Bank Math

Question: In a box, there are 6 white, 4 black, and 2 yellow balls. If two balls are drawn one after the other without replacement, what is the probability that the first one is black and the second one is yellow?

Solution:
মোট বলের সংখ্যা = 6 (সাদা) + 4 (কালো) + 2 (হলুদ) = 12টি

প্রথম বলটি কালো হওয়ার সম্ভাবনা = 4/12 = 1/3

প্রথম বলটি তোলার পর বাক্সে মোট বলের সংখ্যা = 12 - 1 = 11টি

দ্বিতীয় বলটি হলুদ হওয়ার সম্ভাবনা = 2/11

∴ প্রথমটি কালো এবং দ্বিতীয়টি হলুদ হওয়ার সম্ভাবনা:
= (প্রথমটি কালো হওয়ার সম্ভাবনা) × (দ্বিতীয়টি হলুদ হওয়ার সম্ভাবনা)
= (1/3) × (2/11)
= 2/33

Speed downstream = 96/8 = 12 km/hr
Speed of current = 4 km/hr
Speed of the boatman in still water = 12 − 4 = 8 km/hr
Speed upstream = 8 − 4 = 4 km/hr
Time taken to cover 8 km upstream = 8/4 = 2 hrs

Question: 7q² - (5q - 2)(4q + 3) = ?

Solution:
7q² - (5q - 2)(4q + 3)
= 7q² - {(5q × 4q) + (5q × 3) - (2 × 4q) - (2 × 3)}
= 7q² - (20q² + 15q - 8q - 6)
= 7q² - (20q² + 7q - 6)
= 7q² - 20q² - 7q + 6
= - 13q² - 7q + 6

(0.11)³+(0.22)³+....+(0.99)³
=(0.11)³(1³+2³+....+9³)
=0.001331×2025
=2.695275
≈2.695

Question: In how many ways can 3 guests from a group of 8 guests be seated around a circular table?

Solution:
8 জন থেকে 3 জন নির্বাচন করার উপায়:
⁸C₃ = 8!/(3! × 5!)
= (8 × 7 × 6)/(3 × 2 × 1)
= 336/6
= 56

3 জন ব্যক্তিকে একটি গোলাকার টেবিলে সাজানোর উপায় = (3 - 1)!
= 2!
= 2

∴ মোট উপায় = 56 × 2 = 112

Question: In a class of 60, 32 studied English, 28 studied Bengali and 6 did not study either. How many of the students studied both?

Solution:
মোট শিক্ষার্থীর সংখ্যা: 60
ইংরেজিতে পড়ে এমন শিক্ষার্থীর সংখ্যা n(E) = 32
বাংলায় পড়ে এমন শিক্ষার্থীর সংখ্যা n(B) = 28
কোনটিই পড়ে না এমন শিক্ষার্থীর সংখ্যা = 6

অন্তত একটি বিষয় পড়ে এমন শিক্ষার্থীর সংখ্যা = 60 - 6
= 54

উভয় বিষয়ে পড়ে = ইংরেজিতে পড়ে + বাংলায় পড়ে - অন্তত একটি বিষয় পড়ে
= 32 + 28 - 54
= 6  

Question: Which of the following is equivalent to the pair of inequalities 2x - 5 ≤ 7 and 3x + 4 > 10?

Solution:
Solve the first inequality,
2x - 5 ≤ 7 
⇒ 2x ≤ 7 + 5
⇒ 2x ≤ 12
∴ x ≤ 6
And,
Solve the second inequality,
3x + 4 > 10 
⇒ 3x > 10 - 4
⇒ 3x > 6
∴ x > 2

∴ We get 2 < x ≤ 6

Question: How many 4 letter words with or without meaning can be formed out of the letters of the word 'TRIANGLE', where repetition of letters is not allowed?

Solution:
Here,
'TRIANGLE' contains 8 different letters.

So the number of words = Number of arrangements of 8 letters, taking 4 at a time
= ⁸P₄
= (8 × 7 × 6 × 5)
= 1680

As it is given that food is there for 54 days for 2000 students,
But after 15 days, remaining food was sufficient for 2000 students for 39 days only ( as 54-15 = 39 )
Let x number of students came in hostel after 15 days,
=> Total number of students after 15 days = 2000 + x
=> Remaining food was enough for (2000 + x) students for 20 days
So here is indirect proportion, as more people and less food
=> 2000 : ( 2000 + x ) :: 20 : 39
=> 2000× 39 = (2000 + x )× 20
=> 3900 = 2000 + x
=> x = 3900 - 2000
=> x = 1900

1) multiples of 2 till 100 = 100/2 = 50
2) Multiples of 3 till 100 = 100/3 = 33.33 = 33
add the two 50 + 33 = 83; subtract common terms that are multiple of both 2 and 3

LCM of 2 and 3 = 6
Multiples of 6 till 100 = 100/6 = 16.66 = 16
so total multiples of 2 and 3 = 83 - 16 = 67

∴ Number of positive integers less than 100 are neither multiples of 2 or 3 = 100 - 67 = 33

Question:

Solution:

Question: The area of a rectangle and square are equal. The side of the square is 12 cm and the smaller side of the rectangle is one-third that of the square. The length of the other side of the rectangle would be-

Solution:
given that,
Side of the square = 12 cm
Smaller side of the rectangle = one-third of the square’s side = 12/3 = 4 cm
And The area of a rectangle and a square are equal.

Now,
Area of the square = 12² = 144  cm²

∴ Area of rectangle = 144cm² [The area of a rectangle and a square are equal]

Let the other side of rectangle = L
Now,
4 × L = 144
⇒ L = 144/4
∴ L = 36 cm

So the other side of the rectangle is 36 cm.

In right angled triangle ABC,
cos 45° = BC/AC
Or, 1/√2 = 15/AC [As, opposite angle of AB and BC is equal, length of both line is also equal]
∴ AC = 15√2

Therefore, the length of the ladder is 15√2 m.

Total salary of 65 workers = 65 × 5680 = Tk. 369200
Total salary of 31 workers = 31 × 2356 = Tk. 73036
Total salary of 23 workers = 23 × 4589 = Tk. 105547
No. of remaining workers = 65 - 31 – 23 = 65 – 54 = 11
Total Salary of 11 workers = 369200 – 73036 – 105547
= 369200 – 178583
= Tk. 190617
Required average = 190617/11
= Tk. 17328.81
Hence, the required average is Tk. 17328.81

There are 7 digits 1, 2, 0, 2, 4, 2, 4 in which 2 occurs 3 times, 4 occurs 2 times.
Number of 7 digit numbers = 7!3! × 2! = 420
But out of these 420 numbers,
there are some numbers which begin with '0' and they are not 7-digit numbers. The number of such numbers beginning with '0'.
= 6!3! × 2! = 60
Hence the required number of 7 digits numbers = 420 - 60 = 360

Question: If , then x = ?

Solution:

Question: A tank can be filled by pipe A in 4 hours and pipe B in 8 hours. At 2 pm pipe A was opened. At what time will the tank be filled if pipe B is opened at 3 pm?

Solution:
পাইপ A এর কাজের হার = 1/4 অংশ/ঘন্টা
পাইপ B এর কাজের হার = 1/8 অংশ/ঘন্টা

বিকাল 2টা থেকে 3টা পর্যন্ত (1 ঘন্টায়), পাইপ A একা ট্যাঙ্কটির 1/4 অংশ পূর্ণ করে।

বাকি কাজ = 1 - 1/4 = 3/4 অংশ

বিকাল 3টার পর পাইপ A ও B একসাথে কাজ করবে।
একসাথে তাদের কাজের হার = (1/4 + 1/8) অংশ/ঘন্টা
= (2 + 1)/8 অংশ/ঘন্টা
= 3/8 অংশ/ঘন্টা

বাকি 3/4 অংশ পূর্ণ করতে সময় লাগবে = (বাকি কাজ)/(একসাথে কাজের হার)
= (3/4)/(3/8) ঘণ্টা
= 3/4 × 8/3 ঘণ্টা
= 2 ঘণ্টা

সুতরাং, বিকাল 3টার পর আরও 2 ঘন্টা সময় লাগবে।
অতএব, ট্যাঙ্কটি পূর্ণ হবে বিকাল 5 টায়।

Let,
The original fraction be x/y
Let us assume that,
The value of x is 100 and the value of y is 100
Then the numerator of the resultant fraction is 250% of x,
That is {100 + (150/100) × 100}
Similarly the denominator of the resultant fraction is 450%
Given that,
(250% of x)/(450% of y)=25/51
⇒ (x/y) × (250/450) = 25/51
⇒ (x/y) = (25/51)/(250/450)
⇒ (x/y) = (25/51) × (450/250)
⇒ x/y = 15/17.

A number divisible by 4 formed using the digits 1, 2, 3, 4 and 5 has to have the last two digits 12 or 24 or 32 or 52.
In each of these cases, the five digits number can be formed using the remaining 3 digits in 3! = 6 ways.
A number divisible by 4 can be formed in 6 × 4 = 24 ways.
Total number that can be formed using the digits 1, 2, 3, 4 and 5 without repetition
= 5! = 120
Required probability,
= 24/120
= 1/5

Question: A sum of money is to be divided among P, Q, R, S in the ratio 7 : 3 : 5 : 2. If R gets Tk. 2000 more than S, what is Q's share?

Solution:
Let their shares be 7x, 3x, 5x, and 2x respectively.

ATC,
5x - 2x = 2000
⇒ 3x = 2000
⇒ x = 2000/3

Therefore,
Q's share = 3x
= 3 × (2000/3)
= 2000 taka.

Question: P can complete a work in 10 days, Q in 15 days, and R in 30 days. P stops working 2 days before the completion of the work, and Q stops 3 days before completion. R continues working alone till the end. What was the total number of days taken to complete the entire work?

Solution:
ধরি, সম্পূর্ণ কাজটি শেষ হতে মোট সময় লাগে y দিন।

∴ P কাজ করেছে (y - 2) দিন
P-এর কাজের অংশ = (y - 2)/10

Q কাজ করেছে (y - 3) দিন
Q-এর কাজের অংশ = (y - 3)/15

R পুরো y দিন কাজ করেছে, তাই তার কাজের অংশ = y/30

শর্তমতে,
(y - 2)/10 + (y - 3)/15 + y/30 = 1
⇒ {3(y - 2) + 2(y - 3) + y} / 30 = 1
⇒ 3y - 6 + 2y - 6 + y = 30
⇒ 6y - 12 = 30
⇒ 6y = 42
∴ y = 7

∴ সম্পূর্ণ কাজ সম্পন্ন করতে 7 দিন সময় লেগেছে।

Question: Which of the following numbers is not divisible by 3?

Solution:
A number is divisible by 3 if the sum of its digits is divisible by 3.
Checking each number:
729: 7 + 2 + 9 = 18, and 18/3 = 6 → divisible 
567: 5 + 6 + 7 = 18, and 18/3 = 6 → divisible 
1458: 1 + 4 + 5 + 8 = 18, and 18/3 = 6→ divisible 
1376: 1 + 3 + 7 + 6 = 17, and 17/3 = 5.66 → not fully divisible

∴ 1376 can not be divided by 3

Question: A rectangular sheet of paper, 10cm long and 8cm wide has squares of side 2cm cut from each of its corner. The sheet is then folded to form a tray of depth 2cm. What is the volume of this tray?

Solution: 
Length of tray = 10 - (2 × 2) = 10 - 4 = 6 cm.
Breadth of tray = 8 - (2 × 2) = 4 cm.
Depth of tray = 2 cm.

∴ Volume of tray = 6 × 4 × 2 = 48 cm³

We are given that,
age of mother 10 years ago was 3 times the age of her son
So,
let the age of the son be x and as the mother's age is 3 times the age of her son, let it be 3x, three years ago.
At present,
Mother's age will be (3x + 10) and son's age will be (x + 10)
After 10 years,
Mother's age will be (3x + 10) +10 and son’s age will be (x + 10) + 10

Mother's age is twice that of son
(3x + 10) +10 = 2 [(x + 10) + 10]
(3x + 20) = 2[x + 20]
Solving the equation, we get x = 20
We are asked to find the present ratio.
(3x + 10) : (x + 10) = 70 : 30
(3x + 10) : (x + 10) = 7 : 3.

Question: On a certain sum of money, the simple interest for 2 years is Tk. 300 at the rate of 6% per annum. If the same sum is invested at compound interest at the same rate for the same period, how much more interest would be earned?

Solution:
We know,
SI = (P × r × n)/100
⇒ 300 = (P × 6 × 2)/100
⇒ 300 = 12P/100
⇒ P = (300 × 100)/12
∴ P = Tk. 2500

Compound Interest = P(1 + r/100)ⁿ - P
= 2500(1 + 6/100)² - 2500
= 2500 × (1.06)² - 2500
= 2500 × 1.1236 - 2500
= 2809 - 2500
= Tk. 309

∴ Extra interest earned = 309 - 300 = Tk. 9

Man's/Boat's Speed = X
Stream/Current/River Speed = Y

∴ Downstream speed = X + Y
Upstream speed = X - Y

∴ X+Y = 4.5+1.5 = 6 km/hr and X-Y = 4.5-1.5 = 3 km/hr
Let distance be D km

Downstream time = Distance/Speed = D/6
Upstream time = D/3

Average speed = Total distance/Time taken = (D + D)/(D/6 + D/3)
= (6 × 2D)/3D
= 4 km/hr.

Question: A two-digit number is 7 times the sum of its digits. The number that is formed by reversing its digits is 36 less than original number. What is number? 

Solution:
Let the ten's digit be x and the unit's digit be y
Then, number = 10x + y
∴ 10x + y = 7(x + y)
⇔ 3x = 6y
⇔ x = 2y

Number formed by reversing the digits = 10y + x
∴ (10x + y) - (10y + x) = 36
⇒ 9x - 9y = 36
⇒ x - y = 4
⇒ 2y - y = 4
∴ y = 4

So, x = 2y = 2 × 4 = 8
Hence, the number is = 10 × 8 + 4 = 84.

According to the question,
(x + y)/2 = 6.5,
or, x + y = 13 and
xy = 36,
Now substitute the value of x and y from the option and find the answer.

Question: How many times are the hands of a clock at right angle in a day?
 
Solution: 
In 12 hours, they are at right angles 22 times.
∴ In 24 hours, they are at right angles = 2 × 22 = 44 times.

So the hands of a clock are at a right angle 44 times in a day.

The perimeter of the square = 160 m.
Side of square = (160/4) m
= 40 m.
Area of square = (40 × 40) m² = 1600 m²
Area of rectangle = (1600 - 100) m² = 1500 m²
Let the length and breadth of the rectangle be 'l' and 'b' respectively.
Then, 2(l + b) = 160
⇒ (l + b) = 80
⇒ b = 80 - l.

∴ lb = 1500
⇒ l(80 - l) = 1500
⇒ 80l - l² = 1500
⇒ l² - 80 l + 1500 = 0
⇒ (l - 50)(l - 30) = 0
⇒ l = 50. or l = 30

Hence, length = 50 m, breadth = 30 m.

Question: A geometric series has its first term as 1 divided by square root of 3, and its common ratio is √3. Which term in the sequence is 81√3?

Solution:
First term, a = 1/√3
Common ratio, r = √3
Let, the n-th term = ar^{n - 1}
⇒ (1/√3) . (√3)^{n - 1} = 81√3
⇒ (√3)^{n - 1} = 81√3 × √3
⇒ (√3)^{n - 1} = 243
⇒ (√3)^{n - 1} = (√3)¹⁰
⇒ n - 1 = 10
∴ n = 11

So, the 11th term is 81√3.

b & c - এর বর্ণগুলোকে সাজালে গঠিত শব্দ দুটি যথআক্রমে LIRAS (তুরস্কের মুদ্রা) ও MOTTO।
b- এর থেকে LIARS (মিথ্যেবাদী) শব্দটিও পাওয়া যায়।

Question: A person 1.5 meters tall sees the top of a building in a small mirror placed on the ground. The mirror is 2 meters away from the person's feet and 80 meters away from the base of the building. What is the height of the building?

Solution:

ধরি, মানুষের উচ্চতা, AB = 1.5 m
মানুষ এবং আয়নার দূরত্ব, BC = 2 m
ভবনের উচ্চতা, ED = h
ভবন এবং আয়নার দূরত্ব, CD = 80 m

আলোর প্রতিফলনের সূত্র অনুসারে, ∠ACB = ∠ECD (আপতন কোণ = প্রতিফলন কোণ)।
∴ ΔABC এবং ΔEDC সদৃশ।

সদৃশ ত্রিভুজের ধর্ম অনুসারে:
AB/ED = BC/CD
⇒ 1.5/h = 2/80
⇒ h × 2 = 1.5 × 80
⇒ 2h = 120
∴ h = 60 m

অতএব, ভবনটির উচ্চতা = 60 meters

Speed upstream = 8/(24/60)
= 20 km/hr.
Speed of the stream = 4 km/hr.
Speed of boat in still water = (20 + 4)
= 24 km/hr.

A:: - G : S = 40 : 60
B:: - G : S : C = 35 : 40 : 25
New, G : S = {(1×40) + (4×35)} : {(40×4) + (1×60)}
= 180 : 220
= 18 : 22
= 9 : 11 [Answer. ]

Let, 8 years ago,
the age of A and B was 4x and 5x respectively. Age of C = y
so, the present age of A and B is 4x + 8 and 5x + 8 respectively 

A/Q,
5x + 8 : y = 4 : 5 ------------ (1)
y - (4x + 8) = 20 ----------- (2)
Solving this equation we will get x = 8 and y = 60

So, present age of Age of A = 4×8 + 8
= 40
and present age of B = 5×8 + 8 
= 48
From equation (1),
C = {(5x + 8) × 5} ÷ 4 = 60

So, the total age of A, B & C= 40 + 48 + 60
= 148

Given that,
The speed of the train = 50.4km/hr
= 50.4 x 5/18 m/sec
= 14 m/sec
The train crosses a bike(standing object) in 25 seconds.
Then,
Length of the train = (14 x 25)m
= 350 m.
Now,
let, the length of the bridge is X m.
the train crosses the bridge in 40 seconds.
Then, (X + 350)/40 = 14
⇒ X + 350 = 14 x 40
⇒ X = 560 - 350
⇒ X = 210
Hence the bridge is 210m long.

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 diamond (including one king) and there are three more kings.
Let E = event of getting a diamond or a king
Then, n(E) = (13 + 3) = 16
∴P(E) = n(E)/(S) = 16/52 = 4/13