Bank Math

As we know that volume of wire is volume of cylinder.
And in given question silver wire is drawn from 66 cubic centimetres of silver.
So we can equivalent the volume of cylinder with 66 cubic centimetres.
we know that volume of cylinder is Π×r²×l. [where l = length and r = radius]
Given, r = 1/2mm.
i.e. r = 0.5mm.
i.e. r = 0.05cm.
According to question Π×r²×l = 66
i.e. 22/7 × (0.05)²× l = 66
by solving this question we get l = 8400cm.
So, length of wire in metres is 84m.

Question: The measurement of a rectangle is 8 feet by 6 feet. What is the area of the smallest circle that can cover this rectangle entirely (so that no part of the rectangle is outside the circle)?

Solution: 
Here,
Diameter of the circle = √(8² + 6²)
= √(64 +36)
= √100
= 10 feet
∴ Radius of the circle = 10/2 = 5 feet

∴ Area of the circle = π × 5² = π × 25 = 25π sq. feet

Question: If 5374Q is divisible by 9, what is the value of Q?

Solution:
একটি সংখ্যা 9 দ্বারা বিভাজ্য হবে যদি সংখ্যাটির অঙ্কগুলোর সমষ্টি 9 দ্বারা বিভাজ্য হয়।

5 + 3 + 7 + 4 = 19; এর সাথে Q যোগ করলে (19 + Q) হবে, যা 9 দ্বারা বিভাজ্য হতে হবে।

19 + Q = 27 (যা 9 দ্বারা বিভাজ্য নিকটতম সংখ্যা)
⇒ Q = 27 - 19
⇒ Q = 8
∴ Q = 8

Question: The ratio of two numbers is 5:7 and their H.C.F is 6. What is their L.C.M?

Solution:
Let the numbers be 5x and 7x.
∴ HCF = x = 6

We know,
HCF × LCM = Product of two numbers
⇒ 6 × LCM = (5x) × (7x)
⇒ 6 × LCM = (5 × 6) × (7 × 6)
⇒ LCM = (30 × 42)/6
⇒ LCM = 1260/6
∴ LCM = 210

Question: A rectangular block measuring 8 cm by 12 cm by 16 cm is cut into an exact number of cubes of equal size. What will be the least possible number of cubes?

Solution:
Given that,
Sides of the rectangular block = 8 cm, 12 cm, 16 cm

Let n be the number of cubes.
HCF( 8, 12, 16) = 4 cm
And side of the square = 4 cm

We know, 
Volume of the cuboid = n × Volume of the cube
⇒ Length × breadth × height = n × side³
⇒ 8 × 12 × 16 = n ×  4 ×  4 ×  4
⇒ 64n = 8 × 12 × 16
⇒ n = (8 × 12 × 16)/64
∴ n = 24

So the number of cubes is 24.

Question: A group of 8 men can paint a house in 20 days. If they want to complete the painting in 16 days, how many more men should they add to the group?

Solution:
২০ দিনে কাজ শেষ করতে পারে ৮ জন
১ দিনে কাজ শেষ করতে পারে (২০ × ৮) জন
১৬ দিনে কাজ শেষ করতে পারে (২০ × ৮)/১৬ জন
= ১০ জন

∴ অতিরিক্ত (১০ - ৮) বা ২ জন যোগ করতে হবে।

Question: A shopkeeper marks his items 40% above the cost price and then gives a 20% discount on the marked price. What is his gain percentage?

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

Marked Price = 40% more than cost price
= 100 + 40
= 140 Tk.

Discount = 20% of 140
= (20/100) ​× 140
= 28 Tk.

Selling Price (SP)= 140 - 28 = 112 Tk.

∴ Profit = SP - CP = 112 - 100 = 12 Tk.

∴ Gain percent = 12%

Question: Introducing a lady, Jamal said, "Her mother is the only daughter of my mother-in-law." What is Jamal to the lady?

Solution:
জামালের শাশুড়ির একমাত্র কন্যা হলো জামালের নিজের স্ত্রী। আর ঐ মহিলার মা যদি জামালের স্ত্রী হন, তবে জামাল ঐ মহিলার বাবা।

∴ জামাল ঐ মহিলার বাবা (Father)।

Let,
the daily wage of a man, a woman and a boy be Tk. 4x, Tk. 3x and Tk. 2x respectively.
Then, 15 × 4x + 18 × 3x + 12 × 2x = 2070
⇒ 60x + 54x + 24x = 2070
⇒ 138x = 2070
⇒ x = 15
∴ daily wages of 1 man, 2 women and 3 boys
= Tk. (4x + 2 × 3x + 3 × 2x)
= Tk. (4x + 6x + 6x)
= Tk. 16x
= Tk. (16 × 15)
= Tk. 240.

Question: The speed of a boat in still water is 9 km/h. The time it takes to travel downstream is half the time it takes to travel upstream. What is the speed of the stream?

Solution:
Let the speed of the current be = x km/h

Then,
Downstream speed = (9 + x) km/h
Upstream speed = (9 - x) km/h

According to the question:
(9 + x) = 2(9 - x)
⇒ 9 + x = 18 - 2x
⇒ 2x + x = 18 - 9
⇒ 3x = 9
⇒ x = 9/3
⇒ x = 3

∴ Speed of the stream = 3 km/h

Let A = x
B = x + 2
C = x + 4
D = x + 6 and
E = x + 8
Then,
⇒A+C=146
⇒x+(x+4)=146
⇒2x=142
⇒x=71
∴ E=x+8=71+8=79

Question: A man can row 12 km/h in still water. He finds that it takes him twice as long to row upstream as to row downstream. What is the speed of the stream?

Solution: 
Speed in still water = 12 km/h
Speed of the stream = x km/h

Let the distance travelled is = D km
Time taken upstream = 2 × (Time taken downstream)
D/(12-x) = 2 × {D/(12+x)}
⇒ 1/(12-x) = 2/(12+x)
⇒ 12 + x = 24 - 2x
⇒ 3x = 12
∴ x = 4

So, Speed of the stream = 4 km/h

Speed of boat in still water = 25 km/hr
Speed upstream
= 10/1
= 10 km/hr
Speed of the stream = (25-10) = 15 km/hr
Speed downstream = (25+15) = 40 km/hr
Time taken to travel 10 km downstream
= 10/40 hours
= (10×60)/40
= 15 minutes

The numbers 102 and 210 are themselves multiples of 3.
also a multiple of 3 exists once in every three consecutive integers.
Counting the multiples of 3 starting with 1 for 102,
2 {= 1 + (105 - 102)/3 = 1+ 1 = 2 } for 105,
3 { = 1 +(108 - 102)/3 = 1 + 2 = 3} for 108,
and so on, the count we get for 210 equals 1+(210 - 102)/3 = 1 + 36 = 37,
hence the answer is 37

Question: Of the following which is the closet to (6.01 × 501) ÷ (25.05 × 19.97)?

Solution:
(6.01 × 501) ÷ (25.05 × 19.97)
= (601 × 501)/100 ÷ (2505 × 1997)/10000
= (601 × 501)/100 × 10000/(2505 × 1997)
= 3011.01 × 0.00199906
= 6.0192
∴ নিকটতম মান 6

বিকল্প সমাধান:
(6.01 × 501) ÷ (25.05 × 19.97)
= (6 × 500) ÷ (25 × 20) [ যেহেতু নিকটতম মান চাওয়া হয়েছে তাই সংখ্যাগুলোকে নিকটতম পূর্ণ সংখ্যায় রূপান্তর করা হয়েছে]
= 3000 ÷ 500
= 6

Question: A train 240 meters long passes a pole in 16 seconds. How long will it take to pass a platform that is 360 meters long?

Solution:
Train's speed = Distance/Time = 240/16 = 15 m/s

Total distance to pass the platform,
= Length of train + Length of platform
= 240 m + 360 m
= 600 m

∴ Required time = Distance/Speed
= 600/15
= 40 seconds

∴ The train will take 40 seconds to pass the platform.

Suppose B invested Tk. x for y months.
Then, A invested Tk. 3x for 2y months.

So,
A : B = (3x × 2y) : (x × y)
= 6xy : xy
= 6 : 1.
B's profit : Total profit = 1 : 7.

Let the total profit be Tk. x
Then, 1/7 = 4000/x
⇒ x = Tk. 28000.

Question: A takes 2 hours 30 minutes more than B to walk 40 km. If A doubles his speed, then he can make it in 1 hour less than B. What is the average time taken by A and B to walk a 40 km distance?

Solution: 
(40/A) - (40/B) = 5/2 . . . . . . . (i)
(40/B) - (40/2A) = 1 . . . . . . . (ii)

Equation (i) + Equation (ii)
(40/A) - (40/2A) = (5/2) - 1
⇒ (40/A) - (40/2A) = 7/2
⇒ 40/2A = 7/2
⇒ A = 40/7

Putting the value of A in equation (i)
7 - (40/B) = 5/2
⇒ 7 - (5/2) = 40/B
⇒ 9/2 = 40/B
⇒ B = 80/9

Now, 
Total time = [40/(40/7)] + [40/(80/9)]
= 7 + (9/2)
= 23/2

Required average time = (23/2)/2
= 23/4
= 5 hours 45 minutes. 

Let the missing number be x
Given, 48.2 × 2.5 × 2.2 + x = 270
⇒ x = 270 - 48.2 × 2.5 × 2.2
⇒ x = 270 - 265.1
⇒ x = 4.9
Hence, the number is 4.9

Question: Find an equation of the vertical line containing the point (8, - 4)

Solution:

Given,
The given point is (8, -4).

A key property of a vertical line is that the x-coordinate of every point on the line always remains the same.
Since the line passes through the point (8, -4), the value of x for every point on the line will be 8.

Therefore, the required equation is:
x = 8

Question: State the order of the matrix-

Solution:
ম্যাট্রিক্সের মাত্রা বা ক্রম(Order of Matrix): একটি ম্যাট্রিক্সের সারি ও কলামের সংখ্যা যথাক্রমে m ও n হলে, ঐ ম্যাট্রিক্সকে m × n ক্রমের বা আকারের ম্যাট্রিক্স বলা হয়।
অর্থাৎ ম্যাট্রিক্সের আকার বা মাত্রা বোঝাতে প্রথমে সারি এবং পরে কলাম উল্লেখ করা হয়।
প্রদত্ত ম্যাট্রিক্সটি একটি আয়তাকার ম্যাট্রিক্স কারণ এর সারি ও কলাম অসমান।
এখানে,
সারি m = 3 এবং কলাম n = 2

∴ প্রদত্ত ম্যাট্রিক্সটি একটি 3 × 2 আকারের ম্যাট্রিক্স।

Question: A 60-meter cable is attached from the top of a vertical pole down to the ground. If the cable makes an angle of 30 degrees with the ground, find the height of the pole.

 Solution:
 
Let,
Height, AB = h 

Given that,
AC = 60m
∠ACB = 30°

∴ sin30°= AB/AC
⇒ 1/2 = h/60 
⇒ h = 60 × 1/2 
∴ h = 30 m

The average age of a group of 15 employees is 24 years.
Therefore, the sum of the ages of all 15 of them is, 15 × 24 = 360
When 5 more employees join the group, the average age increases by 2 years.
So, New average = 26.
Now, there are 20 employees.
The sum of the ages of everyone = 20×26 = 520
Therefore, the sum of the ages of the 5 new employees = 520 - 360 = 160
∴ The average age of the 5 new employees = 160/5 = 32 years.

Question: In how many different ways can the letters of the word 'ENGINE' be arranged?

Solution:
'ENGINE' শব্দটিতে মোট বর্ণ আছে 6টি।

এখানে,
E আছে ২টি
N আছে ২টি

আমরা জানি, পুনরাবৃত্ত বর্ণ থাকলে বিন্যাস সংখ্যা = n!/(p! × q!)
∴ বিন্যাস সংখ্যা = 6!/(2! × 2!)
= (6 × 5 × 4 × 3 × 2 × 1)/{(2 × 1) × (2 × 1)}
= 720/(2 × 2)
= 720/4
= 180