Bank Math

Question: A cube has a total surface area of 294 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 = 294
⇒ a² = 294/6
⇒ a² = 49
∴ a = 7 মিটার

এখন,
ঘনকের আয়তন = a³
= 7³
= 343 ঘন মিটার

অতএব, ঘনকটির আয়তন = 343 ঘন মিটার।

Let the speed of A = x kmph and that of B = y kmph

According to the question,
(x × 6) + (y × 6) = 60
⇒ x + y = 10 --------- (i)
And,
(2x/3) × 5 + (2y × 5) = 60
⇒ 10x + 30y = 180
⇒ x + 3y = 18 ---------- (ii)
From equation (i) × 3 - (ii)
3x + 3y - x - 3y = 30 - 18
⇒ 2x = 12
Hence, x = 6 kmph.

Question: If x + (2/x) = 4, what is the value of x³ + (8/x³)?

Solution: 
Here, x + (2/x) = 4

Now, 
x³ + (8/x³)
= (x)³ + (2/x)³
= {(x + (2/x)}³ - 3 . x . 2/x {x + (2/x)}
= 4³ - 3 . 2 . 4
= 64 - 24
= 40

Question: If a : b = 3 : 1, find ratio (3a + 5b) : (3a - 5b).

Solution: 
(3a + 5b) : (3a - 5b)
= b(3a/b + 5) : b (3a/b - 5)
= (3 × 3/1 + 5) : (3 × 3/1 - 5)
= (9 + 5) : (9 - 5)
= 14 : 4
= 7 : 2

Question: Anika is taller than Bristi but shorter than Kabir. Rafi is shorter than Bristi. Shawon is taller than Anika but shorter than Kabir. Who is the tallest among them?

Solution: 
From the first statement:
Kabir > Anika > Bristi

From the second statement:
Bristi > Rafi

From the third statement:
Kabir > Shawon > Anika

Putting everything together:
Kabir > Shawon > Anika > Bristi > Rafi

∴ Kabir is the tallest.

Question: In a lottery, there are 8 prizes and 12 blanks. A lottery is drawn at random. What is the probability of getting a prize?

Solution: 
Here, 
Number of prizes = 8
Number of blanks = 12
∴ Total outcome = 8 + 12
= 20

∴ Probability of getting a prize = Number of prizes/Total outcome 
= 8/20
= 2/5 

Question: Find the value of (x + 1)!/(x - 2)! 

Solution:
Here,
(x + 1)!/(x - 2)!
= [x(x + 1)(x - 1)(x - 2)!]/(x - 2)! 
= x(x + 1)(x - 1)
= x(x² - 1) 

Question: Mr. Khan's salary is Tk. 5000.00 and he gets 10% commission of his salary. If his salary increased by 10%, by what percent his commission will increase?

Solution:
পূর্বের কমিশন:
= 5000 এর 10% টাকা
= 5000 × 10/100
= 500 টাকা

১০% বৃদ্ধিতে বর্তমান বেতন = 5000 + 5000 এর 10%
= 5000 + 5000 × 10/100
= 5000 + 500 = 5500 টাকা

নতুন কমিশন = 5500 এর 10% = 5500 × 10/100 = 550 টাকা

কমিশন বৃদ্ধি পায় = 550 - 500 = 50 টাকা

শতকরা কমিশন বৃদ্ধি পায় = (50/500) × 100% = 10%

Question: Find the domain of f(m) = 1/(4m + 3).

Solution:
দেওয়া আছে,
f(m) = 1/(4m + 3)

আমরা জানি,
একটি ভগ্নাংশের হর(denominator) শূন্য হতে পারবে না।
অর্থাৎ,
4m + 3 ≠ ০
or, 4m ≠ - 3
or, m ≠ - (3/4)

∴ f(m) এর ডোমেইন = R - {- 3/4}

Question: What is the probability of rolling an odd number on a standard six-sided die?
(Officer Cash 2022 অনুযায়ী)

Solution:
A standard six-sided die has the numbers,
1, 2, 3, 4, 5, 6
Odd numbers between 1 and 6 are = 1, 3, 5
So, favorable outcomes = 3
And total possible outcomes = 6

∴ Probability = Favorable outcomes/Total outcomes ​
= 3/6 ​
= 1/​2

Question: What percent of 700 is 2.1?

Solution: 
700 এর x% = 2.1
⇒ 700 এর x/100 = 2.1
⇒ 7x = 2.1
⇒ x = 2.1/7
x = 0.3

Let Principle = P.
Then, S.I = 2P and T = 20 years
∴ Rate = {(100 × 2P)/(P × 10)}
Now, principle = P, S.I. = P, R = 10%

∴ Time = {(100 × P)/(P × 10)} yrs
= 10 yrs.

Question: Which of the following fractions is the largest? 

Solution: 
7/4 = 1.75
6/5 = 1.2 
3/2 = 1.5
5/3 = 1.66

Hence the largest fraction is 7/4 

Let the number be x.
Then, x/5 + 4 = x/4 - 10
⇔ x/4 - x/5 = 14
⇔ x/20 = 14
⇔ x = 20 × 14 = 280.
Answer: 280.

Question: Two pipes P and Q can fill a tank in 12 hours and 24 hours, respectively. If both pipes are opened together, how long will it take to fill the tank?

Solution:
Part filled by P in 1 hour = 1/12
Part filled by Q in 1 hour = 1/24

Part filled by (P + Q) in 1 hour
= (1/12) + (1/24)
= (2 + 1)/24
= 3/24
= 1/8

∴ Time to fill the tank = 1/(1/8) = 8 hours

∴ Both pipes can fill the tank in 8 hours

Let the age of Robin be y
Roman is 15 years older than Robin = (y + 15).
So Roman's age 5 years ago = (y + 15 – 5)
Robin's age before 5 years = (y – 5)

5 years ago, Roman is 3 times as old as Robin
(y + 15 – 5) = 3 (y – 5)
(y + 10) = (3y – 15)
2y = 25
y = 12.5
Robin's age = 12.5 years
Roman's age = (y + 15)
= (12.5 + 15)
= 27.5 years.

Question: A 15% stock yields 12.5%. What is the market value of the stock?

Solution:
Earn Tk. 12.5 when market value = Tk. 100
Earn Tk. 1 when market value = Tk. (100 / 12.5)
Earn Tk. 15 when market value = Tk. (100 × 15)/12.5
= 1500/12.5
= 120

∴ The market value of the stock is Tk. 120.

Question: The area of a trapezium is 72 square cm. The lengths of its parallel sides are 12 cm and 6 cm. What is the distance between the parallel sides?

Solution:
We know,
The area of a trapezium = (1/2) × Sum of the lengths of the parallel sides × Distance between the parallel sides.

Let, the distance between the parallel sides be d.
Then,
d = (2 × Area of the trapezium)/Sum of the lengths of the parallel sides
= (2 × 72)/(12 + 6)
= 144/18
= 8 cm

Thus, the distance between the parallel sides 8 cm.

Question: Kobita runs 5/2 times as fast as Babita. In a race, if Kobita gives a lead of 40 m to Babita, find the distance from the starting point where both of them will meet (correct up to two decimal places).

Solution:
Given that,
Kobita runs 5/2 times as fast as Babita
Kobita gives a lead of 40 m to Babita 

We know,
Distance = Speed × Time

Let the speed of Babita be = 2x
∴ Speed of Kobita = (5/2) × 2x = 5x

And,
Let the distance covered by Kobita be y meters
∴ Distance covered by Babita = (y - 40) meters

As time is constant, distance is directly proportional to speed,
2x/5x = (y - 40)/y
⇒ 2/5 = (y - 40)/y
⇒ 2y = 5y - 200
⇒ 3y = 200
⇒ y = 200/3
∴ y = 66.67 m

∴ The distance from the starting point where both of them will meet is 66.67 m.

Question: For a geometric sequence, the first term a = 5 and the common ratio r = 3. What is the sum of the first 4 terms?

Solution:
প্রদত্ত গুণোত্তর ধারাটির, প্রথম পদ, a = 5
সাধারণ অনুপাত, r = 3
পদের সংখ্যা, n = 4
যেহেতু r = 3 > 1,

∴ n সংখ্যক পদের সমষ্টির সূত্র:
S_n = a(rⁿ - 1)/(r - 1)
∴ S₄ = 5(3⁴ - 1)/(3 - 1)
= 5(81 - 1)/2
= 5 × 80/2
= 5 × 40
= 200

অতএব, প্রথম 4টি পদের সমষ্টি হলো 200।

Question: How many possible two-digit numbers can be formed by using the digits 3, 5 and 7 (repetition of digits is allowed)?

Solution:
Number of possible two-digit numbers which can be formed by using the digits 3, 5 and 7 = 3 × 3.

∴ 9 possible two-digit numbers can be formed.

The 9 possible two-digit numbers are-
33, 35, 37, 53, 55, 57, 73, 75, 77 

Question:


Solution:

√a =  √0.67 = 0.81 
1/a = 1/0.67 = 1.49
a² = 0.67²  = 0.4489
1/a² = 1/0.67²  = 1/0.4489 = 2.23

So, a² is the smaller than a

Let the longer side of the right angled triangle be x
Then, hypotenuse = (x + 2) and shorter side = (x - 7)
Here,
• h = x + 2
• p = x
• b = x - 7
Using Pythagoras Theorem :
 (Hypotenuse)² = (Perpendicular)² + (Base)²
⇒ (x + 2)² = (x)² + (x - 7)²
⇒ x² + 2(x)(2) + 2² = x² + x² - 2(x)(7) + (7)²
⇒ x² + 4x + 4 = x² + x² - 14x  + 49
⇒ x² - 18x  +  45 = 0
⇒ x² - 15x - 3x + 45 = 0
⇒ x(x - 15) - 3(x - 15) = 0
⇒ (x - 3) (x - 15)
The value of x can't be 3 as (x - 7) would be negative.
Hence, the value of the longer side is 15 cm.
So, the hypotenuse is 15 + 2 = 17 cm

r = √(5² - 4²)
= 3

So, the volume is V = 1/3∏r²h
= 1/3Π × 3² × 4
= 12Π cm³

Question: A cube has a total surface area of 72 m². Determine the length of its diagonal.

Solution:
মনে করি,
ঘনকটির ধার = a 
ঘনকের সম্পূর্ণ পৃষ্ঠের ক্ষেত্রফল = 6a²

প্রশ্নমতে,
6a²  = 72
⇒ a² = 72/6
⇒ a² = 12
⇒ a² = (4 × 3)
⇒ a = 2√3 

∴ ঘনকটির কর্ণের দৈর্ঘ্য = a√3
= 2√3 × √3 
= 2 × (√3)²
= 2 × 3
= 6

∴ ঘনকটির কর্ণের দৈর্ঘ্য = 6 মিটার।

Question: Sarah sold her bicycle for Taka 5000 while making a profit of Taka 570. Then what is the price at which she bought that cycle?

Solution:
Given,
Selling Price = Taka 5000
Profit = 570 Taka

Now, 
Cost Price = Selling Price - Profit
⇒ Cost Price = 5000 - 570
⇒ Cost Price = 4430 Taka
Thus, cost price of the bicycle is Taka 4430

Question: The height of a cylinder is five times the radius of the cylinder. If the volume of the cylinder is 625π cm^3, what is the height of the cylinder?

Solution: 
Given, radius = r and
Height = 5r

ATQ, πr² × 5r = 625π
⇒ r³ = 125
⇒ r = 5

∴ Height = 5 × 5
= 25 cm 
= 0.25 m

Suppose the slower pipe alone can fill the tank in x minutes.
Then the faster pipe can fill the tank in x/4 minutes.

Part filled by the slower pipe in 1 minute = 1/x
Part filled by the faster pipe in 1 minute = 4/x
Part filled by both the pipes in 1 minute = 1/x + 4/x

Given that both the pipes together can fill the tank in 36 minutes.
Part filled by both the pipes in 1 minute = 1/36

According to the question,
1/x + 4/x = 1/36
5/x = 1/36
x = 180

Question: Carlos & Co. generated revenue of BDT 1,250 in 2006. This was 12.5% of its gross revenue. In 2007, the gross revenue grew by BDT 2,500. What is the percentage increase in the revenue in 2007? 

Solution: 
We are given that Carlos & Co. generated revenue of BDT 1,250 in 2006 and that this was 12.5% of the gross revenue.
Hence, if 1250 is 12.5% of the revenue, then 100% (gross revenue) is (100/12.5) × (1250) = BDT 10,000 
Hence, the total revenue by the end of 2007 is BDT 10,000.

In 2006, revenue grew by BDT 2500. 
∴ Percentage increase in the revenue = (2500/10000) × 100 
= 25%

Question: Arif and Samir started a business in partnership by investing Tk. 20,000 and Tk. 15,000 respectively. After six months, Tanvir joined them with Tk. 20,000. If the total profit earned at the end of 2 years from the start of the business is Tk. 25,000, what is Samir’s share?

Solution:
Here,
Arif : Samir : Tanvir
= (20,000 x 24) : (15,000 x 24) : (20,000 x 18)
= 4 : 3 : 3.

So, Samir's share = 25000 x 3/10 = 7,500.

Question: A shopkeeper mixes 15 kg of tea costing Tk. 280 per kg with 10 kg of tea costing Tk. 400 per kg. He then adds some inferior tea costing Tk. 200 per kg so that the average price of the mixture becomes Tk. 300 per kg. How many kg of inferior tea is added?

Solution:
Let the quantity of inferior tea added be X kg.
Total cost = (15 × 280) + (10 × 400) + (X × 200)
= 4200 + 4000 + 200X 
= 8200 + 200X Tk.

Total weight = 15 + 10 + X
= 25 + X kg.

∴ Average price = 300 Tk. per kg.

(8200 + 200X)/(25 + X) = 300
⇒ 8200 + 200X = 300 × (25 + X)
⇒ 8200 + 200X = 7500 + 300X
⇒ 8200 - 7500 = 300x - 200X
⇒ 700 = 100X
⇒ X = 7

Thus, 7 kg of inferior tea is added.