Bank Math

Question: Five times the first of three consecutive even integers is 4 more than two times the third. The sum of the integers is-

Solution:
Let,
The three even integers = x, x + 2 and x + 4

ATQ,
5x = 2(x + 4) + 4
⇒ 5x = 2x + 8 + 4
⇒ 5x - 2x = 12
⇒ 3x = 12
∴ x = 4

∴ First integer = 4
Second integer = x + 2 = 4 + 2 = 6
And, third integer = x + 4 = 4 + 4 = 8

∴ Sum = 4 + 6 + 8 = 18

Speed = 40 km/hr = (40 × 5/18) m/s = 100/9 m/s
Time = 18 seconds
Distance Covered = 100/9 × 18 = 200 m
Therefore,
Length of the train = 200 m

Given one star(S) equals four circles(C) and three circles equal four diamonds(D).
Then, S = 4C and 3C = 4D
Then, C = (4/3)D
∴ S = 4×(4/3)D
⇒ S = (16/3)D
Hence, S : D = 16 : 3

Let x be the amount of per day consumption of each man.
Let y is the reinforcement.
Assuming men in reinforcement also has the same amount of daily consumption

Total consumption = 2000 × x × 66
Provision left after 14 days = 2000 × x × 66 - 2000 × x × 14 = 2000 × x × 52
This would be consumed by total men including reinforcement.

Thus,
2000 × x × 52 = (2000 + y) × x × 20
⇒ 2000 × 32 = 20y
⇒ y = 3200.

Question: In a class, 30 students play basketball, 20 students play volleyball, and 8 students play both. 12 students play neither basketball nor volleyball. What is the total number of students in the class?

​​Solution:
Number of students who play basketball, n(B) = 30
Number of students who play volleyball, n(V) = 20
Number of students who play both basketball and volleyball, n(B ∩ V) = 8
Number of students who play neither = 12

n(B ∪ V) = n(B) + n(V) - n(B ∩ V)
= 30 + 20 - 8 = 42

Total students in the class = students who play basketball or volleyball + students who play neither
n(U) = n(B ∪ V) + neither = 42 + 12 = 54

∴ Total 54 students in the class.

Given, 36 + 81 + 144
The series is = 6² + 9² + 12² + 15² + 18²
So, next term is 18² = 324

Question: A cylinder with a diameter of 16 cm and height of 2 cm is melted to make 12 spheres. What is the radius of each sphere?

Solution:
Given that, 
Diameter of cylinder = 16 cm,
so radius = 16/2 = 8 cm

Let the radius of each sphere is r cm.

We know,
Volume of a sphere = (4/3)πr³
Volume of a cylinder = πR²h

According to the question, 
Volume of 12 spheres = Volume of the cylinder
⇒ 12 × (4/3) × π × r³ = π × (8)² × 2
⇒ 12 × (4/3) × π × r³ = π × 64 × 2
⇒ 12 × (4/3) × π × r³ = 128π
⇒ 16r³ = 128
⇒ r³ = 128/16
⇒ r³ = 8
⇒ r³ = 2³
∴ r = 2

Therefore, the radius of each sphere is 2 cm.

Question: A cistern measures 5 meters in length, 2.5 meters in breadth, and 3 meters in height. Find its capacity in liters.

Solution:
দেওয়া আছে,
চৌবাচ্চাটির দৈর্ঘ্য = 5 মিটার
চৌবাচ্চাটির প্রস্থ = 2.5 মিটার 
চৌবাচ্চাটির উচ্চতা = 3 মিটার

আমরা জানি,
চৌবাচ্চাটির আয়তন = (5 × 2.5 × 3) ঘন মিটার
= 37.5 ঘন মিটার
= (37.5 × 1000) লিটার [১০০০ লিটার = ১ ঘন মিটার]
= 37500 লিটার

∴ চৌবাচ্চাটির পানি ধারণক্ষমতা = 37500 লিটার।

Question: Which of the following is the smallest?

Solution:
ক) 1/2 = 0.5
খ) 3/8 = 0.375
গ) 11/25 = 0.44
ঘ) 5/12 = 0.4167

∴ The smallest value is 3/8.

Question: A bike travels at a speed twice as fast as walking. A person spends the same amount of time walking and riding the bike, covering a total distance of 180 km. How many kilometers does he walk?

Solution:
Let the walking speed be = x km/h.
Then the bike speed is = 2x km/h ; [since the bike is 2 times faster than walking]
Total distance covered = 180 km.

Let the distance walked = d km.
Then the distance covered by bike = (180 - d) km.

We know,
Time = distance/speed
⇒ d/x = (180 - d)/2x
⇒ d = (180 - d)/2
⇒ 2d = 180 - d
⇒ 3d = 180
⇒ d = 180/3
∴ d = 60

So, the person covers 60 km by walking.

We know, Area of rhombus = 1/2 × x × y [Here, x and y are two diagonals of the rhombus]
Or, x = (96 × 2) / 16 = 12 cm

Question: If 12 workers can complete a project in 15 days, how many days will it take for 9 workers to complete the same project assuming they all work at the same rate?

Solution: 
12 workers can complete work in 15 days
∴ 1 workers can complete work in  = 12 × 15 days 
∴ 9 workers can complete work in = (12 × 15)/9 days 
= 20 days 

It will take 9 workers 20 days to complete the same project. 

Question: The surface area of a sphere is 324π cm². What is its radius? 

Solution: 
Let, r is the square of the sphere. 

A.T.Q,
4πr² = 324π
r² = 81
∴ r = 9

The radius of the sphere is 9 cm.

Question: If 60 men can complete a job in 50 days, how many extra men need to be hired to finish the same work 10 days earlier?

Solution:
Let the total work = 60 × 50 = 3000 man-days
New time to finish work = 50 - 10 = 40 days

Number of men required = Total work ÷ New time
= 3000 ÷ 40
= 75 men

Extra men needed = 75 - 60 = 15 men

Let the six numbers be, x, (x + 1), (x + 2), (x + 3), (x + 4) and (x + 5)
Then,
⇔ x + (x + 1) + (x + 2) = 27
⇔ 3x + 3 = 27
Required sum :
= (x + 3) + (x + 4) + (x + 5)
= 3x + 12 
= (3x + 3) + 9
= 27 + 9
= 36

Given that,
A takes 20 minutes to fill and B takes 12 minutes to empty
Clearly,
tap B is faster than tap A.
And so, the tank will be emptied.
Half of the tank or 1/2 part of the tank is already filled.
Therefore,
we have to find the time taken to empty that 1/2 part.
Part filled by A in 1 minute = 1/20
Part emptied by B in 1 minute = 1/12.
Part emptied by (A + B) in 1 minute
= (1/12) – (1/20)
= (5 - 3)/60
= 2/30.
Therefore,
The time taken by (A + B) to empty the full tank is 15 minutes.
Time taken to empty 1/2 part of the tank is 30/2
= (30/2) × (1/2) minutes.
= 15/2 minutes.

Let BA be the ladder and AC be the wall as shown above.
Then the distance of the foot of the ladder from the wall = BC

Given that BA = 10 m , BAC = 60°
sin 60° = BC/BA
√3/2 = BC/10
BC = 10√3/2
= 5√3.
= 5 × 1.71
= 8.65 m.

Question: A rectangular prism has dimensions 10 inches by 12 inches by 15 inches. A cylinder with a radius of inches 5 and a height of 14 inches is placed inside the prism. To the nearest cubic inch, what is the volume of the space in the prism not taken up by the cylinder?

Solution:
Given that,
Rectangular prism are 10 in × 12 in × 15 in
And
Cylinder inside prism radius, r = 5 in and height, h = 14 in

Now, 
Volume of the prism = length × width × height
= 10 × 12 × 15 = 1800 cubic inch

And, Volume of the cylinder = πr²h
= (22/7) × 5² × 14
= 22 × 25 × 2
= 1100 cubic inch

So the volume of the empty space = Volume of the prism - Volume of the cylinder
= 1800 - 1100 = 700 cubic inch

L.C.M of A and B is ''B'';
L.C.M of B and C is ''B';
⇒ L.C.M of A, B, C is ''B''
Answer : B

Question: A person deposited Tk. 2500 for 4 years and Tk. 3600 for 3 years at the same rate of simple interest in a bank. If he received a total of Tk. 1040 as interest, what was the rate of interest per annum?

Solution:
ধরি, বার্ষিক সরল মুনাফার হার = r%

আমরা জানি, মুনাফা = (আসল × সময় × হার)/100

প্রথম ক্ষেত্রে মুনাফা = (2500 × 4 × r)/100 = 100r টাকা
দ্বিতীয় ক্ষেত্রে মুনাফা = (3600 × 3 × r)/100 = 108r টাকা

প্রশ্নমতে,
100r + 108r = 1040
⇒ 208r = 1040
⇒ r = 1040/208
⇒ r = 5

∴ বার্ষিক মুনাফার হার 5%

Question: A man completes 5/8 of a job in 10 days. At this rate, how many more days will it takes him to finish the job?

Solution:
Work done = 5/8
Balance work = 1 - (5/8) = 3/8

Let the required number of days be x
Then,
(5/8) : (3/8) :: 10 : x
⇒ (5/8) × x = (3/8) × 10
⇒ x = (3/8) × 10 × (8/5)
∴ x = 6

We are given that,
1) Total population of city = 6500
2) Increase in male and female population = 5% & 10% respectively.
3) Final population of city = 7000

Hence,
Let’s assume that number of males = x
Number of female = 6500 – x

Therefore, after an increase in 5% male and 10% female, the population becomes 7000
5% male +10% female = Difference between new and original population
⇒ (5/100)x + (10/100) (6500 - x) = 7000 - 6500
⇒ 5x + 65000 - 10x = 50000
⇒ 5x = 15000
⇒ x = 3000

Number of males = 3000
Number of females = 3500.

Question: Due to bad weather, a car reduced its speed by 15 km/hr and reached a destination 300 km away 1 hour late. What was the car's original speed?

Solution:
ধরি,
গাড়িটির আসল গতিবেগ ছিল S কিমি/ঘন্টা
এবং আসল সময় ছিল T ঘন্টা।

আমরা জানি, সময় = দূরত্ব/গতিবেগ
∴ T = 300/S

গতিবেগ 15 কিমি/ঘন্টা কমালে,
নতুন গতিবেগ হয় (S - 15) কিমি/ঘন্টা।

গন্তব্যে পৌঁছাতে 1 ঘন্টা বেশি লাগে,
অর্থাৎ নতুন সময় হয় (T + 1) ঘন্টা।
সুতরাং, (T + 1) = 300/(S - 15)

এখন,
(T + 1) - T = 300/(S - 15) - 300/S
⇒ 1 = 300 × {1/(S - 15) - 1/S}
⇒ 1 = 300 × {(S - (S - 15)}/{S(S - 15)}
⇒ 1 = 300 × 15/(S² - 15S)
⇒ S² - 15S = 4500
⇒ S² - 15S - 4500 = 0
⇒ (S - 75)(S + 60) = 0

যেহেতু গতিবেগ ঋণাত্মক হতে পারে না, তাই S = 75 কিমি/ঘন্টা।
সুতরাং, গাড়িটির আসল গতিবেগ ছিল 75 কিমি/ঘন্টা।

Required average = (50.25 × 16 + 45.15 × 8)/(16 + 8)
= (804 + 361.20)/24
= 1165.20/24
= 48.55

Sum of ages of A and B, 3 Years ago =(18 x 2) = 36 Years
Sum of ages of A, B, and C,
Now = (22 x 3) = 66 Years
Sum of ages of A and B,
Now = (36 +6)Years = 42 Years

Therefore,
C's age = (66 - 42)Years = 24 Years.

Let, speed of boat in still water = 16x, speed of stream = 5x
Upstream speed = 16x – 5x = 11x
S = D/t
Or, 11x = 16.5 × 60/45 = 22
Or, x = 2
So, speed of boat in still water = 32 km/h, speed of stream = 10 km/h
Downstream speed = 32 + 10 = 42 km/h
Distance = 17.5 km
Required time = 17.5 / 42 = 5/12 hour = 5 × 60/12 = 25 minutes

Question: Aysha and Jara started a business by investing Tk. 85000 and Tk. 15000 each. What be will the ratio of profit earned after 2 years between Aysha and Jara respectively?

Solution:
Given,
Aysha's investment = Tk. 85,000
Jara's investment = Tk. 15,000
Time = 2 years for both

The ratio of profit earned after 2 years between Aysha and Jara respectively = (85000 × 2) : (15000 × 2)
= 170000 : 30000
= 17 : 3

Question: Starting from point P, Kamal walked 25 m towards West. He turned left and walked 30 m. He then turned left and walked 25 m. After this, he turned to his right and walked 15 m. How far and in which direction is Kamal now from point P?

Solution:

- কামাল P বিন্দু থেকে পশ্চিমে 25 মিটার গেল।
- সেখান থেকে বামে মোড় নিয়ে দক্ষিণে 30 মিটার গেল।
- পুনরায় বামে মোড় নিয়ে পূর্বে 25 মিটার গেল। যেহেতু সে পশ্চিমে 25 মিটার গিয়েছিল এবং পুনরায় পূর্বে 25 মিটার ফিরে এসেছে, তাই সে এখন P বিন্দুর ঠিক সোজাসুজি দক্ষিণ দিকে অবস্থান করছে।
- সবশেষে সে ডানে মোড় নিয়ে দক্ষিণে আরও 15 মিটার গেল।

দূরত্ব নির্ণয়:
P বিন্দু থেকে বর্তমান অবস্থানের মোট দূরত্ব = (30 + 15) মিটার = 45 মিটার।
দিক নির্ণয়:
সে P বিন্দুর সাপেক্ষে এখন দক্ষিণ দিকে রয়েছে।

∴ কামাল এখন P বিন্দু থেকে 45 মিটার দক্ষিণে অবস্থান করছে।

Question:

Solution:

Question: What is the greatest 4-digit number which is divisible by 15, 25, 40, and 75? 

Solution:
Greatest number of 4-digits is 9999.
L.C.M. of 15, 25, 40 and 75 is 600.
On dividing 9999 by 600, the remainder is 399.
∴ Required number (9999 - 399) = 9600.

Question: In an election between two candidates, the winner got 65% of the total votes cast and won the election by a majority of 2748 votes. What is the total number of votes cast if no vote is declared invalid?

Solution:
Winner gets 65% of valid votes and loser gets 35% of votes
Difference between this two = 2748
(65 - 35)% = 2748
⇒ 30% = 2748

Total number of voters, 100% = (2748 × 100)/30
= 9160

Question: In one hour, a boat goes 13 km along the stream and 5 km against the stream. What is the speed of the stream? 

Solution:
Let x be the boat speed.
and y be the stream speed.

Down stream speed,
x + y = 13 ............................. (i)
Upper stream speed,
x - y = 5 ................................ (ii) 

Now (i) + (ii); we get,
 x + y + x - y =13 + 5
⇒ 2x = 18
⇒ x = 9

From (i),
⇒ 9 + y = 13
⇒ y = 13 - 9
⇒ y = 4

∴ The speed of the stream is 4 km/h.

Let,
C's income = 100
So, B's income = 100-25 = 75
And,
A's income = 75 + 75×(20/100) = 75+15 = 90
∴ A’s income less from C’s income is = {(100 - 90)/100}×100 = 10% 

Question: In how many ways can 5 examination papers be arranged so that the best and the worst papers never come together?

Solution:
The number of ways in which 5 papers can be arranged is 5! ways.
When the best and the worst papers come together, regarding the two as one paper, we have only 4 papers.
These 4 papers can be arranged in 4! ways.
And two papers can be arranged themselves in 2! ways.

∴ Number of arrangements when the best and worst papers do not come together,
= 5! - (4! × 2!)
= (5 × 4 × 3 × 2 × 1) - (4 × 3 × 2 × 1 × 2)
= 120 - 48
= 72

Question: If the diameter of a circle is 6π, then what is the ratio between radius and the circumference of the circle?

Solution:
Here,
The diameter of the circle is d = 6π
So the radius of the circle r = 3π

∴ Circumference of circle = 2. π. 3π
= 6π²

So the ratio between radius and Circumference of circle = 3π : 6π²
= 3π/6π²
= 1 : 2π

ধরি, সংখ্যাগুলো x - 3, x - 2, x - 1, x, x + 1, x + 2, x + 3, x + 4
শেষ চারটি সংখ্যার যোগফল = 4x + 10
প্রথম চারটি সংখ্যার যোগফল = 4x - 6
∴ ( - ) পার্থক্য = 16

Question: A fruit salad is made by mixing 3 kgs of mango costing Tk 120 per kg and 2 kgs of papaya costing Tk 100 per kg and 2 kgs of grapes costing Tk 140 per kg. At what price (in Taka) per kg should the mixture be sold to make profit of 25 percent?

Solution:
3 কেজি আমের মূল্য = 120 × 3 = 360 টাকা
2 কেজি পেঁপের মূল্য = 100 × 2 = 200 টাকা
2 কেজি আঙ্গুরের মূল্য = 140 × 2 = 280 টাকা

মোট = 360 + 200 + 280 = 840 টাকা

25% লাভে 
বিক্রয়মূল্য = 840 + 840 এর 25%
= 840 + 840 এর 25/100
= 840 + 210
= 1,050

7 কেজি বিক্রয় করতে হবে = 1050 টাকা 
1 কেজি বিক্রয় করতে হবে = 1050/7 টাকা 
= 150 টাকা

Question: If √x + √3 = √48, than what is the value of x.

Solution: 
Given that, 
√x + √3 = √48
⇒ √x + √3 = √(16 × 3)
⇒ √x + √3 = 4√3
⇒ √x = 4√3 - √3
⇒ √x = 3√3
⇒ (√x)² = (3√3)²   ;[Square both sides]
∴ x = 27

Question: Which is the area of this triangle? 

Solution:
দেওয়া আছে, 
ত্রিভুজের বাহুগুলো হলো 5x, 5x, 8x
সুতরাং, এটি একটি সমদ্বিবাহু ত্রিভুজ।

এখন, ভূমি = 8x = 8x/2 = 4x  ; [ভূমি 8x-এর ওপর উচ্চতা আঁকলে, এটি ভূমিকে সমান দুইভাগে ভাগ করে।]
সুতরাং, প্রতিটি অংশ = 4x

এখন, 
আমরা জানি, 
পিথাগোরাসের উপপাদ্য অনুযায়ী,
(অতিভুজ)² = (উচ্চতা)² + (ভূমি)²
⇒ (উচ্চতা)² = (অতিভুজ)² - (ভূমি)²
 = (5x)² - (4x)² = 25x² - 16x²
= 9x²
⇒ (উচ্চতা)² = 9x²
⇒ উচ্চতা = √9x² = 3x
∴ উচ্চতা = 3x

আমরা জানি, 
ত্রিভুজের ক্ষেত্রফল = (1/2) ​× ভূমি × উচ্চতা 
= (1/2) ​× 8x × 3x
= 12x² 

সুতরাং, ত্রিভুজের ক্ষেত্রফল = 12x²