Simple and Compound Interest, Stock

Question: If Tk. 3000 is loaned for 4 months at a 4.5% annual rate, how much interest is earned?

Solution:
P = 3000
n = 4 months = 4/12 years = 1/3 year
r = 4.5%

I = Pnr
= {3000 × (1/3) × (4.5)}/100
= 45

Question: At the rate of 15% simple interest per annum, how much profit will be earned on Tk. 900 as principal in 7 years?

Solution:
Given,
Sum P = Tk. 900
Time n = 7 years
Rate r = 15%
= 15/100
= 3/20

We know,
Profit = Prn
= 900 × 3/20 × 7
= 945

Question: A company increases salary of an officer at 20% per year. In 2025 an employee receives Tk. 43,200. What was his salary in 2023?

Solution:
Let, his salary in 2023 was 100 Tk.

At 20% increment,
Salary of 2024 is = 100(1 + 20/100)
= 120 Tk.

Salary of 2025 is = 120(1 + 20/100)
= 144 Tk.

Now,
144 Tk. corresponds to = 43,200 Tk.
1 Tk. corresponds to = 43,200/144 Tk.
∴ 100 Tk. corresponds to = (43,200 × 100)/144
= 30,000 Tk.

∴ His salary in 2023 was 30,000 Tk.

Simple interest for 2 years is Tk. 320
⇒ Simple interest for first year = 320/2 = 160
⇒ Similarly, simple interest for the second year is also 160
Compound Interest for first year = 160
Compound Interest for second year = 340-160 = 180
we can see that compound Interest for the second year is more than
simple interest for the second year by 180-160 = 20
i.e., Tk. 20 is the simple interest on Tk. 160 for 1 year
R = 100 × S.I./PT
= (100 × 20)/(160 × 1)
= 12.5%

Question: The simple interest on a sum of money at 10% per annum for 4 years is half the sum. What is the value of the sum?

Solution:
ধরি, আসল = P
মুনাফার হার, r = 10% 
সময়, n = 4 বছর
মুনাফা, I = P/2

আমরা জানি,
I = Pnr/100
⇒ P/2 = (P × 4 × 10)/100
⇒ P/2 = 40P/100
⇒ P/2 = 2P/5
⇒ (P/2) - (2P/5) = 0
⇒ (5P - 4P)/10 = 0
⇒ P/10 = 0
⇒ P = 0

যেহেতু আসলের মান 0 হতে পারে না, তাই প্রদত্ত তথ্যটি সঠিক নয় বা অপর্যাপ্ত।

BG = Tk. 360
T = 3 years
R = 12%
TD = (BG×100)/TR
= (360×100)/(3×12)
= Tk. 1000
BG = BD - TD
⇒ BD = BG + TD = 360 + 1000 = Tk. 1360

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: If the simple interest on Tk. M at M% per annum for 4 years is Tk. M, what is the value of M?

Solution:
Given,
P = M
r = M% = M/100
n = 4 years
I = M

We know,
I = Prn
⇒ M = M × (M/100) × 4
⇒ M = M × M/25
∴ M = 25

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

S.I. on Tk. 1800 = T.D. on Tk. 1872
P.W. of Tk. 1872 is Tk. 1800
Tk. 72 is S.I. on Tk. 1800 at 12%
Time = (100×72)/(12×1800)
= 1/3 years = 4months

From the question, you know that R = 6%, T = 4 years, S.I. = Tk.1600

If you apply the above values in the simple interest formula S.I. = PRT/100, you will get
1600 = (P x 4 x 6)/100
⇒ P = (1600 x 100)/6 × 4
⇒ P = 6333.33

Using the above value of P, you have to now calculate C.I. as shown below:
CI = [P(1 + R/100)ⁿ] – P
= [6333.33(1 + 6/100)⁴] - 6333.33
= [6333.33 (106/100)⁴] - 6333.33
= [6333.33× 53/50 × 53/50 × 53/50 × 53/50] - 6333.33
= 7995.68 - 6333.33
= Tk.1662.35

Question: A sum of money becomes 4 times itself in 20 years at simple interest. What is the annual interest rate?

Solution:
Let, Principal amount = P
Sum of amount = 4P
∴ Interest, I = 4P - P = 3P
Time, n = 20 years
Rate of interest = r

We know, I = Pnr/100
⇒ Pnr/100 = 3P
⇒ nr/100 = 3
⇒ 20 × r/100 = 3
⇒ 20r = 300
⇒ r = 300/20
⇒ r = 15

∴ The annual interest rate is 15%.

Here, interest = 34300 - 24500 = 9800 taka
We know, I = pnr
Or, r = (I×100)/pn
= (9800×100) / (24500×5)
= 8%

Question: A sum of Tk. 30,000 yields a compound interest of Tk. 4347 when invested at 7% per annum. What is the investment period in years?

Solution:
Given,
Principal, P = 30000
Rate, r = 7% per annum
Compound Interest, CI = 4347

We know,
Amount, A = P + CI = 30000 + 4347 = 34347

Using the compound amount formula:
A = P(1 + r/100)ⁿ
⇒ 34347 = 30000 × (1 + 7/100)ⁿ
⇒ 34347 = 30000 × (107/100)ⁿ
⇒ (107/100)ⁿ = 34347/30000
⇒ (1.07)ⁿ = 1.1449
⇒ (1.07)ⁿ = (1.07)²
∴ n = 2 years

P = (100 × S.I)/RT
= (100 × 6200)/(8 × 4)
= 620000/32
= 775 × 25
= Tk. 19375.

Simple interest is the same as the rate of interest.
Hence,
Rate of interest = R% and Time = R years

S.I. = (P × R × R)/100
⇒ 60 = (1500 × R²)/100
⇒ 15R²= 540
⇒ R² = 36
⇒ R=6 %

Rate of Interest = 6%.

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: A man invested Tk. 36,000 when he bought Tk. 100 shares at Tk. 144. If 16% dividend is declared, find his annual income.

Solution:
For Tk. 144 he gets Tk. 16
∴ For Tk. 1 he gets Tk. 16/144
∴ For Tk. 36,000 he gets Tk. (16 × 36,000)/144
= Tk. (1/9) × 36,000
= Tk. 4,000

∴ The man's annual income is Tk. 4000.

Question: An amount of Tk. 8,000 yields a simple interest of Tk. 1,440 in 3 years. What is the annual rate of interest?

Solution:
Given,
Principal, P = 8000
Simple Interest, SI = 1440
Time, n = 3 years
Rate of interest, r = ?

We know,
I = Pnr/100
⇒ r = (I × 100)/(P × n)
⇒ r = (1440 × 100)/(8000 × 3) 
⇒ r = 144000/24000 
∴ r = 6%

So, the annual rate of interest is 6%.

Question: Find the simple interest on BDT 12000 at 4% per annum for 8 months.

Solution:
Principal, P = 12000 Taka
Time, n = 8 months = 8/12 = 2/3 years
Rate of interest, r = 4% = 4/100

Simple Interest, I = P × n × r
= 12000 × (2/3) × (4/100)
= 40 × 2 × 4
= 320

∴ The simple interest is Tk. 320.

Question: A sum of 20,000 Taka is invested at 8% per annum. If the interest is compounded quarterly, what is the amount after 9 months?

Solution:
এখানে, আসল (P) = 20,000 টাকা
বার্ষিক সুদের হার = 8%
সময় = 9 মাস
যেহেতু সুদ ত্রৈমাসিক (quarterly) ভিত্তিতে গণনা করা হয়,
∴ ত্রৈমাসিক সুদের হার = 8% ÷ 4 = 2%
9 মাসে চক্রবৃদ্ধির সংখ্যা = 9 মাস ÷ 3 মাস = 3 বার

প্রথম ত্রৈমাসিক:
সুদ =  (20,000 এর 2%) = 400 টাকা
নতুন মূল = 20,000 + 400
= 20,400 টাকা

দ্বিতীয় ত্রৈমাসিক:
সুদ = (20,400 এর 2%) = 408 টাকা
নতুন মূল = 20,400 + 408
= 20,808 টাকা

তৃতীয় ত্রৈমাসিক:
সুদ = (20,808 এর 2%) = 416.16 টাকা
নতুন মূল = 20,808 + 416.16
= 21,224.16 টাকা

∴ 9 মাস পর চক্রবৃদ্ধি মূল হবে 21,224.16 টাকা।

সরল মুনাফার ক্ষেত্রে,
I = Pnr
Or, H = H × 4 × H/100
Or, H = 100/4
Or,  H = 25

Question: A sum of money at simple interest amounts to Tk. 815 in 3 years and to Tk. 854 in 4 years. The sum is-

Solution:
Simple interest for 1 year = Tk. (854 - 815)
= Tk. 39

∴ Simple interest for 3 years = Tk.(39 × 3)
= Tk. 117

∴ Sum = (815 - 117)
= Tk. 698

Let the sum of money be P(which is the principal value) and rate of interest be R.
According to the question,
Amount (principal +simple interest) = 8P/5
Time (T) = 5 years
Simple interest (SI) = PRT/100
SI = 5PR/100
Amount = P + SI
8P/5 = P + (5PR/100)
8P/5 = P(100 + 5R)/100
160 = 100 + 5R
5R = 60
R = 12
Therefore, rate of interest = 12% p.a.

Alternative Method:
Given A = 8/5p (8/5 of the sum) .
Time = 5years .
Let the sum be P = 100.
Then, the amount = 8/5p = (value of p is 100)
= 8/5 × 100
= 160 .
S. I. = A-P
= 160 - 100
= 60 .
Hence,
Rate = S.I. × 100/P × T
= 60 × 100/100 × 5
= 12% p.a.

Question: What is the interest for 2 years on Tk. 600 at a simple interest rate of 9.5%?

Solution: 
Interest rate, R = 9.5% 
Principal amount, P = 600 tk
Time, T = 2 years

We Know, SI = PRT/100
= (600 × 2 × 9.5)/100
= 114 Tk.

∴ The interest for 2 years is Tk. 114.

Question: A sum of money invested at compound interest triples itself in 3 years. In how many years will it become 27 times itself?

Solution:
Since the sum triples in 3 years, we can write:

Thus, it will take 9 years for the amount to become 27 times itself.

Question: A woman borrowed a certain amount of money for 8 months. She paid Tk. 1,200 as simple interest at a rate of 12% per year. What was the original amount (principal) she borrowed?

Solution:
Simple Interest (SI) formula:
SI = [P (principal) × R (rate) × T (time)] / 100

Here,
SI = 1200
R = 12% per annum
T = 8 months = 8/12 = 2/3 year

Now,
1200 = [P × 12 × (2/3)] / 100
⇒ 1200 = (P × 8) / 100
⇒ P = (1200 × 100) / 8
⇒ P = 150000 / 8
⇒ P = Tk. 15000

To earn Tk. 135, investment = Tk. 1620.
To earn Tk. 8, investment = Tk.(1620/135 × 8)
= Tk 96
So, Market value of Tk. 100 stock = Tk. 96.

Question: A sum of money amounts to Tk. 18000 in 5 years at 20% simple interest per annum. Find the sum.

Solution:
দেওয়া আছে,
সুদ-আসল (Amount), A = 18000 টাকা
সময় (Time), n = 5 বছর
সুদের হার (Rate), r = 20%
মূলধন (Principal), P = ?

আমরা জানি,
সুদ (Interest), I = A - P
আবার, I = (Pnr)/100
সুতরাং, A - P = (Pnr)/100
⇒ 18000 - P = (P × 5 × 20)/100
⇒ 18000 - P = (100P)/100
⇒ 18000 - P = P
⇒ 18000 = P + P
⇒ 18000 = 2P
⇒ P = 18000/2
⇒ P = 9000
সুতরাং, নির্ণেয় মূলধন 9000 টাকা।

Question: What is the difference between simple and compound interest at 10% per annum on a sum of Tk. 3000 at the end of 2 years?

Solution:
Principal (P) = Tk. 3000
Rate (r) = 10% per annum
Time (n) = 2 years

Simple Interest (SI):
SI = (P × R × T)/100
= (3000 × 10 × 2)/100
= 60000/100
= Tk. 600

Compound Interest (CI):
Amount (A) = P × (1 + r/100)ⁿ
= 3000 × (1 + 10/100)²
= 3000 × (1.1)²
= 3000 × 1.21
= Tk. 3630

∴ CI = A - P = 3630 - 3000
= Tk. 630

∴ Difference between CI and SI = 630 - 600
= Tk. 30

Let the capital be Tk. x 
Then according to the question, we have
 [{(x/3)× (7/100) × 1} + [{(x/4)×(8/100) × 1} + [{(5x/12)× (10/100) × 1}= 561
⇒(7x/300) ​+ (8x​/400) + (50x​/1200) = 561
⇒(28x + 24x + 50x)/1200 = 561
⇒102x​/1200 = 561
x = (561 × 1200)/102
x = 6600

Let the amount invested in scheme A be Tk. x and that in B be Tk. 3x.
Then,
Then,
{(x × 4 × 8)/100 + (3x × 2 × 13)/100} = 1320
⇒ 32x/100 + 78x/100 = 1320
⇒ 110x/100 = 1320
⇒ x = (1320 × 100)/110
⇒ x = Tk. 1200

Hence, Tk. 1200 was invested in scheme A.

Question: Some principal becomes Tk. 1,750 as profit-principal in 5 years and Tk. 2,250 as profit-principal in 10 years. Find the rate of profit.

Solution:
Given,
Principal + Profit for 5 years = 1,750
Principal + Profit for 10 years = 2,250

So the profit of 5 years, I = 2250 - 1750 = Tk. 500
So the principal, P= 1750 - 500 = Tk. 1250

∴ The rate of profit, r = (I × 100)/(P × n)
= (500 × 100)/(1250 × 5)
= 50000/6250
= 8%

Question: A bank offers 10% compound interest calculated half-yearly. A customer deposits Tk. 2000 on 1st January and another Tk. 2000 on 1st July of the same year. How much interest will he earn at the end of the year?

Solution:
Here,
Half-yearly interest rate = 10% ÷ 2 = 5%
Now,
The first deposit of Tk. 2000 was made on 1st January.
It stays for 12 months, so it earns interest twice (i.e., 2 half-years).   
∴ A₁ = P(1 + r/100)ⁿ
= 2000 × {1 + 5/100}²
= 2000 × (1.05)²
= 2000 × 1.1025
= 2205

Now,
The second deposit of Tk. 2000 was made on 1st July.
It stays for 6 months, so it earns interest only once (1 half-year).
∴ A₂ = P(1 + r/100)ⁿ
= 2000 × (1 + 5/100)¹
= 2000 × 1.05
= 2100

Total amount = 2205 + 2100 = 4305
Total money deposited = 2000 + 2000 = 4000

∴ Interest earned = 4305 - 4000 = Tk. 305

∴ The customer would have gained Tk. 305 by way of interest.

Question: The compound interest on Taka 12,000 for 2 years at 10% per annum compounded half-yearly is -

Solution: 
Given, 
P = 12,000 (Principal)
r = 10% = 0.1
Compounded half-yearly means, n = 2
Time, t = 2 years

Compound Interest = A - P
= 14586 - 12000
= 2586 Taka

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% & n = 2 

Now, S.I. = (x × 5 × 2)/100
= x/10

And, C.I. = [x{1 + (5/100)}² - x]
= [x{1 + (1/20)}² - x]
=[x(21/20)² - x]
= (441x - 400x)/400
= 41x/400

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

Here, the sum becomes 4 times that means 100, becomes 400.
Rate of such question is given by
R = interest/time = 300/10 = 30%

Let the investor invests Tk. 100

Rebate given = 4%

So, the actual investment = 100 - 4
= 96

The bank pays interest of 14% on the investment which is Tk. 100 without rebate and Tk. 96 with rebate.
Therefore required rate of interest = (114 - 96)/96 × 100
= 18.75%

Question: A sum of Tk. 4000 will amount to TK. 4410 in 2 years if the interest is calculated every year. What is the rate of compound interest?

Solution:
Here,
Principal, P = 4000 Tk.
Amount, A = 4410 Tk.
Time, n = 2 years
Let, Rate = r

By using formula,
A = P[1 + (r/100)]ⁿ
∴ 4410 = 4000 [1 + (r/100)]² 
⇒ 441/400 = [1 + (r/100)]²
⇒ 21/20 = 1 + (r/100)                        [Taking square root of both sides]
⇒ 21/20 = (100 + r)/100
⇒ 20r + 2000 = 2100
⇒ 20r = 2100 - 2000
⇒ 20r = 100
⇒ r = 100/20
⇒ r = 5

∴ Rate of compound interest is 5%

For an income of Tk. 756, investment
= Tk. 9000

For an income of Tk. (21/2), investment
= Tk. {(9000/756) × (21/2)}
= Tk. 125

∴ For a Tk. 100 stock, investment = Tk. 125

The market value of Tk. 100 stock
= Tk. {125 - (1/4)}
= Tk. 124.75