Bank Math

Man's speed with the current = 15 km/hr
=> speed of the man + speed of the current = 15 km/hr
Speed of the current is 2.5 km/hr
Hence, speed of the man
= 15-2.5
= 12.5 km/hr
Man's speed against the current = speed of the man - speed of the current
= 12.5-2.5
= 10 km/hr

14 জনের দল থেকে 1 জনকে ঠিক রেখে বাকি 13 জন থেকে (11 - 1) = 10 জনের টিম ¹³c₁₀ রুপে গঠন করা যাবে
= (13 × 12 × 11 × 10!)/(13 - 10)! × 10!
= (13 × 12 × 11)/3!
= (13 × 12 × 11)/(3 × 2 × 1)
= 13 × 2 × 11)
= 286
Answer: 286 উপায়ে টিম গঠন করা যাবে ।

Given that, principal = P = Tk. 33600 and
R = 25/4 %
Time duration = From September 2019 to May 2020
= 9 months
= 9/12 year
= 3/4 year
S.I = (Principle × Interest × Time)/100
= PRT/100
= Tk. (33600 × 25/4 × 3/4) × 1/100
= 21 × 25 × 3
= Tk. 1575.
Hence, the answer is Tk. 1575.

Question: Five identical cubes, each with a side of 5 cm, are placed next to each other. What is the volume of the resulting solid?

Solution:
The new formed is a cuboid of length = 5 × 5 = 25 cm
breadth = 5 cm and height = 5 cm

∴ Volume = (25 × 5 × 5) cm3
= 625 cm3

Question: In a class of 98 students, 41 are taking Bengali, 22 are taking English and 9 are taking both courses. How many students are not enrolled in either course?
(Officer Cash 2022 অনুযায়ী)

Solution:
Total students = 98
Students taking Bengali n(B) = 41
Students taking English n(E) = 22
Students taking both Bengali and English = 9

We know,
n(B ∪ E) = n(B) + n(E) - n(B ∩ E)
n(B ∪ E) = 41 + 22 - 9 = 54

∴ Not enrolled = Total students - n(B ∪ E) = 98 - 54 = 44

Question: Find the smallest number that is a multiple of 11 and leaves a remainder of 5 when divided by 8, 12, 16, and 24.

Solution:
L.C.M. of 8, 12, 16 and 24 is 48.
Let required number be 48k + 5, which is multiple of 11.

Least value of k for which (48k + 5) is divisible by 11 is k = 7.

Required number = (48 × 7) + 5 = 336 + 5 = 341.

∴ নির্ণেয় ক্ষুদ্রতম সংখ্যাটি হলো 341

Question: A cube has a total surface area of 1,350 square meters. What is the volume of the cube?

Solution:
ধরি, ঘনকের বাহুর দৈর্ঘ্য = a মিটার।

আমরা জানি, ঘনকের সম্পূর্ণ পৃষ্ঠের ক্ষেত্রফল = 6a²
প্রশ্নমতে,
6a² = 1350
⇒ a² = 1350/6
⇒ a² = 225
⇒ a = √225
∴ a = 15 মিটার

এখন, ঘনকের আয়তন = a³
= 15³
= 3375 ঘন মিটার

অতএব, ঘনকটির আয়তন = 3375 ঘন মিটার

Question: If 2^x = 3^y = 6^-z, find the value of (1/x) + (1/y) + (1/z).

Solution: 
Let, 
2^x = 3^y = 6^-z = k
Now, 
2^x = k 
2 = k^1/x ......(1)
Similarly, 
3 = k^1/y ....(2)
And
6 = k^-1/z
⇒ 2 × 3 = k^-1/z
⇒ k^1/x × k^1/y = k^-1/z  ; [From (1) and (2)]
⇒ k^{(1/x + 1/y)} = k^-1/z
⇒ (1/x + 1/y) = - 1/z
∴ 1/x + 1/y + 1/z = 0

Question: Mr. Shamim wants to arrange three out of his four saplings in a row on a shelf. If each sapling is in a pot of a different color, in how many different ways can he arrange the saplings?

Solution:
এখানে ভিন্ন ভিন্ন রঙ্গের পাত্রে থাকে, তার মানে নির্দিষ্ট রং ধারণ করে, তাই বিন্যাস হবে।
4 টি চারাগাছ হতে 3 টি নিয়ে সাজানো যায় = ⁴P₃ উপায়ে
= 4!/(4 - 3)! 
= 4! 
= 4 × 3 × 2 × 1 
= 24 উপায়ে

L.C.M. of 21, 36, 66 = 2772.
Now, 2772 = 2 x 2 x 3 x 3 x 7 x 11
To make it a perfect square, it must be multiplied by 7 x 11.
So, required number = 2² x 3² x 7² x 11² = 213444

Let current ages of X and Y correspondingly, is 6A & 5A
Given: 6A + 5A = 44
=> A = 4
Proportion of ages after 0.8 decades will be
6A + 8 : 5A + 8
32:28 (or) 8:7

Question:  The perimeter of a rhombus is 56 m and its height is 5 m. Its area is- 

Solution: 
one side of rhombus is 56/4 = 14 m

area = 14 × 5 
= 70 m²

Suppose 100 be the principal.
In 4 years it doubles and becomes 200.
In next 4 years 200 double to 400.
In next 4 years 400 doubles to 800.
Hence the period for 8 times is 12 years.

Let,
Length of smaller field = x
And, Width of smaller field = y
So, Length of larger field = 2x
and, Width of larger filed = 4y
Area of smaller field = xy = K
Area of larger field = 2x × 4y = 8xy = 8K

∴ Difference of the Larger and Smaller field = 8K – K = 7K

Question: In a mixture of milk and water, the ratio is 5 : 3. If 4 liters of water is added, the new ratio becomes 5 : 4. What was the original amount of milk in the mixture?

Solution:
ধরি, শুরুতে দুধ ছিল = 5x লিটার,
পানি ছিল = 3x লিটার।

এখন ৪ লিটার পানি যোগ করলে,
নতুন পানি = 3x + 4 লিটার

ATQ,
5x/(3x + 4) = 5/4
⇒ 4 × 5x = 5 × (3x + 4)
⇒ 20x = 15x + 20
⇒ 5x = 20
⇒ x = 4

∴ দুধের পরিমাণ = 5x = 5 × 4 = 20 লিটার

12×6×24×x = 18×8×36×460
x =(18×8×36×460)/(12×6×24)
x =1380
∴ Required number of chairs = 1380

Apply formula of
M1D1H1/W1= M2D2H2/W2
Let 'P' pumps are required to empty the reservoir.
(12pumps×6hours×15days)/1 reservoir
= (P×9hours×12days)/1 reservoir
P = 10 pumps

Question: In what ratio must sugar at Tk. 12 per kg be mixed with sugar at Tk. 18 per kg so that the mixture be worth Tk. 15 per kg?

Solution:
Let x kg of sugar at Tk. 12 and y kg of sugar at Tk. 18 be mixed.

ATQ,
⇒ 12x + 18y = 15(x + y)
⇒ 12x + 18y = 15x + 15y
⇒ 15x - 12x = 18y - 15y
⇒ 3x = 3y
∴ x : y = 1 : 1

∴ required ratio 1 : 1.

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:  

For a rectangle of length 8 feet and width 6 feet
∴ Diagonal = √(8² + 6²)
= √(64 + 36) 
= √100
= 10



Here, the diagonal is the diameter of the smallest covering circle.
∴ Radius = 10/2
= 5 feet

 ∴ Area of the circle = πr²
= π × (5)²
= 25π sq. feet

Question: If 3/4 of a number is 7 more than 1/6 of the number then 5/3 of the number is-

Solution: 
Let the number be x
According to the question,
⇒ (3x/4) - (x/6) = 7
⇒ (9x - 2x)/12 = 7
⇒ 7x = 7 × 12
∴  x = 12

Then 5/3 of the number will be
= x × (5/3)
= (12 × 5)/3
= 20

A:B = 1000:900
B:C = 400:360 = 100:90 = 900:810
⇒ A:B:C = 1000:900:810
⇒ A:C = 1000:810
⇒ A:C = 500:405
⇒ In a 500 m race, A beats C by (500-405) m = 95 m

Question: A lemonade stand sold only small and large cups of lemonade on Tuesday. 3/5 of the cups sold were small and the rest were large. If the large cups were sold for 7/6 as much as the small cups, what fraction of Tuesday's total revenue was from the sale of large cups?

Solution: 
lets, the total cups sold 15 
small cups = (3/5) × 15 = 9 
large cups = 15 - 9 = 6 

let, small cups were sold 6 taka each, then large cups were sold 7 taka each.

large cup's revenue = 7 × 6 = 42 taka 
small cup's revenue = 6 × 9 = 54 taka 

 fraction of Tuesday's total revenue was from the sale of large cups = 42/(42 + 54)
= 42/96 
= 7/16

Question: A train, 150 m long, passes a pole in 15 seconds and another train of the same length, travelling in the opposite direction in 10 seconds. What is the speed of the second train? 

Solution: 
Given,
Length of the first train &  second train  = 150 m
Time to pass a pole = 15 seconds
Time taken by trains to cross each other = 10 sec

Speed of the first train = 150/15
 = 10 m/s

And, the relative speed of two trains = (150 + 150)/10 
= 30 m/s

Speed of the second train = (30 - 10) × (18/5)
= 20 × (18/5)
= 72 km/h 

প্রশ্ন: যদি P = 5 + √2  হয়, তবে P² এর মান কত?

সমাধান:
P = 5 + √2
∴ P² = (5 + √2)² 
⇒ P² = 5² + 2 . 5 . √2 + (√2)²
⇒ P² = 25 + 10√2 + 2
∴ P² = 27 + 10√2

Question: In a map, 2 cm represents 85 km. The distance between two cities is 9.4 cm on the map. The actual distance between the cities is - 

Solution: 
Since 2 cm = 85 km,
Actual Distance = (85/2) × 9.4 
= 799/2
= 399.5 km 

Question: If tan3A = √3, then A = ?
​
​Solution:
tan3A = √3
⇒ tan3A = tan60°
⇒ 3A = 60°
⇒ A = 60°/3
∴ A = 20°

যেহতু 1.5 সমীকরণটির একটি মূল সেহেতু (1.5)² + 1.5m + 24 = 0
⇒ 2.25 + 1.5m + 24 =0
⇒ m = - 26.25/1.5
⇒ m = - 17.5

Length of the tower AB = h meter.
∠DAC = ∠ACB = 60°
BC = 70 metre

In ABC,
tan 60° = AB/BC
⇒ √3 = h/70
⇒ h = 70√3 meter.

Question: If sin 45° = √2A, then A =?

Solution: 
sin 45° = √2A
1/√2 =√2A
A = 1/(√2)²
A = 1/2

6 = 2 × 3
10 = 3 × 5
35 = 5 × 7
27 = 3 × 9; Here 9 is not a prime number

মনে করি,
প্রত্যেক ধরনের চকলেট কেনার হলো 60 টি করে।
তাহলে,
Total cost = 60 × (1/4) + 60 × (1/6)
= 25।
আবার, selling price = 60 × 2 × (1/5)
= 24।
কাজেই percentage loss = (25 - 24)/25 × 100
= 4%

Question: A sum of TK. 600 amounts to TK. 720 in 4 years at simple interest. What will it amount to if the rate of interest is increased by 2%?

Solution: 
Given that,
Principal, P = Tk. 600 
Amount = Tk. 720
Time, n = 4 years
∴ Simple Interest, SI = Amount - Principal = 720 - 600 = Tk. 120

We know, 
SI = (P × r × n)/100
⇒ 120 = (600 × r × 4)/100
⇒ 120 = (2400 × r)/100
⇒ 120 = 24r
⇒ r = 120/24
∴ r = 5%

Again, 
New rate = original rate + 2% = 5% + 2% = 7%

∴ New SI = (P × r × n)/100
= (600 × 7 × 4)/100
= 16800/100
= Tk. 168

∴ New Amount = Principal + New Interest
= 600 + 168
= Tk. 768

∴ If the rate of interest is increased by 2%, the sum will amount to Tk. 768 in 4 years.

Question: Rifat sold an article for Tk. 528 alter allowing a discount of 12% on its marked price. What was the marked price of the article?

Solution: 
Marked price of a article be  x
According to the question 
x88/100 = 528
88x = 528 × 100
x = (528 × 100)/88
x= 600

Question: The average price of the first five pens out of six is Tk. 12 and the average price of the last five is Tk. 16. If the price of the first pen is Tk. 50, what is the price of the last pen?

Solution:
Given that,
P₁ = 50 taka
Average of first 5 = Tk. 12
Average of last 5 = Tk. 16

Let the prices of the 6 pens are- P₁, P₂, P₃, P₄, P₅, P₆

now,
 50 taka
Average of first 5 = 12 taka
Average of last 5 = 16 taka

Now,
Sum of first 5 pens is
P₁ + P₂ + P₃ + P₄ + P₅ = 5 × 12 = 60 .......(1)
And,
Sum of last 5 pens is
P₂ + P₃ + P₄ + P₅ + P₆ = 5 × 16 = 80 ........(2)

Now, subtract equation (1) from (2),
(P₂ + P₃ + P₄ + P₅ + P₆) - (P₁ + P₂ + P₃ + P₄ + P₅ ) = 80 - 50
⇒ P₆ - P₁ = 20
⇒ P₆ - 50 = 20
⇒ P₆ = 20 + 50
∴ P₆ = 70

So the price of the last pen is Tk. 70