Bank Math

Question: Find the value of  0!/√4.

Solution:
According to the factorial law,
n! = n × (n - 1)!

If n = 1, then
1! = 1 × (1 - 1)!
⇒ 1! = 1 × 0!
⇒ 1 = 0!
∴ 0! = 1

Now,
0!/√4 = 1/√(2²)
= 1/2

Area of the circle =  πr² 
= π(√5)² 
= 5π
= 15.708

log₁₀2500−log₁₀25
=log₁₀(2500 / 25)
=log₁₀100
=log₁₀10²
=2log₁₀10
=2

Question: The ages of Ayesha and Rina are in the ratio 5 : 4. After 10 years, the ratio of their ages will be 7 : 6. What is the present age difference between them?

Solution:
Let their present ages be 5x and 4x

According to the question,
(5x + 10)/(4x + 10) = 7/6
⇒ 6(5x + 10) = 7(4x + 10)
⇒ 30x + 60 = 28x + 70
⇒ 30x - 28x = 70 - 60
⇒ 2x = 10
⇒ x = 5

Ayesha’s age = 5x = 25 years
Rina’s age = 4x = 20 years

Difference = 25 - 20 = 5 years

Question: A man walk at a speed of 9 km/h. After every kilometer, he takes a rest of 4 minutes. How much time will he take to cover a distance of 6 km?

Solution:
Given,
Distance = 6 km
Speed = 9 km/h

∴ The man needs time = (Distance ÷ Speed)
= (6 ÷ 9) hours
= 2/3 hours
= (2/3 × 60) minutes
= 40 minutes

He will also rest 4 times after 1000, 2000, 3000, 4000 and 5000 meters
Total resting time = 4 × 5
= 20 minutes.

Total time = (40+ 20) minutes
= 60 minutes
= 1 hour

Question: A cistern can be filled by two pipes X and Y in 8 hours and 12 hours respectively. When full, the tank can be emptied by a third pipe Z in 24 hours. If all the taps be turned on at the same time, the cistern will be full in?

Solution:
Net filling in 1 hour = (1/8) + (1/12) - (1/24)
= (3 + 2 - 1)/24
= 4/24
∴ Time taken to fill the cistern = (24/4) hrs.
= 6 hrs.

Question: A cube has a total surface area of 72 m². Determine the length of its diagonal.

Solution:
মনে করি,
ঘনকটির ধার = a 
ঘনকের সম্পূর্ণ পৃষ্ঠের ক্ষেত্রফল = 6a²

প্রশ্নমতে,
6a²  = 72
⇒ a² = 72/6
⇒ a² = 12
⇒ a² = (4 × 3)
⇒ a = 2√3 

∴ ঘনকটির কর্ণের দৈর্ঘ্য = a√3
= 2√3 × √3 
= 2 × (√3)2
= 2 × 3
= 6

∴ ঘনকটির কর্ণের দৈর্ঘ্য = 6 মিটার।

Let the new S.P. be Tk . x
Then, 85 : 18700 = 115 : x
⇒ x = (18700 × 115)/85
⇒ x = 25300

While A covers 1000 meters, B can cover 900 meters

While B covers 1000 meters, C can cover 900 meters

Let's assume that all three of them are running the same race.
So when B runs 900 meters,
C can run 900 × (9/10)
= 810

So A can beat C by = 1000 - 810 = 190 meters.

In every two minutes he is able to ascend 1 m.
In this fashion he ascends up to 12 m because when he reaches at the top he does not slip down.
Thus, up to 12 m he takes 12×2=24 min. and for the last 2 m he takes 1 m.
Therefore, total time taken by him is 24+1 = 25 min to reach the top.

Cloth is required for 1 shirt
= 2 m, 60 cm or 260 cm
Cloth is required for 7 shirt
= 260 × 7
= 1820 cm or 18 m 20 cm

Let the numbers are x, y, z
so, x²+ y² + z² = 532
Given,
x : y = 3 : 2 = 9 : 6
y : z = 3 : 2 = 6 : 4
Then x : y : z = 9 : 6 : 4
Now, Let the numbers are 9a, 6a, 4a
So, (9a)² + (6a)² + (4a)² = 532
Or, 81a² + 36a² + 16a² = 532
Or, 133a² = 532
Or, a² = 532/133 = 4
Or, a = 2
∴ the first number = 9×2 = 18

Question: The surface area of a cube is 96 cm². Its volume is-

Solution: 
Here, 
Surface area, 6a² = 96
⇒ a² = 96/6
⇒ a² = 16
⇒ a = √16
⇒ a = 4

∴ Volume = a³
= 4³
= 64 cm³

ধরি, শূণ্য স্থানের সংখ্যাটি x
প্রশ্নমতে, (x - 2763) ÷ 86 × 13 = 208
⇒ (x - 2763) = 208/13 × 86
⇒ x - 2763 = 1376
⇒ x = 1376 + 2763
⇒ x = 4139

First increase by 20%
So, per share value is = 40 + 40×20% = 48
Then, decrease by 25%,
So new value will become = 48 - 48×25% = 36 

যে কোন করমর্দন অথবা কোলাকুলির অংকে শুধু কত জন করমর্দন (Handshake), বা কোলাকুলি করল তা দেয়া থাকবে
এক্ষেত্রে মনে রাখতে হবে যে প্রত্যেক বার করমর্দন বা কোলাকুলি করার সময় মোট ২ জন লোকের প্রয়োজন।
তাই এক্ষেত্রে সূত্রটি হবে ⁿC₂=^{মোট লোক}C_{২ জন সব সময়}
20C₂= 20!/2!(20 - 2)!
= 20!/2!18!
= (20 × 19)/2
= 10 × 19
= 190

Question: Rafi is currently 18 years old, and his uncle is 54 years old. In how many years will Rafi’s age be one-third of his uncle’s age 6 years from that point?

Solution: 
Given,
Rafi’s current age = 18 years.
Uncle’s current age = 54 years.

Let, the required number of years be x

So, after x years,
Rafi's age = 18 + x
Uncle's age = 54 + x 

Uncle's age 6 years from that point = 54 + x + 6
= 60 + x

According to the question,
18 + x = 1/3(60 + x)
⇒ 3(18 + x) = 60 + x
⇒ 54 + 3x = 60 + x
⇒ 2x = 6
∴ x =3

So, in 3 years, Rafi’s age will be one-third of his uncle’s age 6 years after that.

Question: If (4P - 4)/(3P - 3) = (4/P) where P ≠ 1, what is the value of P² - 4P + 3 = ?

Solution:
Given that,
(4P - 4)/(3P - 3) = 4/P
or, P(4P - 4) = 4(3P - 3)
or, 4P²- 4P = 12P - 12
or, 4P²- 4P - 12P + 12 = 0
or, 4P²- 16P + 12 = 0
or, 4(P² - 4P + 3) = 0
or, P² - 4P + 3 = 0/4
∴ P² - 4P + 3 = 0

Question: A sum of money doubles itself in 5 years at a certain rate of simple interest. In how many years will it become five times itself at the same rate of interest?

Solution:
Let the Principal be P.
The amount doubles in 5 years, so Amount = 2P.
Interest (I) for 5 years = 2P - P = P.

Now, the sum needs to become five times itself, so New Amount = 5P.
Required Interest (I) = 5P - P = 4P.

We know that,
P amount of interest is earned in = 5 years.
∴ 4P amount of interest will be earned in = (5 × 4) years = 20 years.

∴ The sum of money will become five times itself in 20 years.

Question: In a group of 135 persons, 50% people contributed tk. 40 each, 30% contributed Tk. 60 each and the remaining persons contributed Tk. 80. Find the average contribution for the group?

Solution:
total people=135

Since
50% of them contributed Tk. 40 
∴ amount contributed by these 50% = (135/2) × 40 = 2700

Since 30% of them contributed TK. 60
∴ amount contributed by these 30% = (30/100) × (135) × 60 = 2430 

and
remaining 20% of them contributed Tk. 80
∴ amount contributed by these 20% = (20/100) × 135 × 80 = 2160

∴ Total amount contributed = 2700 + 2430 + 2160 = 7290 

∴ Average contribution = 7290/135 = 54

Question:

Solution:

Question: Find the number of triangles in the given figure.


Solution:

The simplest triangles are AJF, FBG, GCH, HDI, and IEJ = 5 in number.

The triangles composed of three components each, EBH, AIC, EFC, ADG, and BJD = 5 in number.

Thus, there are 5 + 5 = 10 triangles in the figure.

Question: A television service center can process 8 televisions in 18 minutes. How many televisions can it process in 3 hours at this rate? 

Solution:
3 hours = 3 × 60 = 180 minute

In 18 minutes, the service center can process = 8 tv
In 1 minute, the service center can process = 8/18 tv
In 180 minutes, the service center can process = (8 × 180)/18 tv
= 80 tv

Let, Speed of the man is x Kmph.
Distance covered in 10 minutes at 20 kmph = distance covered in 8 minutes at (20 + x) kmph.
⇒ 20 × (10/60) = 8/60 × (20 + x)
⇒ 200 = 160 + 8x
⇒ 8x = 40
Hence, x = 5 kmph.

Question: If log_m (1/√32) = - 5/2 what is the value of m?

Solution:
দেওয়া আছে,
log_m (1/√32) = - 5/2
⇒ m^{- 5/2} = 1/√32    [যেহেতু log_aM = x হলে, a^x = M]
⇒ m^{- 5/2} = 1/(32^1/2)
⇒ m^{- 5/2} = 32^{- 1/2}
⇒ m^{- 5/2} = (2⁵)^{- 1/2}
⇒ m^{- 5/2} = 2^{- 5/2}
∴ m = 2

Question: Excluding stoppages, a train travels at 72 km/h and including stoppages, its average speed is 60 km/h. For how many minutes does the train stop per hour?

Solution:
At its moving speed of 72 km/h, the time required to cover 60 km is:
∴ Time moving = distance/speed
= 60 km/72 km/h
= 60/72 hours
= 5/6 hours
= (5/6) × 60 minutes
= 50 minutes

We know,
 1 hour = 60 minutes

∴ Stopping time = 60 - 50 = 10 minutes

∴ The train stops for 10 minutes per hour.

a - b ≥ a + b
⇒ a - b - b ≥ a
⇒ -2b ≥ 0
⇒ 2b ≤ 0 [-1 দ্বারা গুণ করে] 
∴ b ≤ 0

Question: Solve the inequality |x - 2| < 5

Solution:
|x - 2| < 5
⇒ - 5 < x - 2 < 5
⇒ - 5 + 2 < x - 2 + 2 < 5 + 2
⇒ - 3 < x < 7

Choosing 4 marbles means 1^st marble AND then 2^nd AND then 3^rd AND then 4^th marble
At least 1 red marble means there can be 1, 2, 3, or 4 red marbles.
So, Probability of choosing red = 1 - Probability of not choosing red
If we remove red marbles, then 15 - 4 red marbles = 11 marbles remain.
We have to choose 4 out of these 11.

So Probability of choosing 4 marbles that are not red = 11/15 × 10/14 × 9/13 × 8/12 = 22/91.

∴ Probability of picking at least 1 red marble = 1 - 22/91 = 69/91.
[Here we reduce the denominator i.e. the total number of marbles because once we remove a marble from the box we do not put it back in the box.
So while removing the 2^nd marble, there are only 15 - 1 = 14 marbles in the box. The same is for the 3^rd and 4^th marbles.]

Question: If England is written as 1234526 and France is written as 785291, then Greece is written as-

Solution: 
England is written as 1234526
E = 1
n = 2
g = 3
l = 4
a = 5
n = 2
d = 6

France is written as 785291
F = 7
r = 8 
a = 5
n = 2
c = 9
e = 1

So, GREECE will be = 381191

Question: The average age of a group is 35. If 5 new members with an average age of 25 join, the overall average becomes 33. How many members were originally in the group?

Solution:
Let
The original number of members be x.
Total age of original members = 35 × x = 35x

Total age of 5 new members = 25 × 5 = 125
Total number of members after adding 5 = x + 5

New average = 33,
So total age after adding 5 members = 33 × (x + 5) = 33x + 165

Set up the equation for total age:
Original total age + New members’ total age = New total age
⇒ 35x + 125 = 33x + 165
⇒ 35x - 33x = 165 - 125
⇒ 2x = 40
⇒ x = 20

Question: P = {x ∈ N : 2 < x ≤ 6} and Q = {x ∈ N : x is an even number and x ≤ 8}. Find the value of P ∩ Q.

Solution:
Given that, 
P = {x ∈ N : 2 < x ≤ 6}
∴ P = {3, 4, 5, 6}

Q = {x ∈ N : x is even and x ≤ 8}
∴ Q = {2, 4, 6, 8}

Now,
P ∩ Q = {3, 4, 5, 6} ∩ {2, 4, 6, 8}
= {4, 6}

Let X hours be the time taken to fill a tank by P.
Let Y hours be the time taken to empty the tank by Q.
Then the time taken to fill the tank when P and Q are switched together : XY/(Y - X) hours.
Here, X = 16 minutes And Y = 32 minutes
Therefore,
Required time = (16 × 32)/(32 - 16)
= (32 × 16)/16
= 32 minutes.

Question: What is the difference between simple and compound interest at 5% per annum on a sum of Tk. 8000 at the end of 2 years?

Solution:
এখানে,
মূলধন (P) = 8000 টাকা
সুদের হার (r) = 5%
সময় (n) = 2 বছর

সরল সুদ (SI):
SI = (P × r × n)/100
= (8000 × 5 × 2)/100
= 800 টাকা

চক্রবৃদ্ধি সুদ (CI):
A = P(1 + r/100)ⁿ
= 8000(1 + 5/100)²
= 8000(105/100)²
= 8000 × (105/100) × (105/100)
= 8820 টাকা

চক্রবৃদ্ধি সুদ (CI) = A - P
= 8820 - 8000
= 820 টাকা

∴ সুদের পার্থক্য = CI - SI
= 820 - 800
= 20 টাকা

Question: Find an equation for the line with x-intercept = 2, Y - intercept = - 1

Solution:
দেওয়া আছে, 
x-অক্ষের ছেদবিন্দু 2   ∴ বিন্দু (2, 0)
y-অক্ষের ছেদবিন্দু - 1  ∴ বিন্দু (0, - 1)

এখন, (2, 0) এবং (0, - 1) বিন্দুর ঢাল, 
m = (y₂ - y₁)/(x₂ - x₁)
⇒ m = (- 1 - 0)/(0 - 2) = 1/2
∴ m = 1/2

এখন, রেখার সমীকরণ
y = mx + b, [যেখানে, ঢাল m এবং y-অক্ষের ছেদবিন্দু, b ].
⇒ y = (1/2)x - 1  ; [m = 1/2, y-অক্ষের ছেদবিন্দু, b = - 1]
⇒ 2y = x - 2
∴ x - 2y = 2

Question: If 60% of a number is equal to three-fourths of another number, what is the ratio of the first number to the second number?

Solution:
ধরি,
প্রথম সংখ্যাটি x এবং দ্বিতীয় সংখ্যাটি y।
প্রশ্নমতে,
60% of x = (3/4) of y
⇒ (60/100) × (x) = (3/4) × y
⇒ (3/5)x = (3/4)y
⇒ x/5 = y/4
⇒ 4x = 5y
⇒ x/y = 5/4

∴x : y = 5 : 4