Bank Math

Speed of the train relative to Person
= (125/25) m/s.
= 5 m/s.

∴ 5 × (18/5) km/hr
= 18 km/hr

Let the speed of the train be x km/hr.
then, relative speed = (x - 8) km/hr.

So, (x - 8) = 18
⇒ x = 26 km/hr.

Let P = 100,
A = 200,
It means I = 100

Now We want A = 300,
means I = 200
Years for I of Tk. 100 = 10
Years for I of Tk. 200 = (10/100) × 200 = 20 Years

Alternative Method:
Assuming simple interest, let the amount be P and rate of interest per annum be R%.

Given that P becomes 2P in 10 years.
=> P + PX10X R/100 = 2P
=> 1 + R/10 = 2
=> R = 10

Let P triple itself in N years:
=> P + PXNX 10/100 = 3P
=> 1 + N/10 = 3
=> N = 20 years

Question: A bag contains 5 green, 3 yellow, and 2 black balls. If one ball is drawn at random, what is the probability that it will not be a yellow ball?

Solution:
Given that,
Green balls = 5
Yellow balls = 3
Black balls = 2

∴ Total balls = 5 + 3 + 2 = 10
∴ Number of non-yellow balls = Green + Black = 5 + 2 = 7

We know,
Probability(not yellow) = favorable outcomes/total outcomes
= 7/10

∴ The probability that the ball drawn is not yellow is 7/10.

Question: If y + (3/y) = 5, what is the value of y³ + (27/y³)?

Solution:
দেওয়া আছে y + 3/y = 5

∴ y³ + 27/y³ = (y + 3/y)³ - 3 × (y + 3/y) × 3
= 5³ - 9 × 5
= 125 - 45
= 80

Question: Which number replaces the question mark?

Solution:
এখানে,
13 - 8 = 5
22 - 13 = 9
9 - 5 = 4
37 - 22 = 15
15 - 9 = 6
6 - 4 = 2

সুতরাং, প্রশ্নবোধক চিহ্নের স্থানে 2 বসবে।

Question: The average income of A, B and C is Tk. 12000 per month and average income of B, C and D is Tk. 15000 per month. If the average salary of D be twice that of A, then the average salary of B and C is in Tk. 

Solution:
Given that,
Average income of A, B, C = 12,000
∴ Total income of A + B + C = 3 × 12000 = 36000 … (1)
Average income of B, C, D = 15,000
 ∴ Total income of B + C + D = 3 × 15000 = 45000 … (2)
And, D = 2A ....(3)

Now, Subtract equation (1) from (2) than we get,
B + C + D - (A + B + C) = 45000 - 36000
⇒ D - A = 9000
⇒ 2A - A = 9000   ; [from (3)]
∴ A = 9000

Now put a = 9,000 in (i)
9000 + B + C = 36000
⇒ B + C = 36000 - 9000 = 27000
∴ B + C = 27000

∴ Average salary of B and C = (B + C)/2 = 27000/2 = 13500

So the average salary of B and C is Tk. 13500.

Question:

Solution:

Question: Find the equation of the vertical line passing through the point (- 3, 5).

Solution:

একটি উল্লম্ব রেখা (vertical line) হলো এমন একটি সরলরেখা যা Y-অক্ষের সমান্তরাল। এই ধরনের রেখার একটি বিশেষ বৈশিষ্ট্য হলো, রেখার উপর অবস্থিত প্রতিটি বিন্দুর x-স্থানাঙ্ক (x-coordinate) একই থাকে, কিন্তু y-স্থানাঙ্ক (y-coordinate) পরিবর্তিত হতে পারে।

উল্লম্ব রেখার সাধারণ সমীকরণ হলো: x = a, যেখানে a একটি ধ্রুবক সংখ্যা এবং রেখার প্রতিটি বিন্দুর x এর মান একই থাকে।

প্রশ্নে বলা হয়েছে রেখাটি (- 3, 5) বিন্দুর মধ্য দিয়ে যায়।
 যেহেতু এই বিন্দুর x-স্থানাঙ্ক হলো - 3,
সুতরাং রেখাটির সমীকরণ হবে: x = - 3

Question: The area of the floor of a museum is 500 square meters. How many unbroken tiles of dimension 10 x 20 cm² will be required to cover the floor completely?

Solution: 
Area of the floor = 500 square meters

Area of one tile = 10 × 20 cm²
=(10/100) × (20/100) m²
= .10 × .20 m²
= 0.02 m²

Therefore,
Tiles required = 500/.02
= 500/(2/100)
= 250 × 100
= 25,000 tiles

১ম 6 মাসে মুনাফা = 5000 × 6/12 × 7/100 = 175
২য় 6 মাসে মুনাফা = 5175 × 6/12 × 7/100 = 362.25/2 = 181.125
∴ বছর শেষে মোট টাকার পরিমান = 5000 + 175 + 181.125 = 5356.125 ≈ 5356

Question: What percent of 10 kg is 50 grams?

Solution:
Required Percentage = {(50gm/10kg) × 100}%
= {(50/10000) × 100}%  [1kg = 1000gm]
= (5000/10000)%
= 0.5%

Question: Arrange the words given below in a meaningful sequence.
1. Crime
2. punishment
3. Police
4. Justice
5. Judgement

Solution:
Crime → 1
Police → 3
Justice → 4
Judgement → 5
punishment → 2

so, meaningful sequence is 1, 3, 4, 5, 2

Marked Price = 2000
SP after first Discount of 20% = 2000 - 20% of 2000 = 1600
SP after second Discount of 10% = 1600 - 10% of 1600 = 1440
Now, the final selling price at cash = 1440 - 5% of 1440 = Tk. 1368

Question: What is the weight of 1 cubic meter of water?

Solution: 
We know, 
The weight of 1 cubic meter of water is 1000 kg.

S = {1, 2, 3,............, 49, 50}
E = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 7, 14, 21, 35, 42, 49}
n(S) = 50
n(E) = 18

P(E) = n(E)/n(S)
= 18/50
= 9/25.

Here, the order of arrangement of digits does matter.
ⁿP_r = n!/(n-r)!
ⁿP_r = 4!/(4-3)!
⁴P₃ = 4!/1!
⁴P₃ = 4!
⁴P₃ = 24

Question: If 3^2x . 9^{x + 1} = 27^{x - 1}, what is the value of x?

Solution:
3^2x . 9^{x + 1} = 27^{x - 1}
⇒ 3^2x . (3²)^{x + 1} = (3³)^{x - 1}
⇒ 3^2x . 3^{2(x + 1)} = 3^{3(x - 1)}
⇒ 3^2x . 3^{2x + 2} = 3^{3x - 3}
⇒ 3^{2x + 2x + 2} = 3^{3x - 3}
⇒ 3^{4x + 2} = 3^{3x - 3}
⇒ 4x + 2 = 3x - 3
⇒ 4x - 3x = - 3 - 2
∴ x = - 5

Question: If x > 2 and x < 5, then which of the following expressions is positive?
I. (x - 2)(x - 5)
II. (2 - x)(x - 5)
III. (2 - x)(5 - x)

Solution:
Given,
x > 2 and x < 5

For expression I: (x - 2)(x - 5)
Since x > 2, (x - 2) will be positive.
Since x < 5, (x - 5) will be negative.
(x - 2)(x - 5) = positive × negative = negative

For expression II: (2 - x)(x - 5)
Since x > 2, (2 - x) will be negative.
Since x < 5, (x - 5) will be negative.
(2 - x)(x - 5) = negative × negative = positive

For expression III: (2 - x)(5 - x)
Since x > 2, (2 - x) will be negative.
Since x < 5, (5 - x) will be positive.
(2 - x)(5 - x) = negative × positive = negative

∴ Only expression II is positive.

Let geese be denoted by 'G' and Dogs by 'D'
Geese have 2 legs; Dogs have 4 legs.

Total Heads = G + D = 84 ------------------------- (1)
Total Legs = 2G + 4D = 282 --------------------- (2)

Divide equation 2 by 2, we get,
G + 2D = 141 -------------------------------------- (3)
Equation 3 - Equation 2
G + 2D - G - D = 141 - 84
∴ D = 57

So, Geese = 84 - 57 = 27

Question: If x² + y² = 40 and xy = 12, what is the value of (x - y)²?
 
Solution:
We are given:
x² + y² = 40
xy = 12

Use the identity:
(x - y)² = x² + y² - 2xy

Substitute the values:
⇒ (x - y)² = x² + y² - 2xy
⇒ (x - y)² = 40 - 2 × 12
⇒ (x - y)² = 40 - 24
∴ (x - y)² = 16

Question: A 60-meter cable is attached from the top of a vertical pole down to the ground. If the cable makes an angle of 30 degrees with the ground, find the height of the pole.

Solution:
 
Let,
Height, AB = h

Given that, 
AC = 60m                    
∠ACB = 30°

∴ sin30°= AB/AC
⇒ 1/2 = h/60
⇒ h = 60 × (1/2)
∴ h = 30 m

So the height of the pole is 30 meters

15 দিনের কাজের মোট বেতন (15 × 90) = 1350 টাকা
১ম 7 দিন ও শেষ 7 দিনের কাজের মোট বেতন (7×87 + 7×92) = 609 + 644 = 1253 টাকা।
∴ অষ্টম দিনের কাজের বেতন = (1350 - 1253) = 97 টাকা

Pump A can empty the pool in 3 hours therefore the rate at which it empties is 1/3 pool/hour
Pump b can empty the pool in 2 hours therefore the rate at which it empties is 1/2 pool/hour.
If they work together, the resulting rate is the addition of both rates (1/3 +1/2)pool/hour = 5/6 pool/hour

Now we have the following: (5/6pool)/60min = 1pool/x
Or, x = 72minutes

Question: In how many ways can 5 books be selected from 12 books, with 2 particular books always left out?

Solution:
মোট বস্তু, n = 12
সর্বদা বাদ বা বর্জন থাকবে, m = 2
এবং প্রতিবার নিতে হবে, r = 5

∴ সমাবেশ = ^{(n - m)}C_r = ^{(12 - 2)}C₅
= ¹⁰C₅
= 10!/[5!(10 - 5)!]
= (10 × 9 × 8 × 7 × 6)/(5 × 4 × 3 × 2 × 1) 
= 252

Let, Breadth = x 
So, length = x + 15% of x = x + 3x/20 = 23x/20
ATQ,
23x/20 × x = 460
Or, x² = (460×20)/23 = 400
Or, x = 20
So, length = (23×20)/20 = 23 m 

Weight of 7 nets = (7 × 0.2) gm = 1.4 gm
And, Weight of 12 poles = (12 × 1.1) gm = 13.2 gm
Total weight of net and pole = (1.4 + 13.2) gm = 14.6 gm
∴ Empty box's weight = (16.25 – 14.6) = 1.65 gm

Question: What is the total interest on Tk 2,000 at 12.5% per annum for 9 months (in taka)?

সমাধান:
আসল, P = 2,000 টাকা
সুদের হার, r = 12.5% = 12.5/100 = 1/8
সময়, n = 9 মাস = 9/12 = 3/4 বছর

সুদ, I = Pnr
⇒ সুদ, I = 2,000 × (3/4) × (1/8)
∴ সুদ, I = 187.5 টাকা

Question: Solve the inequality |1 - 2x| < 7

Solution:
Given that, 
|1 - 2x| < 7
⇒ - 7 < 1 - 2x < 7
⇒ - 7 - 1 < 1 - 1 - 2x < 7 - 1
⇒ - 8 < - 2x < 6
⇒ - 4 < - x < 3    (dividing by - 2 and reversing the inequality signs)
⇒ 4 > x > - 3
∴ - 3 < x < 4

Question: 


Solution:

Question: 2, 8, 14, 20, 26, 32, 38, ... choose which pair of numbers comes next?

Solution: 
This is an arithmetic sequence where each term increases by 6.
Check the differences, 
8 - 2 = 6
14 - 8 = 6
20 - 14 = 6
26 - 20 = 6
32 - 26 = 6
38 - 32 = 6
So the pattern is very clear that add 6 each time.

Next two numbers after 38 is-
38 + 6 = 44, 44 + 6 = 50

Therefore, the next pair is 44, 50.

Question: A machine is sold at a profit of 20%. Had it been sold for Tk. 60 less, there would have been a loss of 20%. What was the cost price?

Solution:
ধরি, মেশিনটির ক্রয়মূল্য = x টাকা

20% লাভে বিক্রয়মূল্য = x + x এর 20%
= x + 20x/100
= 12x/10

20% ক্ষতিতে বিক্রয়মূল্য = x - x এর 20%
= x - 20x/100
= 8x/10

প্রশ্নমতে,
(12x/10) - (8x/10)= 60
⇒ 4x/10 = 60
⇒ 4x = 60 × 10
⇒ x = (60 × 10)/4
∴ x = 150

∴ মেশিনটির ক্রয়মূল্য = 150 টাকা

Ratio of times taken by A and B = 100:130 = 10:13
Suppose B takes x days to do the work.
Then,
10:13 :: 23:x
⇒ x = (23 × 13)/10
⇒ x = 299/10
A's 1 day's work = 1/23
B's 1 day's work = 10/299
(A + B)'s 1day's work = 1/23 + 10/299
= 23/299
= 1/13
∴ A and B together can complete the work in 13 days.

Question: A sum of TK. 800 amounts to TK. 920 in 3 years at simple interest. What will it amount to if the rate of interest is increased by 3%?

Solution:
Principal (P) = 800 TK.
Amount after 3 years = 920 TK.
∴ Simple Interest (SI) for 3 years = 920 - 800 = 120 TK.

Now, SI = (P × n × r)/100
⇒ 120 = (800 × 3 × r)/100
⇒ 120 = 24r
⇒ r = 120/24
∴ r = 5%

New Interest Rate = 5% + 3% = 8%
New simple interest for 3 years = (800 × 3 × 8) / 100 = 192 TK.
New Amount = Principal + New Interest = 800 + 192 = 992 TK.

Alternative Shortcut Method:
Additional Interest for 3% increase in 3 years = (800 × 3 × 3)/100 = 72 TK.
∴ New Amount = 920 + 72 = 992 TK.

Clearly,
n(S) = 6 × 6
= 36.
Let,
E be the event that the sum of the numbers on the two faces is divisible by 4 or 6.
Then, E = {(1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (5, 1), (5, 3), (6, 2), (6, 6)}
∴ n(E) = n(E)/n(S)
= 14/36
= 7/18.