Bank Math

Question: A bag contains 4 white, 5 red, and 6 blue balls. One ball is drawn at random. What is the probability that the ball drawn is neither white nor blue?

Solution:
Total number of balls, n(S) = 4 + 5 + 6 = 15

Let E = event that the ball is neither white nor blue (which means the ball is red)

Number of red balls, n(E) = 5

∴ Probability, P(E) = n(E)/n(S)
= 5/15
 = 1/3

Suppose A, B and C take x, x/2, x/3 days respectively to finish the work
Then,
1/x + 2/x + 3/x = 1/2
⇒ 6/x = 1/2
⇒ x = 12
So, B takes 12/2 = 6 days to finish the work.

Question: A candidate scores 40% marks and fails by 20 marks. Another candidate scores 50% marks and gets 30 marks more than the minimum pass marks. What is the pass percentage?

সমাধান:
ধরি, মোট নম্বর = x

প্রশ্নমতে,
40% of x + 20 = 50% of x - 30
⇒ 0.40x + 20 = 0.50x - 30 
⇒ 0.50x - 0.40x = 20 + 30 
⇒ 0.10x = 50 
⇒ x = 50/0.10
∴ x = 500

∴ পাশ নম্বর = 40% of x + 20
= 0.4 × 500 + 20
= 200 + 20 = 220

∴ পাশের শতকরা হার = (220/500) × 100
= 44%

Question: In a simultaneous throw of a pair of dice, what is the probability of getting a total more than 8?

Solution:
When two fair six-sided dice are thrown together. Then we get
Total outcomes = 6 × 6 = 36

And, Count outcomes with sum > 8

We want sum > 8 ⇒ sums = 9, 10, 11, 12
Sum 9 = (3, 6), (4, 5), (5, 4), (6, 3) ⇒ 4 outcomes
Sum 10 = (4, 6), (5, 5), (6, 4) ⇒ 3 outcomes
Sum 11 = (5, 6), (6, 5) ⇒ 2 outcomes
Sum 12 = (6, 6) ⇒ 1 outcome

∴ Total favorable outcomes =  4 + 3 + 2 + 1 = 10

∴ Probability(sum > 8) = favorable outcomes/total outcomes
= 10/36
= 5/18

So the probability of getting a total more than 8 is 5/18.

Average speed = Total distance / Time
Distance =Time x Speed

Total distance covered by Mithila = Distance covered in first 4 hours + distance covered in next 6 hours
= (80 x 4) + (30 x 6)
= 500 miles / hr

Total time taken to complete the journey = 4 + 6 = 10 hrs

Therefore,
Average speed = Total Distance/Time
= 500 / 10
= 50 miles/hr

Question: A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Solution: 
Given distance = 360 km.
Let the speed of the train be x km/hr.
Speed when increased by 5 km/hr = (x + 5) km/hr

ATQ,
(360/x) - {360/(x + 5)} = 1
⇒ [360x + 1800 - 360x]/x(x + 5) = 1
⇒ 1800/x² + 5x = 1
⇒ x² + 5x = 1800
⇒ x² + 5x - 1800 = 0
⇒ x² + 45x - 40x - 1800 = 0
⇒ x(x + 45) - 40(x + 45) = 0
⇒ (x - 40)(x + 45) =0
∴ x = 40, - 45

Question: If √(7m) = 343, then the value of m is - 

Solution:
Given that, √(7m) = 343
⇒ √(7m) = 7³ (since, 343 = 7³)
⇒ {√(7m)}² = (7³)²
⇒ 7m = 7⁶
⇒ m = 7⁵
⇒ m = 16,807

∴ The value of m is 16,807.

Question: A, B and C jointly thought of engaging themselves in a business venture. It was agreed that A would invest Tk. 6500 for 6 months, B, Tk. 8400 for 5 months and C, Tk. 10,000 for 3 months. B wants to be the working member for which, he was to receive 5% of the profits. The profit earned was Tk. 7400. Calculate the share of C in the profit.

Solution:

For managing, B received = 5% of Tk. 7400
= (5/100) × 7400
= Tk. 370

Balance = Tk. (7400 - 370)
= Tk. 7030

∴ Ratio of their investments = (6500 x 6) : (8400 x 5) : (10000 x 3)

= 39000 : 42000 : 30000

= 13 : 14 : 10

Sum of the ratio = (13 + 14 + 10) = 37

C's share = [7030 × (10/37)]
= Tk. 1900  

If a month beings with Sunday then there are 5 Sundays in that month.
Total number of visitors come on Sunday = 510×5 = 2550
Total number of visitors come on other days = 240×25 = 6000
∴ Average number of visitors per day = (2550+6000)/30 = 8550/30 = 285

Let, cost of each item = 100
At 20% profit, selling price = 120
At 10% loss, selling price = 90
Total selling price = 210
Total cost = 100+100 = 200
Profit = 210 - 200 = 10
Profit % = (10×100)/200 = 5%

Question: P, Q, and R are three consecutive even integers. If P + R = Q + 14, what is the value of P?

Solution:
ধরি, P, Q, এবং R হলো তিনটি ক্রমিক জোড় পূর্ণসংখ্যা। যেখানে,
P = n (জোড় পূর্ণসংখ্যা)
Q = n + 2 (পরবর্তী ক্রমিক জোড় পূর্ণসংখ্যা)
R = n + 4 (তৃতীয় ক্রমিক জোড় পূর্ণসংখ্যা)

দেয়া আছে,
P + R = Q + 14
⇒ n + (n + 4) = (n + 2) + 14
⇒ 2n + 4 = n + 16
⇒ 2n - n = 16 - 4
∴ n = 12

অতএব, P = 12, Q = 14, R = 16
সুতরাং, P এর মান হলো 12.

Question: A man buys an article for 25% more than its value and sells it for 20% less than its value. His gain or loss percentage is –

Solution:
Let the original value of the article = 100 units
∴ Cost Price (CP) = 100 + 25% of 100 
= 100 + 25 = 125 units

∴ Selling Price (SP) = 100 - 20% of 100 
= 100 - 20 = 80 units

Since SP 80 is less than CP 125, there is a Loss.

∴ Loss = CP - SP = 125 - 80 = 45 units
∴ Loss percentage = (Loss/CP) × 100%
= (45/125) × 100%
= 36% loss

1²  + 2²  + 3²  + ........ + 9² 
n সংখ্যক স্বাভাবিক সংখ্যার বর্গের সমষ্টি  = n(n + 1)(2n + 1)/6
= 9(9 + 1)(2×9 + 1)/6
= (9×10×19)/6
= 285

Question: If one fifth of one third of a number is 12, then what is 2/5 of the number?

Solution:
Let the number be x.
According to the question,
(1/5) of (1/3) of x = 12
⇒ (1/5) × (1/3) × x = 12
⇒ (1/15)x = 12
⇒ x = 12 × 15
∴ x = 180

Now,
(2/5) of x = (2/5) × 180
= 360/5
= 72

Question: Every 2 minutes, 5 litres of water are poured into a 1,500 litre tank. After 3 hours, what percent of the tank will be full?

Solution:
In 2 minutes, 5 liters is poured
In 180 minutes = (180 × 5)/2 = 450 liters

So, percentage filled = (450 × 100)/1500
= 30%

13 x 1 = 13 --> অর্থাৎ, 13 + 2 = 15 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।

একইভাবে,
13 x 2 = 26 --> 28 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।
13 x 3 = 39 --> 41 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।
13 x 4 = 52 --> 54 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।
13 x 5 = 65 --> 67 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।
13 x 6 = 78 --> 80 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।
13 x 7 = 91 --> 93 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।

13 x 8 = 104 --> যেহেতু ১০০ এর চেয়ে ছোট চেয়েছে তাই এটা হবে না।

অর্থাৎ, ১৫, ২৮, ৪১, ৫৪, ৬৭, ৮০, ৯৩ কে ১৩ দিয়ে ভাগ করলে ভাগশেষ ২ থাকে।
তাহলে, মোট সংখ্যা হল ৭টি।

কিন্তু আমরা জানি,
যেকোনো সংখ্যা দ্বারা তার চেয়ে ক্ষুদ্রতম কোন পূর্ণ সংখ্যাকে ভাগ করলে ভাগশেষ ক্ষুদ্র সংখ্যাটিই হবে।

উক্ত নিয়মানুসারে, ২ কে ১৩ দিয়ে ভাগ করলে ভাগশেষ ২ হবে। এবং, ২ অবশ্যই ১০০ এর চেয়ে ছোট ধনাত্মক পূর্ণ সংখ্যা (Positive Integer)।

সুতরাং, ১০০ এর চেয়ে ছোট ধনাত্মক পূর্ণ সংখ্যা যাদেরকে ১৩ দিয়ে ভাগ করলে ভাগশেষ ২ হবে এমন সর্বমোট সংখ্যা হল - ২, ১৫, ২৮, ৪১, ৫৪, ৬৭, ৮০, ৯৩ = ৮টি।

Question: If the area of a square garden is 50 sq. meters, what is the maximum distance between two points on its boundary?

Solution:
Given that, 
Area of square = 50 m²
∴ Side of length = √Area = √50 =√(25 × 2)
= 5√2 meters

The maximum distance between any two points on the boundary of a square is the length of the diagonal.
∴ Diagonal of the square = side × √2
= 5√2 × √2
= 5 × 2
= 10 meters

So the maximum distance between two points on the boundary is 10 meters.

Question: Two numbers are in the ratio 5 : 4 and their difference is 10. Find the largest number.

Solution:
Let the two numbers be,
5x and 4x

According to the problem,
5x - 4x = 10
⇒ x = 10

So, the numbers are, 
5x = 5 × 10 = 50 and 4x = 4 × 10 = 40

∴ The largest number = 50

Question: The cost prices of a book and a notebook are in the ratio 3 : 7, while their selling prices are in the ratio 1 : 4. If the loss incurred on selling each item is the same, find the ratio of the cost price to the selling price of the notebook.

Answer: 
Let,
cost price of
a book = 3x
a notebook = 7x

Selling price of
a book = y
a notebook = 4y

According to the question,
3x - y = 7x - 4y
⇒ 4y - y = 7x - 3x
⇒ 3y = 4x
∴ y = 4x/3

∴ The ratio of the cost price of a notebook : the selling price of a notebook = 7x : [4 × (4x/3)]
= 21 : 16

Question: Six printers working together can print 720 documents in 6 days. If 2 printers stop working, how many documents can the remaining printers print in 9 days? 

Solution:
6টি প্রিন্টার 6 দিনে প্রিন্ট করে = 720 টি ডকুমেন্ট
∴ 1টি প্রিন্টার 6 দিনে প্রিন্ট করে = 720/6 = 120টি ডকুমেন্ট
∴ 1টি প্রিন্টার 1 দিনে প্রিন্ট করে = 120/6 = 20টি ডকুমেন্ট

অবশিষ্ট প্রিন্টার = 6 - 2 = 4 টি

∴ 4টি প্রিন্টার 9 দিনে প্রিন্ট করবে = 20 × 4 × 9 = 720 টি ডকুমেন্ট

Question: The ratio of copper and zinc in an alloy is  5 : 2. If 12 kg of zinc is added, the ratio becomes 5 : 4. What is the initial weight of the copper?

Solution:
Given that, 
The initial ratio of copper to zinc is 5 : 2.
Let the initial quantity of copper = 5x kg
and initial quantity of zinc = 2x kg

When 12 kg of zinc is added when,
Copper remains = 5x kg
And zinc becomes = (2x + 12) kg

Now the new ratio becomes 5 : 4, so we get,
⇒ 5x/(2x + 12) = 5/4
⇒ 4 × 5x = 5 × (2x + 12)
⇒ 20x = 10x + 60
⇒ 20x - 10x = 60
⇒ 10x = 60
∴ x = 6
∴ Initial quantity of copper = 5x = 5 × 6 = 30 kg

So the initial weight of the copper is 30 kg.

Question: The combined salaries of M and N amount to Tk. 4000. M spends 75% of his salary, and N spends 85% of his salary. If both M and N save the same amount of money, what is the salary of M?

Solution:
Let M’s salary = x
and N’s salary = (4000 - x)

According to the question,
Savings of M = Savings of N
⇒ (100 - 75)% of M = (100 - 85)% of N
⇒ (25/100)x = (15/100)(4000 - x)
⇒ 25x = 15(4000 - x) [multiply both sides by 100]
⇒ 25x = 60000 - 15x
⇒ 25x + 15x = 60000
⇒ 40x = 60000
⇒ x = 60000/40
⇒ x = 1500

∴ M’s salary is Tk. 1500

১ম রাশি = x² - 1 = (x + 1)(x - 1)
২য় রাশি =  x⁴ - 1 = (x²)² - (1²)² = (x² + 1)(x² - 1) = (x² + 1)(x + 1)(x - 1)
এবং ৩য় রাশি =  x⁴ - x³ + x - 1 = x³(x - 1) + 1(x - 1) = (x³ + 1)(x - 1) = (x + 1)(x² - x + 1)(x - 1)

So, HCF is (x + 1)(x - 1) = x² - 1

Question: In a zoo, each pigeon has 2 legs, and each rabbit has 4 legs. The head count of the two species together is 12, and the leg count is 32. How many pigeons and how many rabbits are there in the zoo?

Solution:
Let,
Number of pigeons = P
Number of rabbits = R

From the problem we get two equations,
Heads,  P + R = 12 ......(1)

And,
Legs, 2P + 4R = 32 ; [pigeons 2 Legs and rabbits 4 Legs]
⇒ 2(P + 2R) = 32
⇒ P + 2R = 32/2
∴ P + 2R = 16 .......(2)

Now, (2) - (1) Then we get,
⇒ P + 2R - P - R = 16 - 12
∴ R = 4

From (1) we get, 
⇒ P + 4 = 12
⇒ P = 12 - 4
∴ P = 8

There are 8 pigeons and 4 rabbits.

এখানে 9 জন বালিকা = 5 জন বালক
তাহলে 1 জন বালিকা = 5/9 জন বালক
সুতরাং 3 জন বালক + 6 জন বালিকা = 3 + 6×5/9 = 3 + 10/3 = 19/3 জন বালক।
প্রশ্নমতে,
5 জন বালক কাজটি করে 19 দিনে
1 জন বালক কাজটি করে 19×5 দিনে
19/3 জন বালক কাজটি করে = 19×5×3/19 = 15 দিনে।
অর্থাৎ 3 জন বালক এবং 6 জন বালিকা কাজটি একত্রে 15 দিনে করতে পারে।

Ratio of rates of working of A and B = 2:1
So, ratio of times taken = 1 : 2
∴ A's 1 day's work = (1/6);
B's 1 day's work = 1/12
(A + B)'s 1 day's work = (1/6) + (1/12)
= 3/12
= 1/4
So, A and B together can finish the work in 4 days.

Question: An iron rod that weights 24 kg is cut into pieces so that one of these pieces weighs 16 kg and is 34m long. If the weight of each piece is proportional to its length, how long is the other piece?

Solution:
Given that,
Total weight of rod = 24 kg
Cut into two pieces, one weighs 16 kg and is 34 m long
Weight ∝ Length
Let the length of the other piece be L₂​.
Its weight is W₂ = 24 - 16 = 8 kg.

Since weight is proportional to length,

W₁/L₁ = W₂/L₂
​⇒ 16/34 = 8/L₂
⇒ 16 × L₂ = 8 × 34
⇒ L₂ = (8 × 34)/16
∴ L₂ = 17 m

So the other piece is 17 meters long.