Bank Math

Let the speed of the water in still water = x
Given that speed of the stream = 3 kmph
Speed downstream
= (x+3) kmph
Speed upstream
= (x−3) kmph
He travels a certain distance downstream in 4 hour and come back in 6 hour.
ie, distance travelled downstream in 4 hour = distance travelled upstream in 6 hour
since distance = speed × time,
we have
(x+3)4= (x−3)6
⇒ (x+3)2= (x−3)3
⇒ 2x+6= 3x−9
⇒ x= 6+9= 15 kmph

Question: The sum of three consecutive odd integers is 44 more than the last of the numbers. What is the middle number?

Solution:
Let the odd is x
So,
2nd odd is x + 2
3rd odd is x + 4

According to the question,
Sum of the odd numbers = (x + 4) + 44
The equation,
x + ( x + 2) + (x + 4) = (x + 4) + 44
⇒ 3x + 6 = x + 48
⇒ 3x - x = 48 - 6
⇒ 2x = 42
⇒ x = 42/2
∴ x = 21
First number is 21
Second number is 23

∴ The middle number is 23

Question: In a class, 25 students play football, 15 students play cricket, and 5 students play both. 10 students play neither football nor cricket. What is the total number of students in the class?

Solution:
Let the number of students who play football = 25
Number of students who play cricket = 15
Number of students who play both football and cricket = 5
Number of students who play neither = 10

Number of students who play football or cricket:
n(F ∪ C) = n(F) + n(C) − n(F ∩ C)
n(F ∪ C) = 25 + 15 − 5 = 35

Add the students who play neither to get the total number of students:
Total students = n(F ∪ C) + neither = 35 + 10 = 45

16 years ago, let V = x years and G = 8x years.
After 8 years from now, V = (x + 16 + 8)years and G = (8x + 16 + 8) years.
∴ 8x + 24 = 3 (x + 24)
⇒ 8x - 3x = 72 - 24
⇒ 5x = 48
8 years ago, V/g = (x + 8)/(8x + 8)
= {(48/5) + 8}/{8 × (48/5) + 8}
= (48 + 40)/(384 + 40) = 88/424
= 11/53
Answer: none of these.

Question: If A = , then the trace of A is-

Solution:
দেয়া আছে,
A = 

ট্রেস (Trace): Matrix- এর Trace হলো একটি বর্গাকার ম্যাট্রিক্সের প্রধান কর্ণের সব উপাদানের যোগফল।
প্রধান কর্ণ: ম্যাট্রিক্সের উপরের বামদিকের কোণ থেকে নিচের ডানদিকের কোণ পর্যন্ত যে উপাদানগুলো থাকে, সেগুলোই প্রধান কর্ণ।

সুতরাং প্রদত্ত ম্যাট্রিক্স এর ট্রেস = 2 + 5 + 8 = 15

Question: Three friends, Titu, Joyonto, and Rajib, start running around a circular track at the same time in the same direction. Titu completes a round in 120 seconds, Joyonto in 150 seconds, and Rajib in 180 seconds, all starting at the same point. After how much time will they meet again at the starting point?

Solution:
L.C.M. of 120, 150 and 180 = 1800
So, Titu, Joyonto, and Rajib will again meet at the starting point in 1800 sec
Now, 1800 sec = 1800/60 minutes
= 30 minutes

Question: A can do a work in 20 days, and B in 30 days. They work together for 6 days. How much of the work is left?

Solution:
A 20 দিনে করতে পারে কাজটির 1 অংশ
A 1 দিনে করতে পারে কাজটির 1/20 অংশ

B 30 দিনে করতে পারে কাজটির 1 অংশ
B 1 দিনে করতে পারে কাজটির 1/30 অংশ

A ও B 1 দিনে করতে পারে কাজটির (1/20) + (1/30) অংশ
= (3 + 2)/60 অংশ
= 5/60 অংশ
= 1/12 অংশ

A ও B 6 দিনে করতে পারে কাজটির 6/12 অংশ
= 1/2 অংশ

কাজ বাকি থাকে = 1 - 1/2 অংশ
= (2 - 1)/2 অংশ
= 1/2 অংশ

Let, Male voter = x  and  Female voter = y
50% of x + 80% of y = 70% of (x+y)
⇒ 50x/100 + 80y/100 = {70(x+y)}/100
⇒ (50x + 80y)/100 = (70x + 70y)/100
⇒ 80y – 70y= 70x – 50x
⇒ 10y = 20x
⇒ x/y = 10/20 = 1/2
∴ x : y = 1 : 2

Question: Kabir sold his wallet at a loss of 4% . Had he sold it for TK 84 more, he would have gained 10%. What was the cost of the wallet?

Solution: 
Let the original price of the wallet is x TK

∴ [x × (100% - 4%)] + 84 = x × (100% + 10%)
⇒ (x × 96%) + 84 = x × 110%
⇒ (96x + 8400)/100 = 110x/100
⇒ 110x - 96x = 8400
⇒ 14x = 8400
⇒ x = 600

Question: a = 2b = 3c and abc = 36, then find the value of c.

Solution:
Given,
a = 2b = 3c
∴ a = 3c
and b = 3c/2

Now, abc = 36
⇒ 3c . (3c/2) . c = 36
⇒ 9c³/2 = 36
⇒ 9c³ = 72
⇒ c³ = 72/9
⇒ c³ = 8
⇒ c³ = 2³
∴ c = 2

Question: If θ = 45°, then find the value of (3 + cot2θ)/(3 - cot2θ). 

Solution:
(3 + cot2θ)/(3 - cot2θ)
= [3 + cot(2 × 45°)]/[3 - cot(2 × 45°)]
= (3 + cot90)°/(3 - cot90)°
= (3 + 0)/(3 - 0)
= 3/3
= 1

Question: In a code, "STONE" is written as 12345 and "FLAME" is written as 67895. How is "FLOAT" written in the same code?

Solution:
STONE is written as 12345
S = 1
T = 2
O = 3
N = 4
E = 5

FLAME is written as 67895
F = 6
L = 7
A = 8
M = 9
E = 5

So, FLOAT will be 
F = 6
L = 7
O = 3
A = 8
T = 2

∴ FLOAT = 67382

Question: Find out the wrong term: 8, 14, 26, 48, 98, 194, 386 

Solution:
Each term = (previous term × 2) - 2
Let’s check term by term.
1st term = 8
2nd = 8 × 2 - 2 = 16 - 2 = 14
3rd = 14 × 2 - 2 = 28 - 2 = 26
4th = 26 × 2 - 2 = 52 - 2 = 50 ; but the given term is 48 (wrong here)
5th = 50 × 2 - 2 = 100 - 2 = 98
6th = 98 × 2 - 2 = 196 - 2 = 194
7th = 194 × 2 - 2 = 388 - 2 = 386

The first mistake occurs at the 4th term. 26 should become 50, but it is written as 48.
So 48 is the wrong term. It should be replaced by 50.
Correct series should be 8, 14, 26, 50, 98, 194, 386

log_a10−log_a(10/a)
=log_a10−[log_a10−log_aa]
=log_a10−log_a10+log_aa
=0+1=1

Question: If 1 + tan²θ = 4 and θ < 90°, than what is the value of θ = ?

Solution:
Given that,
1 + tan²θ = 4 and θ < 90°
⇒ sec²θ = 4    ; [sec²θ = 1 + tan²θ]
⇒ (secθ)² = (2)²
⇒ secθ = 2
⇒ secθ = sec60°
⇒ θ = 60°

Total number of kings is 4 out of 52 cards.
Total number of queens is 4 out of 52 cards
Number of favorable outcomes i.e. ‘a king or a queen’ is 4 + 4 = 8 out of 52 cards.
Therefore, probability of getting ‘a king or a queen’ = Number of favorable outcomes
∴ P(E) = Total number of possible outcomes = 8/52 = 2/13

Question: In an examination, 65% students passed in Mathematics and 60% students in English, 40% passed in both these subjects. If 90 students failed in Mathematics and English both, then what is the total number of students?

Solution: 
Given that, 
P(M) = 65%
P(E) = 60%
P(M ∩ E) = 40%

We know,
P(M U E) = P(M) + P(E) - P(M ∩ E) 
= 65% + 60% - 40% = 85%
∴ P(M U E) = 85%
∴ passed students = 85%
∴ Failed students = 100% - 85% = 15%

Now,
We are given that 90 students failed in both subjects, which represents 15% of the total students. Let N be the total number of students.
⇒ 15% of N = 90 
⇒ (15/100)N = 90
⇒ N = (90 × 100)/15
∴ N = 600

∴ The total number of students is 600.

Question: The greatest value of sin⁴θ + cos⁴θ is?

Solution:
We know,
sin²θ + cos²θ = 1
⇒ (sin²θ + cos²θ)² = 1²    ; Squaring both sides
⇒ sin⁴θ + cos⁴θ + 2sin²θ.cos²θ = 1 
⇒ sin⁴θ + cos⁴θ = 1 - 2sin²θ.cos²θ
⇒ sin⁴θ + cos⁴θ = 1 - 0    ; [Put θ = 90° , cos90° = 0]
∴ sin⁴θ + cos⁴θ = 1 

So the greatest value of sin⁴θ + cos⁴θ is 1.

Given,
(a - 1/a) = 2
∴ a³ - 1/a³ = (a - 1/a)³ + 3.a.1/a(a - 1/a)
= 2³ + 3.2
= 8 + 6 = 14

Question: Fresh apples contain 85% water, while dry apples contain 15% water. If the weight of dry apples is 300 kg, what was the total weight of the apples when fresh?

Solution:
Given that, 
Dry apples contain 15% water
∴ they contain 85% solid matter. 
Fresh apples contain 85% water
∴ they contain 15% solid matter.

The amount of solid matter (pulp) never changes during drying.

Let the weight of fresh apples = x kg

Solid matter in fresh apples = 15% of x = 0.15x kg  
Solid matter in dry apples = 85% of 300 kg = (85/100)× 300 = 255 kg

Since the solid matter remains the same, 
0.15x = 255  
⇒ x = 255 /0.15  
⇒ x = (255 × 100)/15 
∴ x = 1700 kg

The total weight of the apples when fresh was 1700 kg.

Work done by P in 1 day = 1/18
Work done by Q in 1 day = 1/15
Work done by Q in 10 days = 10/15
= 2/3
Remaining work = 1 - 2/3
= 1/3
Number of days in which P can finish the remaining work = (1/3)/(1/18)
= 6.

Question: A and B are two positive integers such that AB = 72. Which of the following cannot be the value of A + B?

Solution:
Factor pairs of 72:
(1, 72) → A + B = 73
(2, 36) → A + B = 38
(3, 24) → A + B = 27
(4, 18) → A + B = 22
(6, 12) → A + B = 18
(8, 9) → A + B = 17

So, possible values of A + B are: 73, 38, 27, 22, 18, 17.

Among the options, 20 is not possible.

Question: If sin 17° = (x/y), then sec 17° is equal to

Solution:
Given,
sin 17° = (x/y)

We know,
sin² θ + cos² θ = 1
∴ cos θ = √(1 - sin θ)

∴ cos 17° = √(1 - sin²17°)
= √{1 - (x²/y²)}
= √{(y² - x²)/y²}
= {√(y² - x²)}/y

∴ sec 17° = 1/cos 17°
= 1/[{√(y² - x²)}/y]
= y/{√(y² - x²)}

Let, the present age of Mira is x
so her sons' present age is x/30
18 year later, the age of Mira will x+18 and the age of her son will (x/30)+18

ATQ,
x + 18 = 3{(x/30)+18}
or, x + 18= (3x/30) + 3.18 = (x/10) + 54
or, 10x + 180 = x + 540 [Multiplying both sides by 10]
or, 10x - x = 540 - 180
or, 9x = 360
or, x = 360/9
or, x = 40
So, The present age of Mira is 40.

Question: A, B and C start together from the same place to walk round a circular path of length 12 km. A walks at the rate of 4 km/h, B 3 km/h and C 3/2 km/h. They will meet together at the starting place at the end of-

Solution:
Given that,
Circumference of circular path = 12 km
Speeds, A = 4 km/h
B = 3 km/h
C = 3/2 = 1.5 km/h

We know, 
Time = Distance/Speed

Than, 
Time for A = 12/4 = 3 hours
Time for B = 12/3 = 4 hours
Time for C = 12/1.5 = 8 hours

∴ Required time,
= LCM of 3, 4, 8.
= 24 hours.

∴ They will meet together at the starting place at the end of 24 hours.

Total age of grandparents = 67 × 2
Total age of parents = 35 × 2
Total age of grandchildren = 6 × 3

Total age of all persons = 67 × 2 + 35 × 2 + 6 × 3 = 222
Total number of persons = 2 + 2 + 3 = 7

The average age of the family
= 222/7
= 31(5/7)

From 1 to 100 = 100/3 = 33.33 ≈ 33 integers are divisible by 3
From 1 to 100 = 100/15 =  6.67 ≈ 6 integers are divisible by both 3 and 5

So, 33 - 6 = 27 integers are divisible by 3 but not by 5

Question: What is the slope of a line perpendicular to the line whose equation is 10x - y = 3?
(Officer Cash 2022 অনুযায়ী)

Solution:
সরল রেখার সাধারণ সমীকরণ,
y = mx + c ......(1) (এখানেm = ঢাল)

যদি কোনো রেখার ঢাল হয় m, তবে তার লম্ব (perpendicular) রেখার ঢাল হবে,
m' = - (1/m)

এখন,
10x - y = 3
y = 10x + 3
(1) নং এর সাথে তুলনা করে পাই,
m = 10

∴ লম্ব (perpendicular) রেখার ঢাল হবে, m' = - (1/10)

Question: A train 150 meters long takes 40 seconds to cross a 350-meter-long bridge. How much time will the train take to cross a 250-meter-long platform?

Solution:
Length of train = 150 m
Length of bridge = 350 m
∴ Total distance to cross bridge = 150 + 350 = 500 m
Time taken = 40 seconds
∴ Speed of train = Total distance/Time
= 500/40 = 12.5 m/s

Length of platform = 250 m
∴ Total distance to cross platform = 150 + 250 = 400 m

∴ Time taken = Total distance/Speed
= 400/12.5 seconds
= 32 seconds

In 25% profit,
If selling price is $125 then cost is $100
So, when selling price is $35 then cost = (100×35)/125 = $28
In 15% profit,
If cost is $100 then selling price = $115
So, when cost is $28, then selling price = (115×28)/100 = $32.20

Question: In how many ways can a leap year have 53 Mondays?

Solution:
In a leap year there are 366 days
 52 weeks + 2 extra days.
So to have 53 Mondays one of these two days must be a Monday.
This can occur in only 2 ways.
 (Sunday and Monday) or (Monday and Tuesday).
Thus number of ways = 2

Question: Find the maximum distance between two points on the perimeter of a rectangular garden whose length and breadth are 24 m and 7 m.

Solution:
একটি আয়তক্ষেত্রের পরিসীমার উপর অবস্থিত দুটি বিন্দুর মধ্যে সর্বাধিক দূরত্ব হলো এর কর্ণের দৈর্ঘ্য। কর্ণের দৈর্ঘ্য পিথাগোরাসের সূত্র ব্যবহার করে নির্ণয় করা যায়।

কর্ণের দৈর্ঘ্য = √(দৈর্ঘ্য^২ + প্রস্থ^২)
= √(24² + 7²)
= √(576 + 49)
= √625
= 25 মিটার

সুতরাং, দুটি বিন্দুর মধ্যে সর্বাধিক দূরত্ব হলো 25 মিটার।

Question: Pointing to a man in a photograph, Mitu said, "His brother's father is the only son of my grandfather." 
How is Mitu related to the man in the photograph?

Solution:
মিতু বলল: "His brother's father is the only son of my grandfather."

"His brother's father" = ছবির লোকটির বাবা
ছবির লোকটির বাবা = মিতুর দাদার একমাত্র ছেলে
ছবির লোকটির বাবা = মিতুর বাবা

অর্থাৎ মিতু এবং ছবির লোকটি ভাই-বোন।
মিতু মেয়ে হওয়ায় সে ছবির লোকটির বোন।

According to the given,
x(x + 1) = 360
⇒ x² + x = 360
⇒x² + x - 360 = 0
⇒x² + 18x - 17x - 360 = 0
⇒(x +18)(x - 17) = 0
x ≠ - 18,
∴ x = 17

One positive interger 17.
Other positive interger 17 + 1 = 18.

x² + x - 360 = 0 is the required quardratic equation. 

Question: In a geometric progression, the 4^th term is 16 and the 7^th term is 128. Find the 10^th term.

Solution:
Let the first term = a
Common ratio = r
We know,
n-term = ar^{n - 1}

Then,
4^th term, ar³ = 16  ........(1)  
7th term, ar⁶ = 128  ........(2)

Now, divide equation (2) by equation (1) then we get,
(ar⁶)/(ar³) = 128/16  
⇒ r³ = 8  
⇒ r³ = 2³  
∴ r = 2  

Then substitute r = 2 into equation (1) 
a.(2)³ = 16   
⇒ a × 8 = 16  
∴ a = 2

Now, 10^th term
= ar⁹  
= 2 × 2⁹ 
= 2 × 2⁹  
= 2¹⁰ 
= 1024

∴ The 10th term is 1024

Question: A circle and a rectangle have the same perimeter. The sides of the rectangle are 16 cm and 50 cm. What is the area of the circle?

Solution:
আয়তক্ষেত্রটির পরিসীমা = 2 × (16 + 50) সেমি
= 2 × 66 সেমি
= 132 সেমি।

যেহেতু বৃত্তের পরিধি ও আয়তক্ষেত্রের পরিসীমা সমান, তাই বৃত্তের পরিধিও 132 সেমি।

∴ বৃত্তের পরিধি, C = 2πr
⇒ 2πr = 132
⇒ r = 132/(2 × 22/7)
∴ r = 21 সেমি।

বৃত্তের ক্ষেত্রফল, A = πr²
= (22/7) × (21)²
= (22/7) × 441
= 1386 বর্গ সেমি।

সুতরাং, বৃত্তটির ক্ষেত্রফল হলো 1386 বর্গ সেমি।

Question: Four people are running around a circular ground from a point on the circumference at 8:00 am. For one round, these four persons take respectively 40, 50, 60 and 30 minutes. At what time will they meet together again ?

Solution:
L.C.M. of 40, 50, 60 and 30
= 600 minutes
= 10 hours
So, they meet again 10 hours after they start.
They meet together again = 8:00 am + 10 hours
= 6:00 pm

So they will meet together again at 6:00 PM.

Question: In an election, 4% of the total votes cast are invalid. Of the valid votes, one candidate secures 55% and wins the election by a margin of 4,800 votes. Find the total number of votes cast.

Solution:
বিজয়ী প্রার্থী ভোট পায়= 55%
বৈধ ভোট = (100 - 4)% = 96%

বিজয়ী প্রার্থী বৈধ ভোটের শতকরা পায় = (55 × 96)/100
= 52.8%

পরাজিত প্রার্থী ভোট পায় = (96 - 52.8)%
= 43.2%

বিজয়ী প্রার্থী ও পরাজিত প্রার্থীর ভোটের পার্থক্য = (52.8 - 43.2)%
= 9.6%

প্রশ্নমতে
 9.6% = 4800
 1% = 4800/9.6
 100% = (4800 × 100)/9.6
= 50000

Question: When we reverse the digits of the number 14, the increases by 27. How many other two digit numbers increases by 27 when their digits are reversed?

Solution: 
Let the numbers = (10x + y),
When the digits of the number are reversed the number becomes (10y + x)

According to question,
(10y + x) - (10x + y) = 27
Or, 9(y - x) = 27
∴ y - x = 3

Possible numbers are = (25, 36, 47, 58, 69)
Total other two digits possible numbers are 5

Question: A cistern 8m long and 5m wide contains water up to a depth of 1m 50cm. The total area of the wet surface is:

Solution:
দেওয়া আছে,
চৌবাচ্চার দৈর্ঘ্য (l) = 8 m, প্রস্থ (b) = 5 m, এবং গভীরতা (h) = 1 m 50 cm = 1.5 m

ভেজা পৃষ্ঠের (wet surface) ক্ষেত্রফল নির্ণয়ের জন্য আমরা সমগ্র পৃষ্ঠের ক্ষেত্রফল থেকে উপরের তলের ক্ষেত্রফল বাদ দেবো।

আমরা জানি,
সমগ্র পৃষ্ঠের ক্ষেত্রফল = 2(lb + bh + lh) বর্গ একক

∴ ভেজা পৃষ্ঠের ক্ষেত্রফল = 2(lb + bh + lh) - lb বর্গ একক
= 2{(8 × 5) + (5 × 1.5) + (8 × 1.5)} - (8 × 5)
= 2(40 + 7.5 + 12) - 40
= 2(59.5) - 40
= 119 - 40
= 79 m²

সুতরাং, ভেজা পৃষ্ঠের (wet surface) মোট ক্ষেত্রফল 79 বর্গ মিটার।

Question: The area of a rectangle R with a width of 4 feet is equal to the area of a square S, which has a perimeter of 32 feet. The perimeter of the rectangle R is:

Solution:
ধরি, আয়তক্ষেত্র R এর দৈর্ঘ্য এবং প্রস্থ যথাক্রমে l, b.
বর্গের এক বাহু = a

প্রশ্নমতে,
4a = 32
∴ a = 8

এখানে,
আয়তক্ষেত্রের ক্ষেত্রফল = বর্গের ক্ষেত্রফল 
∴ l × b = a²
 ⇒ l = a²/b
= 64/4
= 16

∴ আয়তক্ষেত্রের পরিসীমা = 2(l + b)
= 2(16 + 4)
= 40 feet

Question: Water flows through a cylindrical pipe of an internal diameter of 7cm at the rate of 5m/s. The time, in minutes, the pipe would take to fill an empty rectangular tank of 4m × 3m × 2.31m is - 

Solution: 
the total volume of the tank is = 400 × 300 × 231 cc
total water flow per second through the pipe is = πr²h
= (22/7) × (7/2)²× 500

∴ total time = (400 × 300 × 231)/{(22/7) × (7/2)²× 500}
= (400 × 300 × 231 × 4 × 7)/(22 × 49 × 500)
= 1440 s 
= 24 min

এখানে,
28 টি জুতা = 14 জোড়া জুতা 
∴ আবরণ (case) প্রয়োজন হবে = 112/14 = 8 টি

Suppose B takes t hours to walk d km.
Then, A takes (t + 2) hours to walk d km.
With double speed, A will take = 1/2(t+2) hours.
∴ t − 1/2(t+2) = 1
⇒ {2t − (t+2)}/2 = 1
⇒ 2t − (t+2) = 2
⇒ t = 4
∴ B takes 4 hours to walk d km.

Question: Find the value of cosec(- π/6)

Solution:
cosec(- π/6)
= - cosec(π/6)
= - 1/sin(π/6)
= - 1/sin30°
= - 1/(1/2)
= - 2