Bank Math

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%

S.I.=Tk.(15500−12500)=Tk.3000
Rate=(100×3000)/(12500×4)%= 6%

Question: How many 3-digit numbers can be formed using the digits 3, 4, 5, 6 and 7 without repetition?

Solution:
যেহেতু, অঙ্কের সংখ্যা = 5টি
এদের থেকে 3 অঙ্কবিশিষ্ট সংখ্যা গঠন করতে হবে (প্রতিটি অঙ্ক একবারই ব্যবহার করা যাবে)
∴ মোট সংখ্যা = ⁵P₃
= 5!/(5 - 3)!
= 5!/2!
= (5 × 4 × 3 × 2 × 1)/(2 x 1)
= 5 × 4 × 3
= 60

Question: (p² - 7p + 10)/(p² - 8p + 15) = ?

Solution: 
(p² - 7p + 10)/(p² - 8p + 15) 
= (p² - 2p - 5p + 10)/(p² - 3p - 5p + 15) 
= {p(p - 2) - 5(p - 2)}/{p(p - 3) - 5(p - 3)} 
= (p - 2)(p - 5)/(p - 3)(p - 5) 
= (p - 2)/(p - 3)

(n + 2)(n + 4)
= (2m + 2)(2m + 4)
= 2(m + 1)2(m + 2)
= 4(m +1)(m + 2)
= 4 × (product of two consecutive positive integers, one which must be even)
= 4 × (an even number), and this equals a number that is atleast a multiple of 8.
Hence, the answer is 8.

Question: Which of the following statements is not correct?

Solution:
Option ক)
Since log_aa = 1
So, log⁡₁₀10 = 1 
This is correct.

Option খ)
log⁡(2 + 3) = log⁡(2 × 3)
Compute the left side- 2 + 3 = 5, so log⁡(2 + 3) = log⁡5.
Compute the right side- 2 × 3 = 6, so log⁡(2 × 3) = log⁡6.
Logarithm property: log⁡(a⋅b) = log⁡a + log⁡b, not log⁡(a + b).
This is incorrect.

Option গ)
Since log_a1 = 0, so log₁₀1 = 0.
This is correct.

Option ঘ)
log (1 + 2 + 3) = log 1 + log 2 + log 3
Compute the left side- 1 + 2 + 3 = 6, so log⁡(1 + 2 + 3) = log⁡6.
Right side- log⁡1 + log⁡2 + log⁡3 = log(1 × 2 × 3) = log6
Both sides are equal: log⁡6 = log⁡6
This is correct.

Option খ) is the only statement that is not correct.

Consider a rectangular solid of length l, width w and height h. Then,
Total Surface Area
= 2lw + 2lh + 2wh
= 2(lw + lh + wh)
Volume = lwh
In this case, l = 25, w = 12 m and h = 6 m and all surfaces needs to be plastered except the top.
Hence, the total area to be plastered
total surface area - an area of the top face
= 2(lw + lh + wh) - lw
= lw + 2lh + 2hw
= (25 × 12) + 2 × (25 × 6) + 2 × (12 × 6)
= 300 + 300 + 144
= 744
Cost of plastering
= 744 × 75
= 55800 Paisa
= Tk. 558

S.I. in 3 years = 6750 - 5700 = 1050
Interest every year = 1050/3 = 350
So, principal = 5700 - 2 × 350 = 5000

We know, I = pnr
Or, Interest rate, r = I/pn
= 700/(5000×2) × 100
= 7%

Given,
5 < x < 10 and y = x + 5
Possible value of x = 6, 7, 8, 9
Take greatest value of x which is 9
So, y = 9+5 = 14
∴ x + y = 9 + 14 = 23

Question: In a 500 m race, the speeds of two runners, A and B are in the ratio 5 : 6. If A is given a start of 100m, by how many meters does A win the race?

Solution:
Total race length = 500 meters.
A is given a start of 100 meters, so A runs 500 - 100 = 400 meters.

Speed ratio A : B = 5 : 6.

Let, B runs = X meter

Therefore,
400/X = 5/6
⇒ X = (6 × 400)/5
∴ X = 480m

Remaining distance for B = 500 - 480 = 20 meters.
Therefore, A wins by 20 meters.

Question: If 7m² - 16mn + 4n² is divided by 7m - 2n, the result is-

Solution:
দেওয়া আছে,
7m² - 16mn + 4n²
= 7m² - 14mn - 2mn + 4n²
= 7m(m - 2n) - 2n(m - 2n)
= (m - 2n) (7m - 2n)

∴ (m - 2n) (7m - 2n)/(7m - 2n) = (m - 2n)

Question: A man travels from A to B at a speed of 36 km/hr in 74 minutes and he travels a distance from B to C with a speed of 45 km/hr in 111 minutes. Find the average speed of whole journey.

Solution:
Given that,
A man travels from A to B at a speed of 36 km/hr in 74 minutes and he travels a distance from B to C with a speed of 45 km/hr in 111 minutes. 

We know,
Average speed = Total distance/Total time taken

Now,
Time taken = 74 min : 111 min   [given]
Ratio of Time taken = 2 : 3

∴ Average Speed = {(36 × 2) + (45 × 3)}/(2 + 3) = 207/5
 = 41.4 km/hr

∴ So the average speed of whole journey is 41.4 km/h

B's income is less than A's by
[10/(100 + 10) × 100]%
= (100/11)%
= 9(1/11)%

Question: Which one of the following numbers can be removed from the set S = {1, 2, 3, 4, 5, 6, 7} without changing the average of set S?

Solution:
Given the set is  S = {1, 2, 3, 4, 5, 6, 7}
Sum of elements  = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

There are 7 numbers in the set.
 Average = Sum of elements/Number of elements
= 28/7 = 4

And,
Removing a number that is equal to the current average will not change the average of the remaining numbers. The average of the set is 4, which is an element in the set S.

If 4 is removed, the new set is  {1, 2, 3, 5, 6, 7}
∴ New sum  = 1 + 2 + 3 + 5 + 6 + 7 = 24
And New number of elements  = 6

∴ New average = 24/6 = 4

The number that can be removed from the set S = {1, 2, 3, 4, 5, 6, 7} without changing the average of the set is 4.

Question: In a group of 150 people, 90 people read Newspaper A, 65 people read Newspaper B, and 30 people read both Newspaper A and Newspaper B. How many people read neither Newspaper A nor Newspaper B?

Solution:
মোট লোক = 150
A পত্রিকা পড়ে, n(A) = 90
B পত্রিকা পড়ে, n(B) = 65
উভয়টি পড়ে, n(A ∩ B) = 30

∴ কমপক্ষে একটি পত্রিকা পড়ে, n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
= 90 + 65 - 30
= 125

∴ যারা কোনটিই পড়ে না = মোট লোক - যারা কমপক্ষে একটি পড়ে
= 150 - 125
= 25

∴ 25 জন কোন পত্রিকাই পড়ে না।

Question: 40% of a number A is equal to 1/5 of another number B. Find the ratio A : B.

Solution:
ATQ,
40% of A = 1/5 of B
Or, (40/100) × A = (1/5) × B
Or, (2/5) A = (1/5) B
Or, 2A = B
∴ Ratio A : B = 1 : 2

Question: Working alone, Rasel can complete a certain kind of job in 8 hours. Rasel and Shima working together at their respective rates, can complete one of these jobs in 6 hours. In how many hours can Shima working alone, complete one of these jobs?

Solution:
Working together Rasel and Shima can complete the job in 6 hours
which means
(1/Rasel) + (1/Shima) = (1/6)
⇒ (1/8) + (1/Shima) = (1/6)      [Rasel can complete the job in 8 hours]
⇒ (1/Shima) = (1/6) - (1/8)
⇒ (1/Shima) = (4 - 3)/24
∴ (1/Shima) = 1/24

so, Shima working alone can complete the job in 24 hours

প্রশ্ন: The compound interest on a certain sum for 2 years at 5% per annum is Tk. 820. The simple interest on the same sum for half the time at double the rate percent per annum is:

সমাধান:
দেওয়া আছে,
চক্রবৃদ্ধি সুদ (CI) = 820 টাকা
সময় (n) = 2 বছর
সুদের হার (r) = 5%
আমরা জানি,
CI = P(1 + r/100)ⁿ - P
820 = P(1 + 5/100)² - P
⇒ 820 = P(1.05)² - P
⇒ 820 = 1.1025P - P
⇒ 820 = 0.1025P
⇒ P = 820/0.1025
⇒ P = 8000 টাকা

এখন,
আসল (P) = 8000 টাকা
সময় (n) = অর্ধেক সময় = 2 ÷ 2 = 1 বছর
সুদের হার (r) = দ্বিগুণ সুদের হার = 5% × 2 = 10%
আমরা জানি,
SI = (P × r × n)/100
= (8000 × 10 × 1)/100
= (80000)/100
= 800 টাকা
∴ নির্ণেয় সরল সুদ হলো 800 টাকা।

Question: The area of a circle is 49π cm². The circumference is equal to?

Solution:
দেওয়া আছে,
বৃত্তের ক্ষেত্রফল = 49π সেমি²

আমরা জানি,
বৃত্তের ক্ষেত্রফল = πr²

প্রশ্নমতে,
πr² = 49π
⇒ r² = 49
⇒ r = √49
∴ r = 7 সেমি

এখন, বৃত্তের পরিধি = 2πr
= 2π × 7
∴ পরিধি = 14π সেমি।

Question: What is the slope of a line parallel to the line whose equation is 3x + 4y = 12?

Solution:
প্রদত্ত সরলরেখার সমীকরণ: 3x + 4y = 12
সরলরেখার আদর্শ রূপ y = mx + c-এর সাথে তুলনা করার জন্য সমীকরণটিকে সাজাই:
3x + 4y = 12
⇒ 4y = - 3x + 12
⇒ y = (- 3/4)x + (12/4)
⇒ y = (- 3/4)x + 3

এখানে, প্রদত্ত রেখার ঢাল (m₁) = - 3/4
আমরা জানি, দুটি রেখা সমান্তরাল হলে তাদের ঢাল সমান হয় (m₁ = m₂)

∴ সমান্তরাল রেখাটির ঢাল হবে - 3/4

Question: Karim does a job 60 days quicker than Asad. If Karim works three times faster than Asad, how long will it take him to finish the job by himself?

Solution:
ধরি,
কাজটি আসাদ করে = x দিনে 
করিম করে = (x - 60) দিনে 

প্রশ্নমতে,
x - 60 = 3x
⇒ 3x - x = 60
⇒ 2x = 60
⇒ x = 60/2
⇒ x = 30

The sum of three consecutive integers can be written as n + (n + 1) + (n + 2) = 3n + 3
If the sum is 24, we need to solve the equation 3n + 3 = 24;
=> 3n = 21;
=> n = 7
The greatest of the three numbers is therefore 7 + 2 = 9

Question: How many pairs of natural numbers is there the difference of whose squares are 45.

Solution:
Let the two natural numbers be x and y where x > y.
x² - y² = 45
(x + y)(x - y) = 45

And Factor pairs of 45 Both (x + y) and (x - y)must be positive odd integers (since x and y are natural numbers), and x + y > x - y

Thus, the factors of 45 possibles are 1, 3, 5, 9, 15 and 45
Hence, numbers are 9 and 6 or 7 and 2 or 23 and 22
So, There are 3 pairs are (23, 22), (9, 6), (7, 2)

Total Age = 28 + 24 = 52
Average Age = 52/2
= 26

Let,
After T years their age will be the same as before.
ATQ,
24 + T + 28 + T + T + T = 26 × 4 = 104
Or, 4T = 52
Or, T = 52/4
Or, T = 13 years

So after 13 years their age will be the same average as before.

লাইভ পরীক্ষার প্রশ্নে 'z²' এর পরিবর্তে 'x²' এর মান জানতে চাওয়া হয়েছিল। 
প্রশ্ন অনুযায়ী সঠিক উত্তর - ঙ) None of these. 
---------------------------- 

Question: x, y and z are consecutive positive integers such that x < y < z . If the units digit of x² is 6 and the units digit of y² is 9, what is the units digit of x²?

Solution:
Given that x, y, and z are consecutive positive integers such that x < y < z, we are asked to find the units digit of z², based on the following information:
The units digit of x² is 6.
The units digit of y² is 9.

First, let's analyze the possible values of x and y by checking what numbers, when squared, have the given units digits:

For x², the units digit is 6. The numbers whose squares end in 6 are:
4² = 16(units digit is 6) and 6² = 36(units digit is 6).
Thus, x could be either 4 or 6.

For y², the units digit is 9. The numbers whose squares end in 9 are:
3² = 9 (units digit is 9) and 7² = 49(units digit is 9).
Thus, y could be either 3 or 7.

Since x < y, we will now check the possible pairs of x and y:
If x = 4, the next integer y would be 5, but the units digit of 5² is 5, not 9. So this case is not valid.
If x = 6, the next integer y would be 7, and indeed, the units digit of 7² is 9.
Thus, the valid pair is x = 6 and y = 7.

Since z is the next consecutive integer after y = 7, we have z = 8.
Now,
we compute the units digit of z²
8² = 64. The units digit of z² is 4.

Question: A merchant has three different types of milk: 324 litres, 351 litres, and 459 litres. Find the minimum number of casks of equal size that can store the milk without mixing.

Solution:
The size of each cask should be the greatest possible that divides all three quantities, i.e., H.C.F of 324, 351, and 459.

H.C.F(324, 351, 459) = 27 litres

Total milk = 324 + 351 + 459 = 1134 litres

Number of casks required = 1134 ÷ 27 = 42

If the value of the machine at the beginning of year 1 is 10,000 and then it reduces by 10%, the value of the machine at the end of year 1 is:
10,000 × (100%-10%) = 9,000
And again, at the end of Year 2, we follow a similar route:
9000 × (100% - 10%)
= 8,100

The odds numbers are x - 4, x - 2, x, x + 2, x + 4 and x + 6
therefore, x + 6 = 15
so, x = 9, hence fourth number is 11

Question: In a map, 4 cm represents 80 km. The distance between two cities is 9 cm on the map. The actual distance between the cities is - 

Solution: 
Since 4 cm = 80 km,
Actual Distance = (80/4) × 9
= 20 × 9 km
= 180 km 

Question: Which one will complete the series: CPS, EOT, GNU, IMV, — ?

Solution:
প্রতিটি অংশে তিনটি বর্ণ আছে।

১ম বর্ণগুলোতে একটি বর্ণ বাদ দিয়ে পরের বর্ণটি আসবে। C এর এক বর্ণ পর E, D এর এক বর্ণ পর F. 
অতএব, শূন্যস্থানে ১ম বর্ণটি হবে K

২য় বর্ণগুলোতে ক্রমান্বয়ে পূর্বের বর্ণটি আসবে।
P এর আগের বর্ণ O, O এর আগের বর্ণ N, অতএব শূন্যস্থানে ২য় বর্ণটি হবে L

৩য় বর্ণগুলোতে ক্রমান্বয়ে পরের বর্ণটি আসবে।
S এর পরের বর্ণ T, T এর পরের বর্ণ U, অতএব শূন্যস্থানে ৩য় বর্ণটি হবে W

∴ শূন্যস্থানে বসবে KLW

Let x = 13p + 11
and x = 17q + 9
Then,
13p + 11= 17q + 9
17q - 13p = 2
q = (2 + 13p)/17
The least value of p for which q = (2 + 13p)/17 is a whole number, is p = 26.
∴ x = (13 × 26 + 11)
= 338 + 11
= 349.

Question: A right circular cylinder has a curved surface area of 660 sq. cm and a height of 15 cm. Find the radius of the cylinder.

Solution:
দেওয়া আছে,
সিলিন্ডারের বক্রপৃষ্ঠের ক্ষেত্রফল = 660 বর্গ সেমি
এবং সিলিন্ডারের উচ্চতা (h) = 15 সেমি।

ধরা যাক, সিলিন্ডারের ব্যাসার্ধ হল r সেমি।

আমরা জানি,
সিলিন্ডারের বক্রপৃষ্ঠের ক্ষেত্রফল = 2πrh
⇒ 660 = 2 × (22/7) × r × 15
⇒ 660 = (44/7) × 15 × r
⇒ 660 = (660/7) × r
⇒ r = (660 × 7)/660
⇒ r = 7 সেমি

সুতরাং, প্রদত্ত সিলিন্ডারের ব্যাসার্ধ হল 7 সেমি।

Relative speed = ( 45 + 30 )
= 75 km/hr
= (75 × 5)/18 m/s
= 125/6 m/s

We are calculating the time taken by the slower train to pass the driver of the faster one.

Hence, distance = length of the slower train = 500 metre
Time = 500/( 125/6)
= 24 seconds

Question: P = {x ∈ N : x³ < 216}. Then, how many elements are there in set P?

Solution:
Here, N = the set of natural numbers
= {1, 2, 3, 4, 5, 6, 7, 8, ……}

Given that, 
P = {x ∈ N : x³ < 216}

Now check each value.
When x = 1, 1³ = 1 < 216
When x = 2, 2³ = 8 < 216
When x = 3, 3³ = 27 < 216
When x = 4, 4³ = 64 < 216
When x = 5, 5³ = 125 < 216
When x = 6, 6³ = 216 < 216 ; false (not true)

Therefore, the set P contains only the values that satisfy the condition.
P = {1, 2, 3, 4, 5}
∴ The number of elements in set P = 5

সরল মুনাফার ক্ষেত্রে,
I = Pnr
Or, 81 = (450 × n × 4.5)/100
Or,  n = (81 × 100) / (450 × 4.5)
= 4 বছর