Bank Math

10 টি সংখ্যার সমষ্টি = 10 × 7 = 70
8 দ্বারা ওই দশটি সংখ্যার প্রত্যেককে গুণ করার পর সমষ্টি হবে = 70 × 8 = 560
∴ নতুন গড় = 560/10 = 56 

Question: A sum of money doubles itself in 10 years at simple interest. What is the annual interest rate?

Solution: 
Let,
Principal amount = P
Sum of amount = 2P
∴ Interest, I = 2P - P = P
Time, n = 10 years
Rate of interest = r

We know, 
I = Pnr
∴ r = I/Pn
= P/(P × 10)
= 1/10 × 100%
= 10% 

Question: Tk. 720 was divided among A, B, C, D, E. The sum received by them was in ascending order and in arithmetic progression. E received Tk. 40 more than A. How much did B receive? 

Solution:
Given that,
A + B + C + D + E = Tk. 720
And E - A = 40

We know,
Arithmetic progression,
a, a + d, a + 2d, a + 3d, a + 4d
And nth term = a + (n - 1)d

Let, A receive Tk. a and the difference between each consecutive person be Tk. d.
Amount, E = a + 4d
Amount, A = a

According to the question,
⇒ a + 4d - a = 40
⇒ 4d = 40
∴ d = 10

Also,
a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) = 720
⇒ 5a + 10d = 720
⇒ 5a + 10 × 10 = 720
⇒ 5a = 720 - 100
⇒ a = 620/5
∴ a = 124

So, Amount, B = a + d = 124 + 10 = Tk. 134

Let the speed of the first train be x km/hr.
Then,
Sum of lengths of trains = (66 + 88)m = 154 m.

Relative speed of two trains = (x - 30) km/hr
= {(x - 30) × (5/18)} m/s

∴ 154/(x - 30) × (5/18) = 0.168 × 60
⇒ 5(x - 30) = (154 × 18)/10.08
⇒ 5(x - 30) = 275
⇒ x - 30 = 55
⇒ x = 85 km/hr.

Question: A sequence of numbers a₁, a₂, a₃, …, a_n is generated by the rule a_{n + 1} = 3a_n. If a₅ - a₄ = 48, then what is the value of a₅?

সমাধান:
প্রদত্ত অনুক্রমের নিয়মটি হলো: a_n+1 = 3a_n
n = 4 বসালে পাই,
a_{4 + 1} = 3a₄
⇒ a₅ = 3a₄ 

প্রশ্নমতে,
a₅ - a₄ = 48
⇒ 3a₄ - a₄ = 48
⇒ (3 - 1)a₄ = 48
⇒ 2a₄ = 48
⇒ a₄ = 48/2
⇒ a₄ = 24

এখন,
a₅ = 3a₄
⇒ a₅ = 3 × 24
⇒ a₅ = 72

অতএব, a₅ এর মান হলো 72

Let, there be x men originally.
So, x men had provisions for 40 days whereas (x + 500) men consumed it in 35 days.
More men, Less days [Indirect proportion]
∴ (x + 500) : x :: 40 : 35
⇒ 35(x+ 500) = 40x
⇒ 5x = 35 × 500
⇒ x = (35 × 500)/5
= 3500.

Question: In a right triangle, the length of one of the legs is 9 and the length of the hypotenuse is 15. What is the length of the other leg?

Solution:
এখানে, সমকোণী ত্রিভুজের (right triangle) অতিভুজ (hypotenuse) = 15 একক
সমকোণ সংলগ্ন এক বাহু = 9 একক
সমকোণ সংলগ্ন অপর বাহু = a একক

প্রশ্নমতে,
a² + 9² = 15²
⇒ a² + 81 = 225
⇒ a² = 225 - 81
⇒ a² = 144
⇒ a = √144
⇒ a = 12

∴ সমকোণ সংলগ্ন অপর বাহুর দৈর্ঘ্য = 12 একক

Question: The least number by which 150 must be multiplied to make it a perfect square is:

Solution:
Prime factorization of 150:
150 = 2 × 3 × 5 × 5
= 2¹ × 3¹ × 5²

Here, the powers of 2 and 3 are odd.
∴ To make it a perfect square, we need to multiply by 2 × 3 = 6.

Since the numbers are co-prime, their HCF = 1

Product of first two numbers = 119
Product of last two numbers = 391

The middle number is common in both of these products. Hence,
if we take HCF of 119 and 391, we get the common middle number.

HCF of 119 and 391 = 17
⇒ Middle Number = 17
First Number = 119/17 = 7
Last Number = 391/17 = 23

Sum of the three numbers = 7 + 17 + 23 = 47.

Question: Solve for x: log₂(x + 5) = 3.

Solution:
Given,
log₂(x + 5) = 3
⇒ x + 5 = 2³ [log_ax = b ⇒ x = a^b]
⇒ x + 5 = 8
⇒ x = 8 - 5
∴ x = 3

Let, the number be a and b.
When it is reversed and added to itself we get (10a + b) + (10b + a)
= 11a + 11b
= 11(a + b)
We are given,
143 = 11(a + b)
a + b = 143/11
a + b = 13
so the digits are a and 13 - a.
We are given their products as a(13 - a) = 36, which is a quadratic expression.
a(13 - a) = 36
13a - a2 = 36
-a² + 13a - 36 = 0
a² - 13a + 36 = 0
a² - 9a - 4a + 36 = 0
a(a - 9) -4(a - 9) = 0
a - 9)(a - 4) = 0
a = 9 or a = 4
So, the number could be 49 or 94.
Hence the option is 49 then the answer will be 49.

Suppose B joined after x months
21000 × 12 = 36000 × (12 - x)
⇒ 36x = 180
⇒ x = 5

Question: In a survey, it was found that 70% of people read Ittefaq, 60% read Sangbad, and 40% read both newspapers. If a person is chosen at random, find the probability that they read either Ittefaq or Sangbad.

Solution:
ধরি, ইত্তেফাক পড়ার ঘটনা A 
এবং সংবাদ পড়ার ঘটনা B
P(A) = 70/100 = 7/10 
P(B) = 60/100 = 6/10
P(A ∩ B)= 40/100 = 4/10

নিরপেক্ষভাবে বাছাই করলে একজন লোকের ইত্তেফাক বা সংবাদ পড়ার সম্ভাব্যতা P( A ∪ B)

∴ P(A ∪ B) = P(A) + P(B) - P(A ∩ B) 
= (7/10) + (6/10) - (4/10)
= (7 + 6 - 4)/10
= 9/10

Question: A boy rides his bicycle 10 km at an average speed of 10 km/hr and again travels 12 km at an average speed of 10 km/hr. His average speed for the entire trip is-

Solution:
Given that,
Distance₁ = 10 km, Speed₁ = 10 km/hr
Distance₂ = 12 km, Speed₂ = 10 km/hr

∴ Total Distance = 10 + 12 = 22 km

And,
Time₁ = 10 ÷ 10 = 1 hr
Time₂ = 12 ÷ 10 = 1.2 hr

∴ Total Time = 1 + 1.2 = 2.2 hr

We know,
Average Speed = Total Distance ÷ Total Time
= 22/2.2 = 10 km/hr

∴ His average speed for the entire trip is 10 km/hr.

Question: If 0 ≤ x ≤ 4 and y < 6, which of the following cannot be the value of xy?

Solution:
y < 6 হলে y  এর মান 5, 4, 3, 2, 1, 0, - 1,..............
0 ≤ x ≤ 4 হলে x  এর মান 0, 1, 2, 3, 4

এখন
x = 0, y = 1 হলে xy = 0 × 1 = 0
x = 2, y = - 1 হলে xy = 2 × (- 1) = - 2
x = 3, y = 2 হলে xy = 3 × 2 = 6

সঠিক উত্তর: None of these

Question: The probability that a card drawn from a pack of 52 cards will be a diamond or a king is: 

Solution: 
Here, n(S) = 52
There are 13 cards of diamonds (including one king), and there are three more kings.

Let E = the event of getting a diamond or a king
Then, n(E) = (13 + 3) = 16

∴ P(E) = n(E)/n(S)
= 16/52
= 4/13

Question: Two trains A and B are moving in the same direction. A has speed of 8 km/h and B has speed of 13 km/h. What is relative speed of B with respect to A?

Solution: 
Given that,
Speed of train A = 8 km/h
Speed of train B = 13 km/h

Since both are moving in the same direction, and B is faster.
∴ Relative speed of B with respect to A = Speed of B - Speed of A
= 13 km/h - 8 km/h
= 5 km/h

So the relative speed of B with respect to A is 5 km/h.

(X + y)'s 1 day's work = (1/16) + (1/16)
= 2/16
= 1/8
Z's 1 day's work = (X + Y + Z)'s 1 day's work - (X + Y)'s 1 day's work
= 1/6 - 1/8
= 1/24.
∴ Z alone can finish the work in 24 days.

Question: An article has a marked price of Tk. 500. If two successive discounts of x% and 5% reduce the selling price to Tk. 427.50, determine the value of x.

Solution:
Marked Price = Tk. 500
Final Selling Price after two successive discounts = Tk. 427.50
Let the first discount = x%
Second discount = 5%

First discount be:
500 × (1 - x/100​) × (1 - 0.05) = 427.50
⇒ 500 × (1 - x/100​) × 0.95 = 427.50
⇒ 475 × (1 - x/100​) = 427.50 
⇒ (1 - x/100​) = 427.50 / 475
⇒ 1 - x/100 = 0.9
⇒ - x/100 = 0.9 - 1
⇒ - x/100 = - 0.1
⇒ x = 0.1 × 100
⇒ x = 10%

∴ First discount = 10%

Question: A, B and C can do a piece of work in 24, 36 and 72 days respectively. In how many days can A do the work if he is assisted by B and C on every third day? 

Solution:
A's 2 day's work = (1/24) × 2 = 1/12
∴ (A + B + C)'s 1 day's work = 1/24 + 1/36 + 1/72

= 1/24 + 1/36 + 1/72
= (3 +2 +1)/72
= 6/72
= 1/12

Work done in 3 days = 1/12+ 1/12
= 2/12
= 1/6

Now,
1/6 work is done in 3 days.
∴ Whole work will be done in (3 × 6) = 18 days

Let the number be x
As given 65% of x = 4/5 of x -21
So, solving above equation:
65 X x/100= 4 X x/5 - 21
⇒ 65 X x /100 = ( 4x - 105)/5
⇒ (65x) X 5 = (4x- 105) X 100
⇒ 65 x = (4x -105) X 20
⇒ 65 x = 80 x - 2100
⇒ 15x = 2100
∴ x = 140
So, the number is 140.

Question: What is the sum of the squares of the digits from 1 to 11?

Solution: 
আমরা জানি,
n সংখ্যক ক্রমিক সংখ্যার বর্গের যোগফল, Sn = [n(n + 1)(2n + 1)]/6
= [11(11 + 1)(22 + 1)]/6
= (11 × 12 × 23)/6 
= 22 × 23 
= 506

Let s=shirt and t=tie
Need to solve:
3s + 5t = 23 and 5s + 1t = 20
Look for a multiple of one (or both) formula that will match the quantity of the second. One option is to multiply the second equation by 5:
5x (5s + 1t) = 5x 20
Or, 25s + 5t = 100
Subtract the first formula :
25s + 5t - 3s - 5t = 100 - 23
Or, 22s = 77

So shirts are 77 / 22 = Tk. 3.50 each

Here, Total coders = 8 × 3 = 24
∴ Numbers of groups each having 6 coders = 24/6 = 4

Question: A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. The fraction obtained is -

Solution:
Let, the fraction is a/b

Now, (a - 1)/b = 1/3
⇒ b = 3a - 3

Again, a/(b + 8) = 1/4
⇒ b + 8 = 4a
⇒ b = 4a - 8

∴ 3a - 3 = 4a - 8
⇒ 4a - 3a = 8 - 3
⇒ a = 5

And, b = (3 × 5) - 3
= 15 - 3
= 12

∴ The fraction = a/b = 5/12 

​Question: The height of a cylinder is five times the radius of the cylinder. If the volume of the cylinder is 135π cm³, what is the height of the cylinder?

Solution:
Let
The radius of the cylinder is r cm
The height of the cylinder is 5r cm.

We know,
The volume of a cylinder = πr²h cubic units.

ATQ,
πr² × 5r = 135π
⇒ 5r³ = 135
⇒ r³ = 135/5
⇒ r³ = 27
⇒ r³= 3³

∴ r = 3

So the height ot the cylinder = 5 × 3 = 15 cm

Question: A square and a circle have the same perimeter. The length of the side of the square is 44 cm. What is the area of the circle?

Solution:
বর্গের পরিসীমা = 4 × বাহুর দৈর্ঘ্য
= 4 × 44 সেমি
= 176 সেমি

প্রশ্নমতে, বর্গ এবং বৃত্তের পরিসীমা সমান।
সুতরাং, বৃত্তের পরিধি = 176 সেমি

আমরা জানি,
বৃত্তের পরিধি = 2πr
⇒ 2πr = 176
⇒ 2 × (22/7) × r = 176
⇒ (44/7) × r = 176
⇒ r = 176 × (7/44)
∴ r = 28 সেমি

এখন, বৃত্তের ক্ষেত্রফল = πr²
= (22/7) × (28)²
= (22/7) × 784
= 22 × 112
= 2464 বর্গ সেমি

Question: What is the angle between the hour and minute hands of a clock when it is 3 : 15 pm?

Solution:
3টা 15 মিনিট = 3 + (15/60) ঘন্টা = 3 + 1/4 = 13/4 ঘন্টা

আমরা জানি, ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 13/4 ঘণ্টায় ঘোরে = (30° × 13)/4
= 390°/4
= 97.5°

আবার, মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 15 মিনিটে ঘোরে = 15 × 6° = 90°

∴ ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = |97.5° - 90°|
= 7.5°

Question: What is the angle between the hour and minute hands of a clock when it is 3 : 15 pm?

Solution:
3টা 15 মিনিট = 3 + (15/60) ঘন্টা = 3 + 1/4 = 13/4 ঘন্টা

আমরা জানি, ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 13/4 ঘণ্টায় ঘোরে = (30° × 13)/4
= 390°/4
= 97.5°

আবার, মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 15 মিনিটে ঘোরে = 15 × 6° = 90°

∴ ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = |97.5° - 90°|
= 7.5°

Let,
He purchases 6 liters milk.
So, cost of 6 liters = 6x tk.
After mixing 2 liters waters, he sells , (6 + 2) = 8 liters
Now, selling price of 8 liters = 8 × 2x = 16x tk
Profit = 16x – 6x = 10x tk.
∴ Profit percentage = (10x/6x) × 100 = 166.66%

The ratio of original money numbers = 3:8
Common factor helps in finding actual values easily
So, take 'M' as a common factor.
∴ Original numbers will be 3M and 8M
Adding 5 to them, we get (3M + 5) and (8M+5)
∴ (3M + 5)/(8M + 5) = 2/5 ............ (Ratio of new numbers is 2:5)
∴ 15M + 25 = 16M + 10
∴ M = 15
Smaller money value is 3M = 3 x 15 = 45.

Let the cost price x.
Therefore, 90% of 15000 = 108% of x .
Or, x = 90 × 15000108
Or, x = Tk. 12500

Question: Find the equation of the line with x-intercept = 3 and y-intercept = 4.

Solution:
Given
x-intercept = 3 (line passes through point (3, 0)
y-intercept = 4 (line passes through point (0, 4)

We know,
The intercept form of a line is:
x/a + y/b =1 ,where a = x-intercept and b = y-intercept.
⇒ x/3 + y/4 = 1
⇒ (4x + 3y)/12 =1
⇒ 4x + 3y = 12 
⇒ 4x + 3y - 12 = 0

∴ The equation of the line is 4x + 3y - 12 = 0

Question: A square park is surrounded by a path of uniform width 3 meters. If the area of the path is 75 square meters, find the side length of the park.

Solution:
Let the side of the park = x meters.
Then, the side of the park including the path = x + (2 × 3)
= x + 6 meters.

Area of the path = Area of park with path - Area of park
⇒ 75 = (x + 6)² - x²
⇒ 75 = x² + 12x + 36 - x²
⇒ 75 = 12x + 36
⇒ 12x = 75 - 36 = 39
⇒ x = 39/12 = 3.25 meters

Therefore, the side length of the park is 3.25 meters.

Let length of each train = x metre
Total distance covered while passing the slower train = (x + x) = 2x metre
Relative speed = (46 − 36)
= 10 km/hr
= 10 × 5/18
= 50/18 m/s

Time = 36 seconds
⇒ 2 x/36 = 50/18
⇒ x = 50

Question: If x² - √(7). x + 1 = 0 then, x² + 1/x² = ?

Solution:
দেওয়া আছে,
x² - √7x + 1 = 0
⇒ x² + 1 = √7x  
⇒ x²/x + 1/x = √7x/x [উভয় পক্ষকে x দ্বারা ভাগ]
⇒ x + 1/x = √7

এখন,
x² + 1/x²
= (x + 1/x)² - 2 . x . 1/x
= (√7)² - 2
= 7 - 2
= 5

∴ x² + 1/x² এর মান 5।

Question: What is the value of tan240°

Solution: 
tan240°
= tan(180° + 60°)
= tan(180° + θ)
= tanθ
= tan60°
= √3


Note:
240° lies in the 3rd quadrant.
In the 3rd quadrant, tan θ is positive (because both sin θ and cos θ are negative).

Trepidation (noun)
- English Meaning: An uncomfortable feeling of nervousness or worry about something that is happening or might happen in the future.
-  Bangla Meaning: সচকিত উত্তেজিত মনোভাব।  

Synonyms:
• Anxiety - ভবিষ্যৎ বিষয়ে ভয় ও অনিশ্চয়তাবোধ; উদ্বেগ; দুশ্চিন্তা।

• Apprehension - আশঙ্কা; ভবিষ্যৎ বিষয়ে উৎকণ্ঠার অনুভূতি; উপলব্ধি; চেতনা; বোধ।

• Disquietude - মানসিক অস্থিরতা বা উদ্বেগ।

Antonyms:
• Calmness - শান্ততা, বিশ্রান্ততা।

• Equanimity - মনমেজাজের প্রশান্তি।

• Composure - শান্তি; স্থৈর্য; আত্মসংবরণ।

Other options:
খ) Very comfortable situation.
- Translations: খুবই আরামদায়ক অবস্থা। 

গ) Find a solution.
- Translations:  সমাধান বের করা। 

ঘ) Always being confident.
- Translations: সবসময় আত্মবিশ্বাসী।

Source: Live MCQ Lecture.

Since the number is greater than the number obtained on reversing the digits, so the ten's digit is greater than the unit's digit.
Lets, ten's and unit's be 2x and x respectively
Then, (10 times; 2x + x) - (10x + 2x) = 36
⇒ 9x = 36
⇒ x = 4

∴ Required difference (2x + x) - (2x - x ) = 2x = 4 × 2 = 8
Answer : 8

Let, n =  3

n - 1 = 2
n + 1 = 4
4n + 1 =  13
3n + 1 =  10