Bank Math Master

Question: The difference between the compound interest and simple interest on Tk. x at 8.5% per annum for 2 years is Tk. 28.90. The value of x is-

Solution :
Here,
Principal P = x Tk.
Rate of Interest per year r = 8.5%
= 8.5/100
Time n = 2 years 

∴ Simple Interest  =Prn
= x × 8.5/100 × 2
= 17x/100
= 0.17x

∴ Compound Interest = P (1 + r)ⁿ - P
= x × (1 + 8.5/100)² - x
= x × (1 + 0.085 )² - x
= x × (1.085)² - x
= 1.177225 x - x
= (1.177225 - 1)x
= 0.177225x

According to the question,
Compound Interest - Simple Interest = 28.90
⇒ 0.177225 x - 0.17x = 28.90
⇒ (0.177225 - 0.17) x = 28.90
⇒ 0.007225x = 28.90
⇒ x = 28.90/0.007225
∴ x = 4000

So the value of x is = 4000

Question : The sum of three numbers is 116. The second number and the third number are in the ratio of 9 : 16 while the first number and the third number are in the ratio of 1 : 4. Find the second number.

Solution : 
Given,
Second : Third = 9 : 16
Third : first = 4 : 1
= 16 : 4

So the ratio of Second  :Third : first = 9 : 16 : 4

∴ The second number = (116 × 9/29)
= 36

Question : At what rate percent per annum will a sum of Tk. 15,625 amount to Tk. 21,952 in three years, if the interest is compounded annually?

Solution :
Given,
Principal P = Tk. 15,625
Amount  C = Rs. 21,952
Time n = 3 years
Rate of Interest  = r%
= r/100

Compound interest when compounded annually is:
C = P (1 + r / 100)ⁿ
⇒ 21,952 = 15,625 (1 + r / 100)³
⇒ 21,952 / 15,625 = (1 + r/100)³
⇒ (1 + r/100)³ = 1.404928
⇒ (1 + r/100) = ³√1.404928  [ taking the cube root on both sides ]​
⇒ (1 + r/100) ≈ 1.12
⇒ r/100  = 1.12 - 1
⇒ r/100  = 0.12
⇒ r = 0.12 × 100
∴ r = 12

Question : In a college union, there are 48 students. The ratio of the number of boys to the number of girls is 5 : 3. How many number of girls to be added in the union, so that the number of boys to girls in 6 : 5 is ?

Solution :
Let,
The ratio of numbers of boys to the number of girls is = 5x : 3x

According to the question,
⇒ 5x+3x=48
⇒ 8x=48
∴ x = 6

So the number of boys = 5x
= 5 × 6
= 30

and the number of girls = 3x
= 3 × 6
= 18

Let,
The number of girls to be added in the union = y

So,
30/(18 + y) = 6 : 5
⇒ 6(18 + y) = 30 × 5
⇒ 6(18 + y) = 150
⇒ 18 + y = 150/6
⇒ 18 + y = 25
⇒ y = 25 - 18
∴ y = 7

∴ The number of girls to be added in the union = 7

Question : The cost of a table and a chair are in the ratio of 10 : 7. If the cost of table and a chair is increased by 8% and 20% respectively, then what will be the new ratio?

Solution :
Let the cost of the table and chair be Tk. 10x and Tk. 7x respectively

New cost of table = 108% of 10x
= (108 × 10x)/100
= 54x/5

New cost of chair = 120% of 7x
= (120 × 7x)/100
= 42x/5

So the new ratio = 54x/5 : 42x/5
= 54 : 42
= 9 : 7

Question : Present population of a city is 20 lac. What will be the population of the city after 3 years if the growth rate of population of that city is 50 per thousand ?

Solution :
Given that,
The present population of the city is P = 20,00,000
Groth rate of population r = 50/1000
= 5/100
= 1/20
Time n= 3 years

So the population of the city after 3 years will be = P(1 + r)ⁿ
= 20,00,000 × {1 + (1/20)}³
= 20,00,000 × (21/20)³
= 20,00,000 × 9,261/8,000
= 23,15,250

∴ After 3 years, population of the city will be 23,15,250.

Question : Two numbers are in the ratio 5 : 8. If 3 is subtracted to the first number, the ratio becomes 1 : 2. The numbers are = ?

Solution :
Let,
The numbers are = 5x and 8x

∴ According to question,
⇒ (5x - 3) : 8x = 1 : 2
⇒ (5x - 3)/8x = 1/2
⇒ 2(5x - 3) = 8x
⇒ 10x - 6 = 8x
⇒ 10x - 8x = 6
⇒ 2x = 6
∴ x = 3

So,The first numbers =5x
=5×3
=15
The second number = 8x
= 8 × 3
= 24

Question : If 4 years ago the ratio between the ages of Nuha and Naba was 5 : 6 and the sum of their ages at present is 52. What is the ratio of their present ages ?

Solution :
Let
Four years ago, Nuha's age was = 5x
and Naba's age was = 6x

According to the question,
(5x + 4) + (6x + 4) = 52
⇒ 11x + 8 = 52
⇒ 11x = 52 - 8
⇒11x = 44
∴ x = 4

∴ Present age of Nuha = 5x + 4
= 5 × 4 + 4
= 24

∴ Present age of Naba = 6x + 4
= 6 × 4 + 4
= 28

So the ratio of their present ages = 24 : 28
= 6 : 7

Question : The sum for 4 years given a compound interest of Tk. 5992.05 at 15% rate. Then the sum is ?

Solution :
Given,
Compound interest = Tk. 5992.05
Time n = 4 years
Interest rate r = 15%
= 15/100
= 0.15

We know,
Compound interest = P(1 + r)ⁿ - P
⇒ 5992.05 = P{(1 + 0.15)⁴ - 1}
⇒ 5992.05 = P{(1.15)⁴ - 1}
⇒ 5992.05 = P(1.749 - 1)
⇒ 5992.05 = P × 0.749
⇒ P = 5992.05 ÷ 0.749
∴ P = 8000

Question : A piece of ornament is made by mixing bronze, silver and gold. In that piece of ornament, the ratio of bronze and silver is 1 : 2 and the ratio of silver and gold is 3 : 5. Find how many grams of gold there are in an ornament weighing 19 grams.

Solution :
Given,
The ratio of bronze and silver is = 1 : 2
= 3 : 6
The ratio of silver and gold is = 3 : 5
= 6 : 10

So the  ratio of bronze, silver and gold = 3 : 6 : 10

∴ Total unit of bronze, silver and gold = 3 + 6 +10
= 19

So the weight of gold = 10/19 parts of 19 grams
= 10 grams

Question : The profit-principal of some principal is Tk. 8,000 in 3 years. If the profit is 3/5 part of the principal, find the rate of profit ?

Solution :
Given,
The profit-principal A = Tk. 8000
Time n = 3 years

Let,
the principal = P
So the profit = (3/5 × P)
= 3P/5

We know
Principal + profit = profit-principal
⇒ P + (3P/5) = 8000
⇒ 8P/5 = 8000
⇒ 8P = 8000 × 5
⇒ 8P = 40000
⇒ P = 40000/8
∴ P = 5000

So the principal P = Tk. 5000

∴ the profit I = (3/5 × 5000)
= Tk. 3000

We know,
r = (I × 100) ÷ Pn
= (3000 × 100) ÷ (5000 × 3 )
= 300000 ÷ 15000
= 20%

Question : A man bought some rice and wheat for Tk. 490. The ratio of weight of rice and wheat is 8 : 5 and the price equal amount of rice and wheat is in the ratio 3 : 5. The price of total rice = ?

Solution :
Given,
The ratio of weight of rice and wheat is = 8 : 5
The ratio of equal amount of rice and wheat is = 3 : 5

So the ratio of total price of rice and wheat is = 24 : 25
∴Total = 24 + 25
= 49

According to the question,
49 unit = Tk. 490
∴ 1 unit = Tk. 10

∴ Price of total rice = 24 × 10
= Tk. 240

Question : Calculate the simple interest on Tk. 4000 for 15 months at 6 paise per taka per month.

Solution :
Given,
Principal P = Tk. 4000
interest rate r = 6 paise per Taka per month
∴ Tk. 1 = 6 paise
∴ Tk. 100 = (6 × 100) paise
= 600 paise
= 6 taka
= 6% per month
= 6/100

Time t =15 months

We know,
The simple interest I = Prn
= 4000 × (6/100) × 15
= 360000/100
= 3600

So the simple interest = Tk. 3600

Question : If a, b and c are positive numbers such that (a² + b²) : (b² + c²) : (c² + a²) = 34 : 61 : 45, then (b - a) : (c - b) : (c - a) = ?

Solution :
Given,
(a² + b²) : (b² + c²) : (c² + a²) = 34 : 61 : 45

Add all
2(a² + b² + c²) = 34 + 61 + 45
⇒ 2(a² + b² + c²) = 140
⇒ (a² + b² + c²) = 70
⇒ c² = 70 - 34
⇒ c² = 36
∴ c = 6

b² = 70 - 45
⇒ b² = 25
∴ b = 5

a² = 70 - 61
⇒a² = 9
∴ a = 3

(b - a) : (c - b) : (c - a)
= (5 - 3) : (6 - 5) : (6 - 3)
= 2 : 1 : 3

Question : How long will it take for Tk. 6000 to earn Tk. 1800 as simple interest at an annual interest rate of 6% ?

Solutoin :
Given,
Principal P = Tk. 6000
Rate r = 6%
= 6/100
Simple Interest I = Tk. 1800

We know,
I = Prn
⇒ n = I/Pr
= 1800/6000 × (6/100)
= (1800 × 100)/(6000 × 6)
= 180000/36000
= 5

Question : If 3/5 of A = 80% of B = 0.5 of C, then A : B : C is-

Solution :
According to the question,
3/5 of A = 80% of B
⇒ 3A/5 = 80B/100
⇒ 3A/5 = 4B/5
⇒ A/B = 4/3
⇒ A : B = 4 : 3
∴ A : B = 20 : 15 (multiple by 5)

80% of B = 0.5 of C
⇒ 80B/100 = 5C/10
⇒ 4B/5 = C/2
⇒ B/C = 5/8
⇒ B : C = 5 : 8
∴ B : C = 15 : 24 (multiple by 3)

A : B : C = 20 : 15 : 24

Question : The compound interest on Tk. 3000 at 10% per annum is Tk. 630. The period (in years) is:

Solution : 
Amount= Tk. (3000 + 630)
= Tk. 3630

Let, the time = n years

Then,
3000(1 + 10/100)ⁿ = 3630
⇒ 3000(11/10)ⁿ = 3630
⇒ (11/10)ⁿ = 3630/3000
⇒ (11/10)ⁿ = 121/100
⇒ (11/10)ⁿ = (11/10)²
∴ n = 2

Question : If a : b : c = 5 : 2 : 7, then the ratio (a + b + c) : c is equal to

Solution :
Here,
a : b : c = 5 : 2 : 7

Let, a : b : c = 5x : 2x : 7x
So, a + b + c = 5x + 2x + 7x
= 14x

and, c = 7x

(a + b + c) : c = 14x : 7x
= 2 : 1