Bank Math Master

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: B's income is 60% of A's, and the ratio of their expenditures is 9 : 5. If each saves Tk. 4,000, find A’s income.

Solution:
Suppose,
A's income = 100,
B's income 60% of 100 = 60
A : B = 100 : 60 = 5 : 3

So,
A’s income = 5x, B’s income = 3x
A’s expense = 9y, B’s expense = 5y

Then their savings are:
A’s savings = Income - Expense = 5x - 9y.......(1)
B’s savings = Income - Expense = 3x - 5y........(2)

Given that each saves Tk. 4000:
5x - 9y = 4000
3x - 5y = 4000

Subtract equation (2) from (1):
(5x - 9y) - (3x - 5y) = 0 
 ⇒ 2x - 4y = 0 
⇒ x = 2y

Substitute x = 2y into equation (2):
3(2y) - 5y = 4000
⇒ 6y - 5y = 4000
⇒  y = 4000

Then x = 2y = 8000

Finally, A’s income = 5x = 5 × 8000 = 40,000 Taka

∴ A's income = 40,000 Taka

Question: The price of sugar falls by 20%. By how much percent can a person increase consumption without increasing expenditure?

Solution:
Let,
Original price = 100 units
After 20% fall
New price = (100 - 20) = 80 units

With the same money,
Now we can buy = (100/80) × 100
= 1.25 × 100
= 125 units

Increase = 125 - 100
= 25 units

Percentage increase in consumption = (25/100) × 100%
= 0.25 × 100%
= 25%

∴ 25% consumption can be increased.

Question: A person sold a book at 20% profit and another at 12% loss, both at Tk. 330 each. Find his overall profit or loss percent.

Solution:
Let the cost price of the first book = C1​
Let the cost price of the second book = C2​

First book:
Sold at 20% profit, Selling Price = 330
C1 × 1.2 = 330
⇒ C1 = 330 / 1.2 = 275

Second book: Sold at 12% loss,
Selling Price = 330
C2 × 0.88 = 330
⇒ C2 = 330 / 0.88 = 375

Total cost price  = 275 + 375 = 650
Total selling price = 330 + 330 = 660

Profit/Loss = SP - CP
= 660 - 650
= 10

Profit% = (10/650) × 100%
= 1.538%
= 1.54%

∴ His overall profit = 1.54%

Question: A trader sets his products 50% above cost price and gives a discount of 20%. Find his profit percent.

Solution:
Let the cost price (CP) = Tk. 100

Marked Price (MP)
MP = CP + 50% of CP
= 100 + 50
= 150

Selling Price (SP) after discount:
SP = MP - 20% of MP
= 150 - 30
= 120

Profit = SP - CP
= 120 - 100
= 20

Profit percent:
Profit % = (Profit/CP) × 100
= (20/100) × 100
= 20%

∴ Profit = 20%

Question: The ratio of investments of A, B, and C is 2 : 3 : 4, and their profit ratio is 1 : 2 : 3. If A invested for 12 months, find for how many months C invested.

Solution:
Let the investments of A, B, and C be:
I_A : I_B : I_C = 2 : 3 : 4

Let the time periods of investment be:
T_A : T_B : T_C = ?

Profit = investment × time,
P_A : P_B : P_C = 1 : 2 : 3

So:
I_A × T_A : I_B × T_B : I_C × T_C = 1 : 2 : 3

Substitute investments in ratio form:
⇒ 2 × 12 : 3 × T_B : 4 × T_C = 1 : 2 : 3
⇒ 24 : 3T_B : 4T_C = 1 : 2 : 3

Find multiplier
Let k be the factor:
⇒ 24 = 1 × k 
 ⇒ k = 24

Then:
3T_B = 2 × k
⇒ 3T_B = 2 × 24
⇒ 3T_B = 48
⇒ T_B = 48/3
⇒ T_B = 16 months

Again,
4T_C = 3 × k
⇒ 4T_C = 3 × 24
⇒ 4T_C = 72
⇒ T_C = 72/4
⇒ T_C = 18 months

Question: The population of a town increases by 10% in the first year and decreases by 10% in the next year. What is the overall percentage change?

Solution:
Let the initial population = 100

Population after 1st year (10% increase)
Population = 100 + 10% of 100 = 100 + 10 = 110

Population after 2nd year (10% decrease)
Population = 110 - 10% of 110 = 110 - 11 = 99

Overall change = Final population - Initial population
= 99 - 100
= - 1

Overall percentage change = (- 1/100) × 100% = - 1%

∴ Population decreased by 1%

Question: A mixture contains milk and water in the ratio 7 : 3 cost Tk. 500. Then 2 litters of water were added. What is the profit percentage?

Solution:
Let
Milk = 7 liters
Water = 3 liters
Original mixture quantity = 10 liters

Cost of original mixture = 500 Taka
Cost per liter = 500 / 10 = 50 Taka

Water added = 2 liters
New total water = 3 + 2 = 5 liters
Milk remains = 7 liters
Total mixture = 7 + 5 = 12 liters

Total Selling Price = 12 × 50 = 600 Taka

Profit = SP - CP = 600 - 500 = 100
Profit % = (100/500) ​× 100 = 20%

∴ Profit percent = 20%

Question: An investment becomes Tk. 12,100 in 2 years at compound interest, the rate being 10% per annum. Find the principal.

Solution:
Given,
Amount, A = 12,100 Taka
Rate, R = 10%
Time, T = 2 years

Compound Interest Formula:
A = P[1 + 100/R​]^T
⇒ 12,100 = P[1.1]²
⇒ 12,100 = 1.21P
⇒ P = 12,100/1.21
P = 10,000

∴ Principle = 10,000 taka

Question: A’s salary is 25% more than B’s, and B’s salary is 20% less than C’s. If C’s salary is 50,000 Taka, find A’s salary.

Solution:
B’s salary is 20% less than C’s:
B = C × (1 - 20/100) = 50,000 × 0.8 = 40,000 Taka

A’s salary is 25% more than B’s:
A = B × (1 + 25/100) = 40,000 × 1.25 = 50,000 Taka

∴ A’s salary = 50,000 Taka

Question: An investment becomes Tk. 6,720 in 2 years and Tk. 7,392 in 3 years at compound interest. Find the rate of interest per annum.

Solution:
Let the principal = P, rate = R%.

Compound Interest:
A = P[1 + 100/R​]^T

Amount after 2 years = 6,720
6,720 = P[1 + 100/R​]²

Amount after 3 years = 7,392
7,392 = P[1 + 100/R​]³

Divide the 3rd year amount by 2nd year amount:
7,392/6,720 = P(1 + R/100)³/ P(1 + R/100)² 
⇒ 1.1 = 1 + R/100 [7,392/6,720 = 1.1]
⇒ 1.1 - 1 = R/100
⇒ 0.1 = R/100
⇒ R = 0.1 × 100
⇒ R = 10%

∴ R = 10%

Question: If 20% of A = 30% of B, and B = 400, find A.

Solution:
the equation;
20% of A = 30% of B
⇒ (20/100) × A = (30/100) × B
⇒ 0.2A = 0.3 × 400 [Substitute B = 400]
⇒ 0.2A = 120
⇒ A = 120/0.2
⇒ A = 600

∴ A = 600

Question: A shopkeeper sells goods at 25% profit. If he had sold for Tk. 200 more, the profit would have been 30%. What is the cost of that goods?

Solution:
Let the cost price = C Tk.

Selling price at 25% profit:
SP₁ = C + 25% of C = C × 1.25

Selling price at 30% profit:
SP₂ = C + 30% of C = C × 1.30

Difference:
SP₂ - SP₁ = 200
⇒ 1.30C - 1.25C = 200
⇒ 0.05C = 200
⇒ C = 200/0.05
⇒ C = 4,000 Tk.

∴ Cost Price = 4,000 Tk.

Question: A trader sells a bag for Tk. 2,400 at a loss of 20%. If his regular profit is 10%, find the regular selling price.

Solution: 
Selling price = 2,400 Taka
Loss % = 20%

Original cost price = 2400 / 0.8 = 3000 Taka

Regular profit = 10%
Regular Selling price = 3000 × 1.1 = 3300 Taka

∴ Regular Selling price = 3300 Taka

Question: An article has a marked price of Tk. 500. If two successive discounts of x% and 5% reduce the selling price to Tk. 427.50, determine the value of x.

Solution:
Marked Price = Tk. 500
Final Selling Price after two successive discounts = Tk. 427.50
Let the first discount = x%
Second discount = 5%

First discount be:
500 × (1 - x/100​) × (1 - 0.05) = 427.50
⇒ 500 × (1 - x/100​) × 0.95 = 427.50
⇒ 475 × (1 - x/100​) = 427.50 
⇒ (1 - x/100​) = 427.50 / 475
⇒ 1 - x/100 = 0.9
⇒ - x/100 = 0.9 - 1
⇒ - x/100 = - 0.1
⇒ x = 0.1 × 100
⇒ x = 10%

∴ First discount = 10%

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: A man spends 75% of his income. If his income increases by 25% and expenditure increases by 15%, find the percentage increase in his savings.

Solution:
Let,
Original income = 100 Taka
Original expenditure = 75% of 100 = 75 Taka
Original savings = Income - Expenditure = 100 - 75 = 25 Taka

After increase,
New income = 100 + 25% of 100 = 100 + 25 = 125 Taka
New expenditure = 75 + 15% of 75 = 75 + 11.25 = 86.25 Taka
New savings = 125 - 86.25 = 38.75 Taka

Percentage increase = [(New savings - Original savings) / Original savings] ​× 100 
= [(38.75 - 25) / 25] × 100
= [13.75/ 25] × 100
= 0.55 × 100
= 55% 

∴ Saving increased = 55%

Question: Two successive discounts of 25% and 15% are equal to a single discount of___

Solution:
Formula for successive discounts
Single equivalent discount = d₁ + d₂ - (d₁ × d₂)/100
⇒ d = (25 + 15) - (25 × 15)/100
⇒ d = 40 - (375/100)
⇒ d = 40 - 3.75
d = 36.25

∴ Single discount = 36.25%

Question: Tamim and Shakib invest Tk. 15,000 and Tk. 25,000 respectively. Tamim, being the active partner, gets 10% of the total profit as salary. If the annual profit is Tk. 10,000, find Tamim’s total earnings.

Solution:
Total profit = 10,000 Taka.
Tamim gets 10% of total profit as salary.
Tamim’s salary = 10% of 10,000
= (10/100) × 10,000
= 1,000 Taka.

Remaining profit after salary = 10,000 - 1,000
= 9,000 Taka.

Divide remaining profit in ratio of their capitals
Tamim : Shakib = 15,000 : 25,000
= 3 : 5

Tamim’s share = [3/(3 + 5)] × 9,000 
= [3/8] × 9,000
= 3,375 Taka

Tamim’s total earning = Tamim’s salary + Tamim’s profit share
= 1,000 + 3,375
= 4,375 Taka

∴ Tamim's total earning = 4,375 Taka

Question: Two partners invest Tk. 1,20,000 and Tk. 80,000. After 6 months, they admit a new partner with Tk. 1,00,000. What is the ratio of their profits after one year?

Solution:
Profit sharing ratio depends on: Capital × Time.
Let the partners be A, B, and C.

A’s investment: Tk. 1,20,000 for 12 months
⇒ 1,20,000 × 12 = 14,40,000

B’s investment: Tk. 80,000 for 12 months
⇒ 80,000 × 12 = 9,60,000

C’s investment: Tk. 1,00,000 for 6 months
⇒ 1,00,000 × 6 = 6,00,000

Now, the ratio of profits:
14,40,000 : 9,60,000 : 6,00,000

Simplify = 12 : 8 : 5

∴ Ratio = 12 : 8 : 5