Bank Math

Question: 10 women can do a work in 8 days. 8 men can complete the same work in 7 days. What is the ratio between the capacity of a man and a woman?

Question: 
(10 × 8) = 80 women can complete the work in 1 day.
∴ 1 woman's 1 day's work = 1/80 part

(8 × 7) = 56 men can complete the work in 1 day.
1 man's 1 day's work = 1/56 part

So, required ratio = 1/56 : 1/80
= (1/56)/(1/80)
= 80/56
= 10/7
= 10 : 7

• Tag question করার নিয়ম:
- Tag question ব্যবহার করা হয় উক্তিটি সত্য না মিথ্যা তা নিশ্চিত করার জন্য।
- Statement positive হলে tag question টি negative হবে।
- Subject ও tense অনুসারে auxiliary verb দ্বারা tag question হয়।

• Confusing Contraction sentence এর tag question করার নিয়ম:
- যদি subject এরপর 'd থাকে অর্থাৎ, verb টি base form এ আসে, তাহলে এটিকে modal auxiliary হিসেবে would ধরতে হবে।
- Example: If I were a rich man, I'd establish a hospital, wouldn't I?

- তবে 'd এরপরের verb টি যদি past participle form এ থাকে তাহলে তাকে past perfect tense এর had হিসেবে ধরতে হবে।
- Example: I'd reached there in time, hadn't I?

• Correct answer: I had called her yesterday, hadn't I?

Source: Advanced Learner's Communicative English Grammar & Composition By Chowdhury & Hossain.

Question: Among 80 students, the average marks in Mathematics is 65. If the 50 girls scored an average of 68, determine the average score of the remaining boys.

Solution:
Let,
the average marks of the boys = k
Total marks of 80 students = 80 × 65 = 5200 
Total marks of 50 girls = 50 × 68 = 3400

According to the question,
3400 + (80 − 50) × k = 5200
⇒ 3400 + 30k = 5200
⇒ 30k = 5200 − 3400
⇒ 30k = 1800
⇒ k = 1800 / 30
⇒ k = 60

∴ Average marks of the remaining 30 boys = 60

Question: At what rate of simple interest per annum will Tk. 25,000 amount to Tk. 37,500 in 5 years?

Solution:
Principal, P = Tk. 25000
Total Amount, A = Tk. 37500
Time, n = 5 years

Simple Interest, SI = A - P = 37500 - 25000 = Tk. 12500

We know, SI = (P × n × r)/100
⇒ 12500 = (25000 × 5 × r)/100
⇒ 12500 = 250 × 5 × r
⇒ 12500 = 1250r
⇒ r = 12500/1250
⇒ r = 10

∴ Rate of simple interest = 10% per annum.

Question: Given that the diagonal of a square measures 8√2 units, find the perimeter of the square in units.

Solution:
দেয়া আছে, বর্গক্ষেত্রের কর্ণের দৈর্ঘ্য = 8√2 একক

আমরা জানি, বর্গক্ষেত্রের কর্ণের দৈর্ঘ্য = √2 × বাহু

প্রশ্নমতে,
√2 × বাহু = 8√2
⇒ বাহু = 8√2/√2
⇒ বাহু = 8 একক

এখন, বর্গক্ষেত্রের পরিসীমা = 4 × বাহুর দৈর্ঘ্য
= 4 × 8
= 32 একক

∴ বর্গক্ষেত্রের পরিসীমা 32 একক।

Question: If n is a whole number greater then 1, then n² (n² - 1) is always divisible by -

Solution: 
Given that,
n²(n² - 1)
= 2²(2² - 1)     ; [put n = 2]
= 4(4 - 1)
= 4 × 3
= 12
Check the option it is divisible by 12.
Take n = 3
= 3³(3² - 1)
= 9(9 - 1)
= 9 × 8
= 72
It is divisible by 12

Question:

Solution:

Question: If B = {1, 2, 3, 4} and C = {x, y, z}, then B ∪ C = ?

Solution:
B ∪ C = {1, 2, 3, 4} ∪ {x, y, z}
= {1, 2, 3, 4, x, y, z}

Question: If log_a2 = a and log_a5 = b, then log_a50 = ?

Solution:
50 = 2 × 5²

log_a50 = log_a(2 × 5²) = log_a2 + log_a5²
∴ log_a50 = log_a2 + 2 log_a5
∴ log_a50 = a + 2b [log_a2 = a and log_a5 = b]

Question: 


Solution:

Question: A cube has a total surface area of 384 square units. What is the volume of the cube?

Solution:
Given:
Total surface area of the cube, S = 384 square units

We know,
Surface area of a cube, S = 6a²
⇒ 6a² = 384
⇒ a² = 384 / 6
⇒ a² = 64
∴ a = 8

Volume of a cube, V = a³
= 8³
= 512

So, the volume of the cube is 512 cubic units.

Question: If 8 men can reap 80 hectares in 24 days, then how many hectares can 36 men reap in 30 days?
 
Solution:
8 জন লোক 24 দিনে ফসল কাটতে পারে = 80 হেক্টর জমির 
∴ 1 জন লোক 1 দিনে ফসল কাটতে পারে = 80/(8 × 24) হেক্টর জমির 
∴ 36 জন লোক 30 দিনে ফসল কাটতে পারে = (80 × 36 × 30)/(8 × 24)
= 450 হেক্টর জমির 

LCM of 5, 6, 7 and 8 = 840
Hence the number can be written in the form (840k + 3) which is divisible by 9.
If k = 1, number = (840 × 1) + 3 = 843 which is not divisible by 9.
If k = 2, number = (840 × 2) + 3 = 1683 which is divisible by 9.
Hence 1683 is the least number which when divided by 5, 6, 7 and 8 leaves a remainder 3,
but when divided by 9 leaves no remainder.

Question: What is the quotient when (x^{- 1} - 1) is divided by (x - 1)?

Solution: 

Question: A sum of money doubles itself in 10 years at a certain rate of simple interest. In how many years will it become six times itself at the same rate of interest?

Solution:
Given that,
The sum doubles itself in 10 years.
Amount after 10 years = 2P
Simple Interest for 10 years = 2P - P = P

We know,
SI = (P × r × n)/100
⇒ P = (P × r × 10)/100
⇒ 10r = 100
⇒ r = 100 / 10
∴ r = 10% per year

Now, we want to find in how many years the sum becomes six times itself.
Amount = 6P
Interest needed = 6P - P = 5P

We know,
SI = (P × r × n)/100
⇒ 5P = (P × 10 × n)/100
⇒ 5 = (10 × n)/100
⇒ n = (5 × 100)/10
∴ n = 50 years

∴ The sum will become six times itself in 50 years.

Question: The distance between the points (4, 3) and (1, 7) is -

Solution:
Given points (4, 3) and (1, 7).
Now the formula for the distance between (x₁, y₁) and (x_2, y₂) is
= √[(x₂ - x₁)² + (y₂ - y₁)²]

∴ The distance between the points (4, 3) and (1, 7) is
= √[(1 - 4)² + (7 - 3)²]
= √[9 + 16]
= √25
= 5

Question: In a class of 92 students, 40 are taking English, 24 are taking Arabic and 10 are taking both courses. How many students are not enrolled in either course?

Solution:
Total students = 92
Students taking English n(E) = 40
Students taking Arabic n(A) = 24
Students taking both English and Arabic = 10

We know,
n(E ∪ A) = n(E) + n(A) - n(E ∩ A)
n(E ∪ A) = 40 + 24 - 10 = 54

∴ Not enrolled = Total students - n(E ∪ A) = 92 - 54 = 38

Question: The value of the polynomial 2y³ + 5y² - 7 when y = - 2 is?

Solution:
প্রদত্ত বহুপদী: 2y³ + 5y² - 7
যদি y = - 2 হয়, তবে y-কে - 2 দ্বারা প্রতিস্থাপন করি,
= 2 × (- 2)³ + 5 × (- 2)² - 7
= 2 × (- 8) + 5 × 4 - 7
= - 16 + 20 - 7
= 4 - 7
= - 3

∴ বহুপদীটির মান হলো - 3

1470 = 2 x 3 x 5 x 7 x 7
So, the least number is = 2 x 3 x 5 = 30

Question: A camp had provisions to support 560 soldiers for 20 days. After 12 days, 112 soldiers were reassigned to another camp. For how many days can the remaining soldiers continue without receiving more rations?

Solution:
After 12 days, there was support for 560 soldiers for 8 days.
Remaining persons = (560-112) = 448

Less soldiers, more days (inverse proportion)
Let the x is the required number of days

Then, 448 : 560 = 8 : x
Or, x = (560 × 8)/ 448 = 10

Hence, the required number of days is 10.

Question: What is the greatest prime factor of 25¹⁵ − 5²⁸?

Solution:
25¹⁵ − 5²⁸
= (5²)¹⁵ - 5²⁸
= 5³⁰ - 5²⁸
= 5²⁸(5² - 1)
= 5²⁸(25 -1)
= 5²⁸ × 24
= 5²⁸ × 8 × 3
= 5²⁸ × 2³ × 3

সুতরাং, সম্পূর্ণ রাশিটির মৌলিক গুণনীয়কগুলো (prime factors) হলো 5, 2, এবং 3.
এর মধ্যে সবচেয়ে বড় হল 5. 
সুতরাং, বৃহত্তম মৌলিক গুণনীয়ক (prime factors) হলো 5.

Question: Three dice are thrown together. Find the probability of getting a total of at least 6?

Solution:
The total possible ways of sum of 3 dice less than, 6
(1 + 1 + 1), (1 + 1 + 2), (1 + 1 + 3), (1 + 2 + 1), (1 + 2 + 2), (1 + 3 + 1), (2 + 1 + 1), (2 + 1 + 2), (2 + 2 + 1), (3 + 1 + 1)

Total 10 ways.
Total Sample space = 6³ = 216

Probability of getting the sum less than 6 = 10/216
Probability of getting the sum at least 6 = 1 - (10/216)
= (216 - 10)/216
= 206/216
= 103/108

Question: If D is the midpoint of the points P(4, 1) and Q(10, 9), find the length of PD.

Solution:
দেওয়া আছে, P(4, 1) এবং Q(10, 9), এবং D হলো PQ-এর মধ্যবিন্দু।

প্রথমে PQ-এর দৈর্ঘ্য নির্ণয় করি:
PQ = √{(x₂ - x₁)² + (y₂ - y₁)²}
⇒ PQ = √{(10 - 4)² + (9 - 1)²}
⇒ PQ = √(6² + 8²)
⇒ PQ = √(36 + 64)
⇒ PQ = √100
∴ PQ = 10

যেহেতু D হলো PQ-এর মধ্যবিন্দু, তাই PD হবে PQ-এর অর্ধেক।
∴ PD = PQ/2
= 10/2
= 5

Number of ways of selecting (3 consonants out of 7) and (2 vowels out of 4) = (7C3 × 4C2)
= 210.
Number of groups, each having 3 consonants and 2 vowels = 210.
Each group contains 5 letters.
Number of ways of arranging 5 letters among themselves = 5! = 120
Required number of ways = (210 x 120)
= 25200.

Question: A bookseller procures 40 books for 3600 Tk and sells them at a profit equal to the selling price of 4 books. What is the selling price of one quarter dozen books, if the price of each book is same?

Solution:
Cost price of each book = 3600/40 Tk
= 90 Tk

Now,
Selling Price of 40 books = CP of 40 books + SP of 4 books.
Selling Price of 40 books - SP of 4 books = CP of 40 books.
SP of 36 books = 3600 Tk
SP of 1 book = 3600/36 Tk
= 100 Tk

Selling price of one dozen books = 12 × 100 = 1200 Tk
∴ Selling price of one quarter dozen books = (1/4) × 1200 = 300 Tk

Given, Green marbles = 2
Red marbles = 2
Total marbles = 2 + 2 = 4
Probability of randomly picked marbles that both of them are green = ²C₂ / ⁴C₂ = 1/6

Question: Tk. 6000 becomes Tk. 7500 in 5 years at a certain rate of simple interest. If the rate becomes half, what will be the total amount in 6 years on the same principal?

Solution:
Principal = Tk. 6000
Amount after 5 years = Tk. 7500
∴ Interest in 5 years = 7500 - 6000 = Tk. 1500

∴ Interest per year = 1500/5 = Tk. 300

Since the interest rate becomes half, the yearly interest also becomes half,
∴ Interest per year = 300/2 = Tk. 150

∴ Interest for 6 years = 150 × 6 = Tk. 900

∴ New total amount = 6000 + 900 = Tk. 6900

Question: Kamal is shorter than Ruhan, but taller than Anika, Dina is shorter than Kamal. Rubayet is shorter than Ruhan but taller than Kamal. Who is tallest in terms of height?

Solution:
Kamal is shorter than Ruhan, but taller than Anika.
∴ Ruhan > Kamal > Anika

Dina is shorter than Kamal.
∴ Kamal > Dina

Rubayet is shorter than Ruhan but taller than Kamal.
∴ Ruhan > Rubayet > Kamal.

From all of these relational statement we can say that,
Ruhan is the tallest in terms of height.

Question: If a² - √5a + 1 = 0, then the value of a² + a^{- 2} = ?

Solution:
দেয়া আছে, 
a² - √5a + 1 = 0
⇒ a² + 1 = √5a
⇒ a + (1/a) = √5 [উভয়পক্ষকে a দ্বারা ভাগ করে]

প্রদত্ত রাশি= (a² + a^{- 2})
= a² + (1/a²)
= {a + (1/a)²} - 2. a. (1/a)
= (√5)² - 2
= 5 - 2
= 3

∴ নির্ণেয় মান = 3

Question: In a partnership, three friends invested Tk. 2700, Tk. 8100, and Tk. 7200 respectively. After one year, the third partner received Tk. 3600 as his share of the profit. How much total profit did the business make?

Solution:
Let the total profit be x

Here,
ratio of investment,
= first partner : second partner : third partner 
= 2700 : 8100 : 7200
= 3 : 9 : 8

Then, third partner's share = 8x/20 = 2x/5

According to the question,
2x/5 = 3600
⇒ x = (3600 × 5)/2 
⇒ x = 9000

So, The total profit is Tk. 9000

Speed of the boat in still water = 22 km/hr
speed of the stream = 5 km/hr
Speed downstream = (22+5) = 27 km/hr
Distance travelled downstream = 54 km
Time taken = distance/ speed
= 54/27
= 2 hours