60 - Tk. 5 = Tk. 55
Number of shares = 40
Dividend = 12(1/2)%
= (25/2)%

Dividend per share = (60 × 25)/(2 × 100)
= Tk. 7.5

Total dividend = (40 × 7.5)
∴ He got a dividend of (40 × 7.5) for an investment of Tk. (40 × 55)

Interest obtained
= (40 × 7.5 × 100)/(40 × 55)
= 13.64%

Cannot be determined
Let rate = R% and time = R years.
Then, (1200 × R × R/100) = 432
12R² = 432
=> R² = 36
=> R = 6

Question: A sum of Tk. 4000 amounts to Tk. 4840 in 2 years at compound interest. Find the rate of interest per annum.

Solution:
Here, Principal, P = 4000 Tk.
Final amount, A = 4840 Tk.
Time, n = 2 years
Interest rate, r = ?

Now,
A = P × (1 + r/100)ⁿ
⇒ 4840 = 4000 × (1 + r/100)²
⇒ (1 + r/100)² = 4840/4000
⇒ (1 + r/100)² = 121/100
⇒ 1 + r/100 = √(121/100)
⇒ 1 + r/100 = 11/10
⇒ r/100 = 11/10 - 1
⇒ r/100 = 1/10
⇒ r = (1/10) × 100
⇒ r = 10

∴ The annual rate of interest is 10%.

Question: At what rate of simple interest per annum will Tk. 25,000 amount to Tk. 37,500 in 5 years?

Solution:
Principal, P = Tk. 25000
Total Amount, A = Tk. 37500
Time, n = 5 years

Simple Interest, SI = A - P = 37500 - 25000 = Tk. 12500

We know, SI = (P × n × r)/100
⇒ 12500 = (25000 × 5 × r)/100
⇒ 12500 = 250 × 5 × r
⇒ 12500 = 1250r
⇒ r = 12500/1250
⇒ r = 10

∴ Rate of simple interest = 10% per annum.

Question: A sum of money doubles itself in 10 years at a certain rate of simple interest. In how many years will it become six times itself at the same rate of interest?

Solution:
Given that,
The sum doubles itself in 10 years.
Amount after 10 years = 2P
Simple Interest for 10 years = 2P - P = P

We know,
SI = (P × r × n)/100
⇒ P = (P × r × 10)/100
⇒ 10r = 100
⇒ r = 100 / 10
∴ r = 10% per year

Now, we want to find in how many years the sum becomes six times itself.
Amount = 6P
Interest needed = 6P - P = 5P

We know,
SI = (P × r × n)/100
⇒ 5P = (P × 10 × n)/100
⇒ 5 = (10 × n)/100
⇒ n = (5 × 100)/10
∴ n = 50 years

∴ The sum will become six times itself in 50 years.

Question: Tk. 6000 becomes Tk. 7500 in 5 years at a certain rate of simple interest. If the rate becomes half, what will be the total amount in 6 years on the same principal?

Solution:
Principal = Tk. 6000
Amount after 5 years = Tk. 7500
∴ Interest in 5 years = 7500 - 6000 = Tk. 1500

∴ Interest per year = 1500/5 = Tk. 300

Since the interest rate becomes half, the yearly interest also becomes half,
∴ Interest per year = 300/2 = Tk. 150

∴ Interest for 6 years = 150 × 6 = Tk. 900

∴ New total amount = 6000 + 900 = Tk. 6900

Question: The simple interest on a certain sum at 6% p.a. is Tk. 360 in 1 year. What will be the additional interest earned in 1 year if the rate increases to 7% p.a. on the same sum?

Solution:
Given,
Interest at 6% for 1 year:
Simple Interest, SI₁ = Tk. 360
Interest rate, R = 6%
Time, T = 1 year (initial)

We know,
SI₁ = PRT/100
⇒ 360 = (P × 6 × 1)/100
⇒ P = (360 × 100)/6
∴ P = Tk. 6000

So, the original sum (principal) is Tk. 6000.

Now,
Interest at 7% for 1 year:
SI₂ = PRT/100
⇒ SI₂ = (6000 × 7 × 1)/100
⇒ SI₂ = Tk. 420

Additional interest = SI₂ - SI_I
= 420 - 360
= Tk. 60

S.I. in 3 years = 6500 - 5600 = 900
Interest every year = 900/3 = 300
So, principal = 5600 - 2 × 300 = 5000

We know, I = pnr
Or, Interest rate, r = I/pn
= 600/(5000×2) × 100
= 6%