Bank Math

Question: 


Solution: 

Question:

Solution:

=> 3(2x+9) = 75
=> 2x + 9 = 25
=> x = 8
Answer : 8

Question: A circle has a circumference of 24π. A square is inscribed inside the circle. What is the perimeter of the square?

Solution: 
Given that, 
The circumference of the circle is 24π

We know, 
Circumference, 2πr = 24π
⇒ 2r = 24
∴ diameter = 24
and radius = 24/2 = 12
Since the square is inscribed in the circle, the diagonal of the square equals the diameter of the circle.
∴ Diagonal of the square = 24

Let the side length of the square be s.
For a square, diagonal = s√2
So,
⇒ s√2 = 24
⇒ s = 24/√2
⇒ s = 24√2/2
∴ s = 12√2
∴ Perimeter of the square = 4 × side = 4 × 12√2 = 48√2

So the perimeter of the square is 48√2.

Question: In the triangle ABC, AB = 12 cm and AC = 8 cm, and ∠BAC = 60°. What is the value of the length of the side BC? 

Solution:
Given that,
In the triangle, ABC, AB = 12 cm and AC = 8 cm, and ∠BAC = 60°.


We know,
According to the law of cosine, if a, b, and c are three sides of a triangle ΔABC and ∠A is the angle between AC and AB then, a² = b² + c² - 2bc × cos∠A

According to the concept,
BC² = AB² + AC² - 2 × AB × AC × cos60°
⇒ BC² = 12² + 8² - 2 × 12 × 8 × (1/2)
⇒ BC² = 144 + 64 - 96 = 112
⇒ BC = √112 = √(16 × 7)
⇒ BC = 4√7

∴ The measure of BC is 4√7 cm.

Number of ways of choosing 2 black pens from 5 black pens in ⁵C₂ ways.
Number of ways of choosing 2 white pens from 3 white pens in ³C₂ ways.
Number of ways of choosing 2 red pens from 4 red pens in ⁴C₂ ways.
By the Counting Principle, 2 black pens, 2 white pens, and 2 red pens can be chosen in 10 x 3 x 6 = 180 ways

Cost price = Tk 3000
Selling price = [{3600 × 100}/{100 + (10 × 2)}]
= Tk. 3000
Gain = 0%

Question: Six friends Rita, Anika, Zara, Lima, Arif, and Sayeed sit randomly in a row of six chairs. What is the probability that Rita and Anika do not sit next to each other?

Solution:
Total number of possibilities = 6! = 720

Number of possibilities where Rita and Anika sit together = 5! × 2! 
​= 120 × 2
= 240

So the possibilities where Rita and Anika do not sit together = 720 - 240
= 480

∴Probability that Rita and Anika do not sit next to each other = 480/720
= 2/3

Let x km/hr be the speed of the train.

Time required to cover 360 km = 360/x hr.

As per the question given,

⇒ (x + 5)((360/x)- 1) = 360
⇒ (x + 5)(360 – x) = 360x
⇒ 360x – x² + 1800 - 5x = 360x
⇒ x² + 5x – 1800 = 0
⇒ x(x + 45) -40(x + 45) = 0
⇒ (x + 45)(x – 40) = 0
⇒ x = 40, -45

Negative value is not considered for speed, hence the answer is 40km/hr.

Question: If x = 12, which of the following has the least value? 

Solution:
ক. x - 2 = 12 - 2 = 10

খ. x/2 = 12/2 = 6

গ. 2/x  = 2/12 = 1/6 ≈ 0.17

ঘ. 2 - x  = 2 - 12 = - 10

The least value among the calculated options is - 10 or 2 - x

Question: A bag contains 6 red, 4 green, and 5 blue balls. Two balls are drawn without replacement. What is the probability that the first ball is red and the second ball is green or blue?
​
​Solution:
​Given that,
​Red balls = 6
​Green balls = 4
​Blue balls = 5
​Total balls = 6 + 4 + 5 = 15

​We know 
​Probability = Favorable outcomes/Total outcomes

​Now,
​First ball is Red = 6/15 = 2/5
​If first is red, then remaining balls = 14
​Second ball is Green or Blue = (4 + 5)/14 = 9/14
​
​∴ Required probability = (2/5) × (9/14) 
​= 9/35
​
​∴ The required probability is 9/35.

Question: If 45% of (A - B) is equal to 3/10 of (A + B), then what percent of A is B?

Solution:
Given that,
45% of (A - B) = (3/10) of (A + B)
⇒ (45/100) of (A - B) = (3/10) of (A + B)
⇒ (9/20) of (A - B) = (3/10) of (A + B)
⇒ (9/2) of (A - B) = 3 of (A + B)
⇒ 9(A - B) = 6(A + B)
⇒ 9A - 9B = 6A + 6B
⇒ 3A = 15B
⇒ B/A = 3/15 = 1/5
⇒ B = (1/5) × A
∴ B = (1/5) × 100% of A = 20% of A

So B is 20% of A

x - 2y = 4
এখানে অপশন থেকে দেখলে উত্তর বের করা সহজ হবে
অপশন a, b, d থেকে মান বসালে সমাধান 4 হবে না
অপশন c থেকে x = 4 এবং Y = 0 বসালে সমীকরণের সমাধান হবে 4

Let third proportional be x
⇒ 25 : 30 : : 30 : x
⇒ 25 × x = 30 x 30
⇒ x = (30 x 30)/25
= 36.

Question: If 7543P is divisible by 9, what is the value of P?

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

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

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

Downstream speed = 55/(5/2) = 11 × 2
= 22 km/hours

Time taken in upstream = 2.2 × 5/2
= 5.5 hours

Upstream speed = 55/5.5
= 10 km/hour

∴ The speed of boat in still water

= (10 + 22)/2
= 32/2
= 16 km/hr.

যেহেতু ট্রাক দুটি বিপরীত দিকে চলছে, সেহেতু এদের গতিবেগ যোগ করলে আপেক্ষিক বেগ পাওয়া যাবে। 

∴ নির্ণেয় সময় = 300 / (70+50) ঘণ্টা
= 300/120 ঘণ্টা
= 2.5 ঘণ্টা

(2a - b)/(2a + b) + 2/9
= ({2a - b)/a}/{(2a + b)/a} + 2/9
= {2 - (b/a)}/{2 + (b/a)} + 2/9
= (2 - 0.25)/(2 + 0.25) + 2/9
= 1.75/2.25 + 2/9
= 7/9 + 2/9
= 9/9
= 1.

Question: A man borrowed some money for six months. He paid Tk. 500 at an interest rate of 10% per annum. What was the amount he borrowed?

Solution:
এখানে,
সরল সুদ (SI) = Tk. 500
সুদের হার, r = 10%
সময়, n = 6 মাস = 6/12 = 1/2 বছর
আসল (Principal), P = ?

আমরা জানি,
I = Pnr/100
⇒ 500 = (P × 1/2 ×10)/100
⇒ 500 = 5P/100
⇒ 500 = P/20
⇒ P = 20 × 500
∴ P = 10000

অতএব, তিনি মোট Tk. 10,000 ধার নেন।

ধরি, Nabila's income = T
এবং Nuru's income = N
প্রশ্নমতে, T এর 35% = S এর 25%
⇒ T(35/100) = N (25/100)
⇒ T/N = 25/100 × 100/35 = 5/7
T : N = 5 : 7

ধরি, C বিনিয়োগ করেছিল x টাকা, B বিনিয়োগ করেছিল (x + 5,000) টাকা
A বিনিয়োগ করেছিল (x + 5,000 + 4,000) = (x + 9,000) টাকা
প্রশ্নমতে, x + x + 5,000 + x + 9,000 = 50,000
⇒ 3x + 14,000 = 50,000
⇒ 3x = 50,000 - 14,000
⇒ 3x = 36,000
⇒ x = 12000 টাকা
অতএব, A, B, C এর বিনিয়োগের অনুপাত
= (12,000 + 9,000) : (12,000 + 5,000) : 12,000
21,000 : 17,000 : 12,000
21 : 17 : 12
∴ A এর মুনাফা = 35000 × (21/50)
= 14,700 টাকা।

দুই জন একত্রে ১ ঘণ্টায় কাজ করতে পারে ১/৬ অংশ।
বিল ১ ঘণ্টায় কাজ করতে পারে ১/১০ অংশ।
বেন ১ ঘণ্টায় কাজ করতে পারে (১/৬ - ১/১০) = ২/৩০ = ১/১৫ অংশ।
∴ বেন একা পুরো কাজটি করতে পারবে  = ১৫ দিনে

We know that,
sec²θ - tan²θ = 1
⇒ (secθ + tanθ)(secθ - tanθ) =1
⇒ x (secθ - tanθ) = 1
⇒ secθ - tanθ = 1/x ...... (i)
again, secθ + tanθ = x ...... (ii)
From (ii) - (i)
2tanθ = x - 1/x = (x² - 1)/x
⇒ tanθ = (x² - 1)/2x

Question: A school has 8 basketball players. A 5-member team and a captain (selected from the remaining players) will be chosen out of these 8 players. How many different selections can be made?

Solution:
We can select the 5 member team out of the 8 in = ⁸C₅ ways
= 8!/(5! × 3!)
= 56 ways

The captain can be selected from amongst the remaining 3 players in = ³C₁
= 3 ways.

∴ The total ways the selection of 5 players and a captain can be made = 56 × 3 ways
= 168 ways

Question: If 5 workers can paint a house in 8 hours, how long would it take 20 workers to paint the same house?

Solution:
 5 workers can paint a house in 8 hours
∴ 1 worker would take = (8 × 5) hours
∴ 20 workers would take = (8 × 5)/20 hours
= 2 hours

Question: Kabir paid Tk. 9,600 as interest on a loan he took 5 years ago at the rate of 16% simple interest per annum. What was the principal amount (the original loan amount) he borrowed?

Solution: 
SI = Interest paid = Tk. 9600
R = Rate of interest = 16% per annum
T = Time = 5 years
P = Principal (the amount we need to find)

We know, 
SI = (P × R × T)/100
⇒ 9600 = (P × 16 × 5)/100
⇒ 9600 = (P × 80)/100
⇒ P = 96000/8
∴ P = 12000

∴ Kabir took a loan of Tk. 12000.

5% হার সুদে 2000 টাকার 2 বছরের সরল সুদ = (5 × 2000 × 2) / 100 = 200 টাকা
5% হার সুদে 2000 টাকার 2 বছরের চক্রবৃদ্ধি সুদ = [2000(1 + 5/100)² - 2000]
= 2205 - 2000 = 205 টাকা
সুতরাং, সরল সুদ এবং চক্রবৃদ্ধি সুদের পার্থক্য = 205 - 200 = 5 টাকা

Question: Pipe A can fill a tank in 10 minutes, and pipe B can fill it in 20 minutes. If both are opened together into an empty tank, when should pipe A be turned off so that the tank gets filled in exactly 12 minutes?

Solution:
২য় নল দ্বারা,
20 মিনিটে পূর্ণ হয় = 1 অংশ
∴ 1 মিনিটে পূর্ণ হয় = 1/20 অংশ
∴ 12 মিনিটে পূর্ণ হয় = 12/20 অংশ
= 3/5 অংশ

∴  অবশিষ্ট থাকে = 1 - (3/5) অংশ
= 2/5 অংশ

১ম নল দ্বারা
1 বা সম্পূর্ণ অংশ পূর্ণ হয় = 10 মিনিটে
∴ 2/5 অংশ পূর্ণ হয় = (10 × 2)/5 মিনিটে
= 4 মিনিটে

∴  4 মিনিট পর প্রথম নলটি বন্ধ করতে হবে।

Question: If the nth term of an arithmetic progression is 7n + 1, then what is the common difference?

Solution:
The nth term of an arithmetic progression is T_n = 7n + 1
n = 1 then, T₁ = 7 × 1 + 1 = 8
n = 2 then, T₂ = 7 × 2 + 1 = 15
n = 3 then, T₃ = 7 × 3 + 1 = 22
n = 4 then, T₄ = 7 × 4 + 1 = 29
............................

Common difference,
T₂ - T₁ = 15 - 8 = 7
T₄ - T₃ = 29 - 22 = 7

∴ The common difference is 7.

Question: If each side of the square is increased by 20%, what will be the ratio between the new area and the original area of the square?

Solution:
Let,
The side of original square is x
∴ The area of original square is x²

The side of new square is x + 20% of x
= x + (x/5)
= 6x/5

∴ The area of new square is
= (6x/5)²
= (36x²)/25

∴ The ratio between the new area and the original area of the square = {(36x²)/25} : x²
= 36/25 : 1
= 36 : 25