Bank Math

Question: A sum of money doubles itself in 8 years at a certain rate of simple interest. In how many years will it become four times itself at the same rate of interest?

Solution: 
Given that, 
The sum doubles itself in 8 years.
Amount after 8 years = 2P
Simple Interest for 8 years = P

We know,
SI = (P × r × n)/100
⇒ P = (P × r × 8)/100
⇒ 8r = 100
⇒ r = 100/8
∴ r = 12.5% per year

Now, we want to find in how many years the sum becomes four times itself.
Amount = 4P
Interest needed = 4P - P = 3P

We know,
SI = (P × r × n)/100 
⇒ 3P = (P × 12.5 × n)/100.  ; [r = 12.5%]
⇒ 3 = (12.5 × n)/100
⇒ n = (3 × 100)/12.5
∴ n = 24 years

∴ The sum will become four times itself in 24 years.

It is a G.P (general process) with r i.e;
2¹, 2², 2³,...
If number of term is n.Then,
2× 2^n-1 = 1024
2^n-1 = 512
2^n-1 = 2⁹
n -1 = 9
n = 10
Answer : 10

Let,
Fahim's age = x
Bilash's age = y and
Wasim's age = z.

According to the given information
Age of Bilash - Age of Fahim = Age of Fahim - Age of Wasim
⇒ y - x = x - z
⇒ 2x = y + z ......(i)
Also, y + z = 66
from (i) x = 33 years

but, y + z = 66 means, there can be many sets of values that will satisfy the equation.
01 - 65
02 - 64
... ... ...
32 - 34

We want to determine x - z but as we can't get a fixed value for z; hence, the answer can not be determined.

Question: The difference between two numbers is 840. When the larger number is divided by the smaller, the quotient is 6 and the remainder is 10. What is the smaller number?

Solution:
Given that,
The difference of two numbers = 840
Quotient when the larger number is divided by the smaller number = 6
Remainder when the larger number is divided by the smaller number = 10

Now, Let the smaller number be x.
Larger number = 6x + 10

ATQ,
⇒ 6x + 10 - x = 840
⇒ 5x + 10 = 840
⇒ 5x = 840 - 10
⇒ 5x = 830
⇒ x = 830/5
⇒ x = 166

So the smaller number is 166.

Total number of votes polled = (1136 + 7636 + 11628) = 20400.
Required percentage =(11628/20400)x 100%
= 57%.

Question: 


Solution:

প্রশ্ন: যদি একটি ধনাত্মক পূর্ণসংখ্যা n কে 18 দ্বারা ভাগ করলে ভাগশেষ 7 থাকে, তাহলে n কে 6 দ্বারা ভাগ করলে ভাগশেষ কত হবে?

সমাধান:
এখানে, ভাজ্য = n
ভাজক = 18
ভাগশেষ = 7
ধরি, ভাগফল = q

ভাজ্য = (ভাজক × ভাগফল) + ভাগশেষ 
∴ n = 18q + 7
⇒ n = (18q + 6) + 1
⇒ n = 6(3q + 1) + 1

যেহেতু 6(3q + 1) সংখ্যা, 6 দ্বারা নিঃশেষে বিভাজ্য হবে, তাই ভাগশেষ থাকবে শুধু 1।

∴ n কে 6 দ্বারা ভাগ করলে ভাগশেষ হবে = 1

Question: Solve: (73.48 + 126.52) ÷ 5² = 1.6 × 3 + ?

Solution:
Given that, (73.48 + 126.52) ÷ 5² = 1.6 × 3 + ?
⇒ 200 ÷ 25 = 1.6 × 3 + ?
⇒ 8 = 4.8 + ?
⇒ ? = 8 - 4.8
∴ ? = 3.2

Question: A man rows a boat a certain distance downstream in 9 hours, while it takes 18 hours to row the same distance upstream. How many hours will it take him to row three-fifth of the same distance in still water?

Solution: 
Let distance = d
So downstream speed = d/9​
And upstream speed = d/18​

We know, 
Speed in still water = (Downstream + Upstream)/2
= (d/9 + d/18)/2 
= {(2d + d)/18}/2
= (d/6)/2
= d/12
So, still water speed = d/12
Distance to travel = 3d/5

∴ Time = (3d/5)/(d/12) = (3 × 12)/5 = 36/5 = 7.2 hours

So it will take the man 7.2 hours to row three-fifth of the distance in still water. 

Question: If X and Y can complete a work together in 12 days, Y and Z together in 20 days, and X and Z together in 15 days, then in how many days can Y alone complete the work?

Solution:
মনে করি, সম্পূর্ণ কাজ = 1 অংশ

∴ (X + Y) একদিনে করে = 1/12 অংশ ..........(1)
(Y + Z) একদিনে করে = 1/20 অংশ ..........(2)
(X + Z) একদিনে করে = 1/15 অংশ ..........(3)

(1), (2), (3) যোগ করে পাই,
2 × (X + Y + Z) = (1/12) + (1/20) + (1/15)
⇒ 2 × (X + Y + Z) = (5 + 3 + 4)/60 = 12/60
⇒ 2 × (X + Y + Z) = 1/5
⇒ (X + Y + Z) = 1/10

∴ X, Y এবং Z একসাথে একদিনে করে = 1/10 অংশ

(X + Z) একসাথে একদিনে করে = 1/15 অংশ
∴ Y-এর একদিনের কাজ = (1/10) - (1/15) অংশ
= (3 - 2)/30 
= 1/30

অর্থাৎ, Y সম্পূর্ণ কাজ করে = 1 ÷ (1/30) = 30 দিনে

When m is divided by n, the remainder is 8.
⇒m = nk + 8, where k is the quotient and n>8 (since Divisor > Remainder)
⇒m/n = (nk + 8)/n
⇒m/n = k + 8/n … (i)
In the above relation, 8/n must be a fractional value such that:
0 < 8/n < 1 ..… (ii)
We know that:
m/n = 89.32 = 89 + 0.32 ..… (iii)
Thus, from (i), (ii) and (iii), we have:
k = 89
8/n = 0.32
⇒ n = 8/0.32 = 100/4
⇒ n = 25

Question: If 500 laborers can complete a work in 48 days, determine the number of extra laborers required to finish the same work in 40 days.

Solution:
48 দিনে কাজটি করতে শ্রমিক লাগে = 500 জন
∴ 1 দিনে কাজটি করতে শ্রমিক লাগে = (500 × 48) জন লোক
∴ 40 দিনে কাজটি করতে শ্রমিক লাগে = (500 × 48)/40 জন
= 600 জন

∴  অতিরিক্ত শ্রমিক লাগবে = (600 - 500) জন = 100 জন

Question: A person who pays income tax at the rate of 4 paise per tk finds that a fall in the interest rate from 4% to 3.75% diminishes his net yearly income by tk 48. What is his capital?

Solution:
If the capital after tax deduction be p, then
p × (4 - 3.75) % = 48
⇒ (p × 0.25)/100 = 48
⇒ (p × 25)/10000 = 48
⇒ p/400 = 48
⇒ p = 48 × 400 = tk 19200

∴ Required capital = (19200 × 100)/96
= 20000 tk

Given: Speed of the person = 5 km/hr, length of train = 100 m, speed of train = 60 km/hr
Speed of train relative to walking person = (60–5) = 55 km/hr
Convert km/hr into m/s
55 km/hr = 55 x(5/18) = 15.27 m/s
Distance to be covered by the train = 200 + 100 = 300 m
Therefore, time taken by the train to cross the person
= Distance over speed =300/15.27 = 19.64 sec

Let the numbers be x and (x + 3)
Then,
⇔x²+(x+3)²=369
⇔x²+x²+9+6x=369
⇔2x²+6x−360=0
⇔x²+3x−180=0
⇔(x+15)(x−12)=0
⇔x=12 
So, the numbers are 12 and 15
∴ Required sum = (12 + 15) = 27

Amount of milk left after 3 operations = [40(1 - 4/40)³] litres
= 40 × 0.729 liters
= 29.16 litres

Number of students who passed = 120 - 12 = 108
Passed student in percent = 108/120 × 100 = 90%
Passed in both subject = (70 + 80)% - 90% = 60%
∴ Only passed in English = (80 - 60)% of 120
                                        = 20/100 × 120
                                        = 24

Question: In how many ways can we select a team of 5 students from a given choice of 20?

Solution:
The number of possible ways of selection is given by,
20C5
​= 20!/5!(20 - 5)!
= (20 × 19 × 18 × 17 × 16 × 15!)/(5 × 4 × 3 × 2 × 15!)
​= 15504

So, the number of ways to select 5 students from 20 is 15504.

Question: Two dice are rolled together. What is the probability that the sum is at least 10?

Solution:
 দুইটি ছক্কা নিক্ষেপ করা হলে নমুনা বিন্দুর সংখ্যা = 6² = 36

প্রশ্নমতে,
দুইটি ছক্কায় উঠা সংখ্যাদ্বয়ের যোগফল ≥ 10 
 
এখন,
যোগফল 10 হলে অনুকূল ঘটনা = (4, 6), (5, 5), (6, 4) অর্থাৎ 3 টি। 
যোগফল 11 হলে অনুকূল ঘটনা = (5, 6), (6, 5) অর্থাৎ 2 টি । 
যোগফল 12 হলে অনুকূল ঘটনা = (6, 6) অর্থাৎ 1 টি। 

∴ মোট অনুকূল ঘটনা = 3 + 2 + 1 = 6

সম্ভাবনা = 6/36
= 1/6

Question: M and N are two positive integers such that MN = 72. Which of the following cannot be the value of M + N?

Solution:
72-এর উৎপাদক জোড়াগুলো হল:
1 × 72 = 72 ⇒ M + N = 1 + 72 = 73
2 × 36 = 72 ⇒ M + N = 2 + 36 = 38
3 × 24 = 72 ⇒ M + N = 3 + 24 = 27
4 × 18 = 72 ⇒ M + N = 4 + 18 = 22
6 × 12 = 72 ⇒ M + N = 6 + 12 = 18
8 × 9 = 72 ⇒ M + N = 8 + 9 = 17

সুতরাং, M + N এর সম্ভাব্য মানগুলো: 17, 18, 22, 27, 38, 73

তাই, M + N = 25 হতে পারে না।

Question: Three numbers A, B, and C are in the ratio 1 : 2 : 3. Their average is 600. If A is increased by 10% and B is decreased by 20%, by what amount should C be increased so that the new average becomes 3% higher?

Solution:
Let
A = x, B = 2x, C = 3x.
Then,
x + 2x + 3x = 600 × 3
Or, 6x = 1800
Or, x = 300

So, A = 300, B = 600, C = 900.
New value of A = 110% of 300 = 330
New value of B = 80% of 600 = 480 
New average = 103% of 600 = 618

∴ New value of C = (618 × 3) - (330 + 480)
= 1854 - 810
= 1044

∴ Increase in value of C = 1044 - 900
= 144 Tk.

Question: If , the value of x is = ?

Solution: 
Given that, 
52/x = √(169/289)
⇒ 52/x = 13/17
⇒ x = (52 × 17)/13
⇒ x = 4 × 17
∴ x = 68

Question: If a + (1/a) = 4, what is the value of a³ + (1/a³)?

Solution:
দেওয়া আছে, a + (1/a) = 4

আমরা জানি,
a³ + (1/a³)
= {a + (1/a)}³ - 3 × a × (1/a) × (a + 1/a)
= (4)³ - 3 × 4
= 64 - 12
= 52

Question: A group of 8 boxes has its average weight increased by 3 kg after replacing a 50 kg box with a new one. What is the weight of the new box?

Solution:
Let the weight of the new box be x kg.
Let the total weight of the original 8 boxes = W.
Original average = W/8.

After replacing the 50 kg box with a box weighing x kg, 
The total weight becomes = W - 50 + x. 
∴ The new average = (W - 50 + x)/8.

ATQ,
New average = Old average + 3
(W - 50 + x)/8 = W/8 + 3
⇒ W - 50 + x = W + 24
⇒ x = 24 + 50
⇒ x = 74

∴ The weight of the new box is 74 kg.

(3⁴x¹⁷y¹⁸)^1/4
= (3^4/4x^17/4y^18/4)
= 3x^17/4y^9/2

Question: If a and b are whole numbers such that, a^b = 32; the value of (a + 1)^{2b - 7} is-

Solution:
Here, a^b = 32
a^b = 2⁵
∴ a = 2 and b = 5 

Now,
(2 + 1)^{(2 × 5) - 7} = 3^{(10 -7)}
= 3³
= 27

Question: What is the compound amount of Tk. 4000 for 2 years at a rate of interest 5% per annum?

Solution:
Given,
Principal, P = 4000
Rate, r = 5% = 5/100 = 1/20
Time, n = 2 years

We know,
A = P(1 + r)ⁿ
= 4000 × (1 + 1/20)²
= 4000 × (21/20)²
= (4000 × 21 × 21)/(20 × 20)
= (4000 × 441)/400
= 4410

∴ The compound amount is Tk. 4410.

Question: Three Hikers A, B and C start on a trip with Tk. 50 each and agree to share the expenses equally. If at the end of the trip. A has Tk. 20 left with him, B Tk.30 and C Tk. 40 how must they settle their accounts?

Solution:
Expenses of A = 50 - 20 = 30
Expenses of B = 50 - 30 = 20
Expenses of C = 50 - 40 = 10

Total expenses = 30 + 20 + 10 = 60
Each person's share = 60/3 = 20
Since A overpaid by Tk. 10 and C underpaid by Tk. 10, C should pay Tk. 10 to A to balance the expenses.

Average speed
= (2xy / x+y) km/hr
= (2×50×30 / 50+30) km/hr
= 37.5 km/hr

Given, (1/5)^3y = 0.008 = 8/1000
Or, (1/5)^3y = 1/125 = (1/5)³
Or, 3y = 3
Or, y = 1

So, (0.25)^y = (0.25)¹ = 0.25

Question: A tank is filled to three-fifths of its capacity with water. When 9 liters of water are added, the tank becomes six-sevenths full. Find the total capacity of the tank.

Solution:
ধরি,
ট্যাংকটির ধারণ ক্ষমতা = x লিটার

প্রশ্নমতে,
(3x/5) + 9 = 6x/7
⇒ (6x/7) - (3x/5) = 9
⇒ (30x - 21x)/35 = 9
⇒ 9x/35 = 9
⇒ 9x = 9 × 35
⇒ x = (9 × 35)/9
⇒ x = 35

∴ ট্যাংকটির ধারণ ক্ষমতা = 35  লিটার

To complete work in 18 days we need either 3 men or 6 boys.
∴ 1 man = 2 boys
Take work done = 1
∴ 3 men x 18 days x 1 = (4 men + 4 boys) x ? days x 1
∴ 6 boys x 18 days x 1 = (8 boys + 4 boys) x ? days x 1 [Convert either all men to boys or all boys to men.]
∴ ? = 9 days = they will need these many days.

We know,
Man's/Boat's Speed = X
Stream/Current/River Speed = Y

∴ Downstream speed = X + Y
Upstream speed = X - Y

Downstream speed = Distance covered/Time taken
= 48/8 = 6 km/hr
Upstream Speed = Distance covered/Time taken
= 48/12 = 4 km/hr

X+Y = 6 km/hr and X-Y = 4 km/hr
Adding them we get,
X + Y + X - Y = 10 km/hr

∴ X=5 km/hr = Speed of Boat
Y=6 - 5 = 1 km/hr = Speed of stream

We know,
If loss is A%, then Selling Price = (100 - A)% of cost price.

Selling Price = 3200 = (100 - 20)% of CP = (80 × CP)/100
∴ CP = (3200 × 100)/80

For profit 20% = Selling Price = (100 + 20)% of CP = (120/100) × (3200 × 100)/80
∴ Selling Price = Tk. 4800

Gain% = [ Error/(True weight - Error)]%

Error = True weight - False weight
Error = 1000 - 970
= 30
Gain% = [{30/(1000 - 30)} × 100]%
Gain% = 3.09%

Question: A girl bought a Nepenthes plant for Tk. 1600 after getting a 20% discount. What was the original catalog price of the plant?

Solution:
20% discount,
If the catalog price is Tk. 100 then the purchased price = (100 - 20)
= Tk. 80

Now,
If the purchase price is Tk. 80, the catalog price = 100 taka.
∴ If the purchase price is Tk. 1, the catalog price = 100/80 taka
∴ If the purchase price is Tk. 1600, the catalog price = (100 × 1600)/80
= Tk. 2000

∴ the original catalog price of the plant is  Tk. 2000

Pipe 1 can fill 1/2 of the cistern in 1 hour
Pipe 2 can fill 1/6 of the cistern in 1 hour
Pipe 3 can empty 1/9 of the cistern in 1 hour
According to the question,
Time taken to full the cistern = 1/2 + 1/6 - 1/9
= 5/9
5/9 of the cistern will be filled in 1 hour.
Full cistern will be filled in = (5/9)/1
= 9/5 x 1
= 1.8 hours

We know that Ratio of Investment x Time = Ratio of Profit
∴ To find the ratio of time just divide respective profits ratios by respective investment ratios
So if the profit ratio is P₁: P₂ : P₃ and the investment ratio is I₁ : I₂ : I₃ then,
Time ratio is found by = (P₁/I₁) : (P₂/I₂) : (P₃/I₃)

6 : 4 : 10 can be reduced to 3 : 2 : 5. So, their investments are actually in the ratio 3 : 2 : 5.
The ratio of time periods for A, B, and C is found by = 8/3 : 7/2 : 9/5
Making denominators common by multiplying by 30
∴ The ratio of Time periods for A, B, and C = (30 × 8)/3 : (30 × 7)/2 : (30 × 9)/5
= 80 : 105 : 54