Bank Math Master

Question: A number is 5 more than its one-sixth part. The number is-

Solution:
Given that,
A number is 5 more than its one-sixth part.
Let the number be x.

According to the question,
x = (1/6)x + 5
⇒ x - (1/6)x = 5
⇒ (6x - x)/6 = 5
⇒ 5x/6 = 5
⇒ x = 5 × (6/5)
∴ x = 6

So the number is 6.

Question: Due to a 10% reduction in the price of tea, a dealer can purchase 25 kg more tea for TK. 22500. Find the reduced price per kg.

Solution: 
Let the original price per kg be x.
After a 10% reduction, New price per kg = x - 10% of x = 0.9x
Total money = TK. 22500

According to the question, the dealer can buy 25 kg more tea after the reduction.
So,
⇒ 22500/0.9x = (22500/x) + 25
⇒ 25000/x = (22500/x) + 25
⇒ (25000/x) - (22500/x) = 25
⇒ (25000 - 22500)/x = 25
⇒ 2500/x = 25
⇒ x = 2500/25
∴ x = 100 

So, original price per kg = Tk. 100

∴ Reduced price per kg = 0.9 × 100 = Tk. 90

So the reduced price per kg of tea is Tk. 90.

Question: The average age of P, Q, R, S and T is 32 years. The average age of P and Q is 28 years, and the average age of R and S is 35 years. What is the age of T?

Solution:
Given that, 
There are 5 people P, Q, R, S, T
Average age of all five = 32 years
Total age of all five = 5 × 32 = 160 years

And, 
Average age of P and Q = 28 years
So, age of (P + Q) = 2 × 28 = 56 years

Average age of R and S = 35 years
So, age of (R + S) = 2 × 35 = 70 years

∴ Now, total age of (P + Q + R + S) = 56 + 70 = 126 years

Age of T = Total age of all five - Age of (P + Q + R + S)
= 160 - 126
= 34 years

So the age of T is 34 years.

Question: A train having length 150 m passes a platform of 550 m length. The time taken for it is 56 seconds. In how much time will this train take to pass a platform of 250 m length?

Solution:
We know,
When a train passes a platform, the total distance to be covered is,
Length of train + Length of platform

Case 1:
Distance covered = 150 m + 550 m = 700 m
Time taken = 56 seconds
∴ Speed of the train = Distance/Time
= 700/56
= 12.5 m/s

Case 2: 
Length of new platform = 250 m
∴ Total distance to be covered now = 150 + 250 = 400 m
Time taken to pass the new platform = Distance/Speed
= 400m/12.5 m/s
= 32 seconds

So the train will take 32 seconds to pass the platform of 250 m length.

Question: A room has a length of 10 m, width of 6 m, and height of 4 m. What is the area of the four walls of the room?

Solution:
Given that, 
Length of the room = 10 m
Width = 6 m
Height = 4 m

The area of the four walls = 2 × (length × height) + 2 × (width × height)
= 2 × (10 × 4) + 2 × (6 × 4)
= 2 × 40 + 2 × 24
= (80 + 48)
= 128 m²

So the area of the four walls is 128 m².

Question: What will come at the place of question mark?
3, 7, 23, 95, ?

Solution: 
1st number = 3
2nd number = 3 × 2 + 1 = 7
3rd number = 7 × 3 + 2 = 23
4th number = 23 × 4 + 3 = 95
5th number = 95 × 5 + 4 = 479

So the next number is 479.

Question: The greatest number that divides 86, 182, and 366 leaving the same remainder of 6 in each case is:

Solution:
Given that,
Numbers are 86, 182, 366
And remainder is 6

Now, numbers leaving remainder 6:
86 - 6 = 80
182 - 6 = 176
366 - 6 = 360

HCF of 80, 176, and 360:
Prime factorization:
80 = 2⁴ × 5
176 = 2⁴ × 11
360 = 2³ × 3² × 5
∴ Common factor = 2³ = 8

So the greatest number is 8.

Question: If √(2ⁿ) = 64, then the value of n is-

Solution: 
Given that, 
√(2ⁿ) = 64
⇒ (2ⁿ)^1/2 = 2⁶
⇒ 2^n/2 = 2⁶
⇒ n/2 = 6
⇒ n = 2 × 6
∴ n = 12

Question: In a selection process, the ratio of applicants who were selected to those who were rejected is 10 : 4. If 420 applicants were rejected, what was the total number of applicants?

Solution:
Given ratio,
Selected : Rejected = 10 : 4
This means for every 10 selected applicants, 4 are rejected.

Let, Number of selected applicants = 10k
Number of rejected applicants = 4k

According to the question,
Rejected applicants = 420
So, 4k = 420
⇒ k = 420/4
∴ k = 105

∴ Selected applicants = 10k = 10 × 105 = 1050
∴ Rejected applicants = 4k = 420 (given)

∴ Total number of applicants = Selected + Rejected
= 1050 + 420
= 1470

So the total number of applicants was 1470.

Question: If θ = 60°, then what is the value of (1 - tan²θ)/(1 + tan²θ)?

Solution: 
Given that, 
θ = 60°

Now, 
(1 - tan²θ)/(1 + tan²θ)
= {1 - (tan60°)²}/{1 + (tan60°)²}
= {1 - (√3)²}/{1 + (√3)²}
= (1 - 3)/(1 + 3)
= (- 2)/4
= - 1/2

Question: Rita invested equal amounts at 5% and 7% simple interest for 4 years. If she received total interest of Tk. 1920, what was the amount invested in each scheme?

Solution:
Let the amount invested in each scheme = x taka
So, same amount at 5% and same amount at 7%

Interest at 5% for 4 years, 
I₁ = (x × 5 × 4)/100
= 20x/100
= x/5
= 0.20x

And, Interest at 7% for 4 years,
I₂ = (x × 7 × 4)/100
= 28x/100
= 0.28x

Total interest received = I₁ + I₂
0.20x + 0.28x = 1920
⇒ 0.48x = 1920
⇒ x = 1920/0.48
⇒ x = 1920 × (100/48)
⇒ x = 1920 × (25/12)
⇒ x = 160 × 25
∴ x = 4000

So, the amount invested in each scheme Tk. 4000

Question: If f(y) = y³ + ky² - 4y - 8 , then for what value of k will f(- 2) = 0 ?

Solution: 
Given that, 
f(y) = y³ + ky² - 4y - 8 
f(- 2) = (- 2)³ + k(- 2)² - 4(- 2) - 8
= - 8 + 4k + 8 - 8
∴ f(- 2) = 4k - 8

Set f(- 2) = 0
⇒ 4k - 8 = 0
⇒ 4k = 8
⇒ k = 8/4
∴ k = 2

So the value of k is 2.

Question: Find the equation of the line with x-intercept = 6 and y-intercept = 2.

Solution: 
x- intercept = 6, so, the line passes through (6, 0)
y- intercept = 2, so, the line passes through (0, 2)

We know, the intercept form of a line is:
(x/a) + (y/b) = 1, where a = x- intercept and b = y- intercept  
⇒ (x/6) + (y/2) = 1
⇒ (x + 3y)/6 = 1
⇒ x + 3y = 6
∴ x + 3y - 6 = 0

so, the equation of the line is  x + 3y - 6 = 0.

Question: In an isosceles triangle, each of the equal sides is 16 cm long and the included angle between the equal sides is 30°. What is the area of the triangle?

Solution:
Given that,
The included angle between the equal sides, θ = 30°
Each of the equal sides of the isosceles triangle, a = b = 16 cm

We know that,
Area of the triangle = (1/2) × a × b × sin θ
= (1/2) × 16 × 16 × sin 30°
= (1/2) × 16 × 16 × (1/2)
= 8 × 8
= 64 square cm

∴ The area of the triangle is 64 cm².

Question: In how many ways can 6 books be selected from 10 books such that 2 particular books are always excluded?

Solution:
Since 2 specific books must always be excluded, we are effectively choosing from the remaining books.
Remaining books = 10 - 2 = 8 books

Now, we need to select 6 books from these 8 books.
The number of ways to do so is given by the combination formula = ⁸C₆
= 8!/(6! × (8 - 6)!)
= 8!/(6! × 2!)
= (8 × 7 × 6!)/(6! × 2 × 1)
= (8 × 7)/2
= 56/2
= 28

So, the total number of ways 28. 

Question: What is the angle between the hour and minute hands of a clock when it is 4 : 20?

Solution: 
We know,
The angle between the hour and minute hands is,
Angle = |11M - 60H|/2
= |(11 × 20) - (60 × 4)|/2
= |220 - 240|/2
= |- 20|/2
= 20/2
= 10°

So the angle between the hour and minute hands at 4 : 20 is 10°.

Question: Express the following inequality using absolute value notation:
- 18 < x < - 6

Solution:
Given: - 18 < x < - 6
The midpoint (average) of - 18 and - 6 is,
Midpoint = {- 18 + (- 6)}/2
= - 24/2
= - 12

Now add 12 to all parts of the inequality to center it at zero.
- 18 + 12 < x + 12 < - 6 + 12
⇒ - 6 < x + 12 < 6
This is equivalent to |x + 12| < 6

Question: A cistern is filled by Pipe A and Pipe B together in 2 hours. Pipe A alone can fill the cistern at the rate of 100 litres per hour. Pipe B alone can fill the cistern in 4 hours. What is the capacity of the cistern?

Solution:
Let the capacity of the cistern = x litres.

Pipe A fills at 100 litres per hour
∴ Time taken by A alone = x/100 hours
Pipe B alone fills the cistern in 4 hours
∴ Pipe B's rate = x/4 litres per hour
And Combined rate of A + B = 100 + (x/4) litres per hour

They together fill the cistern in 2 hours, so:
Combined rate = x/2 litres per hour

Therefore, 100 + (x/4) = x/2
⇒ (x/2) - (x/4) = 100
⇒ (2x - x)/4 = 100
⇒ x = 4 × 100
∴ x = 400

So the capacity of the cistern is 400 litres.

Question: What is the length of a chord that is 6 cm away from the center of a circle with a radius of 10 cm?

Solution:
Given that,
Radius of the circle, r = 10 cm
Distance from the center of the circle to the chord, d = 6 cm

Then, the length of the chord = 2 × √(radius² - distance from center²)
= 2 × √(10² - 6²) cm
= 2 × √(100 - 36)
= 2 × √64
= 2 × 8
= 16 cm

So the length of the chord is 16 cm.

Question: A bag contains 6 blue marbles, 4 green marbles, and 5 black marbles. If one marble is drawn at random from the bag, what is the probability that the marble is green?

Solution:
Total number of marbles = 6 + 4 + 5 = 15
Number of green marbles = 4

Therefore, P(green) = Number of green marbles/Total marbles
= 4/15

So the probability is 4/15.

Question: If a + b + c = 15 and a² + b² + c² = 77, what is the value of ab + bc + ca = ?

Solution:
Given that, 
a + b + c = 15 and a² + b² + c² = 77

We know,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ 15² = 77 + 2(ab + bc + ca) ; [Substitute the given values]
⇒ 225 = 77 + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = 225 - 77
⇒ 2(ab + bc + ca) = 148
⇒ ab + bc + ca = 148/2
∴ ab + bc + ca = 74

Question: The second term of a geometric progression is 10 and the fifth term is 80. What is the common ratio?

Solution: 
Let the first term = a
Common ratio = r

We know,
n term of geometric progression = ar^{n - 1}
∴ Second term = ar = 10
And fifth term   = ar⁴ = 80

Now, divide the second equation by the first. Then we get,
(ar⁴)/(ar) = 80/10
⇒ r³ = 8
⇒ r³ = 2³
∴ r = 2

So the common ratio is 2.

Question: Rice and wheat are in a mixture in the ratio 5 : 3. If 16 kg wheat is added to this mixture, the ratio of rice to wheat changes to 5 : 7. How much wheat is in new mixture?

Solution: 
Let the initial amount of rice = 5x kg
Initial amount of wheat = 3x kg

After adding 16 kg wheat,
Rice remains = 5x kg
And wheat becomes = 3x + 16 kg

New ratio of rice : wheat = 5 : 7
So,
⇒ 5x/(3x + 16) = 5/7
⇒ 7 × 5x = 5 × (3x + 16)
⇒ 35x = 15x + 80
⇒ 35x - 15x = 80
⇒ 20x = 80
⇒ x = 80/20
∴ x = 4

Now, wheat in the new mixture = 3x + 16
= 3 × 4 + 16
= 12 + 16
= 28 kg

So, 28 kg of wheat is in the new mixture.

Question: If the length of the face diagonal of a cube is 7√2 meters, what is the volume of the cube?

Solution:
Given that, 
The length of the face diagonal of the cube is 7√2 meters.

Let the edge length of the cube be a meters.
∴ The face diagonal of the cube = a√2 meters

According to the given condition:
a√2 = 7√2
∴ a = 7
Therefore, the volume of the cube = a³ = 7³ = 343 cubic meters

∴ The volume of the cube is 343 m³.

Question: A fruit seller buys lemons at 2 for Tk. 1 and sells them at 5 for Tk. 3. What is his gain percent?

Solution:
Cost price,
2 lemons = Tk. 1
So, cost price of 1 lemon = 1/2

Selling price, 
5 lemons = Tk. 3
So, selling price of 1 lemon = 3/5

Profit per lemon = SP - CP
= (3/5) - (1/2)
= (6 - 5)/10
= 1/10

Gain percent = (Profit/CP) × 100%
= (1/10)/(1/2) × 100%
= (1/10) × (2/1) × 100%
= (2/10) × 100%
= 100/5%
= 20%

So his gain percent is 20%.

Question: If 26 August of a certain year was Tuesday, what day was 29 September of that year?

Solution:
From 26 August to 29 September,
= 6 + 29 days
= 35 days

The same weekday repeats after every 7 days (or multiples of 7).

So, the 36th day would be Tuesday again.
Therefore, the 35th day (29 September) was Monday.

Question: Solve for x;  log_x3 + log_x9 + log_x27 + log_x81 = 10.

Solution: 
Given that, 
log_x3 + log_x9 + log_x27 + log_x81 = 10
⇒ log_x(3 × 9 × 27 × 81) = 10
⇒ log_x(3¹ × 3² × 3³ × 3⁴) = 10
⇒ log_x(3¹⁰) = 10
⇒ 10 log_x3 = 10
⇒ log_x3 = 10/10
⇒ log_x3 = 1
⇒ x¹ = 3
∴ x = 3

Question: Pavel travels 96 km at a speed of 16 km/hr using a bike, 124 km at 31 km/h by car and another 105 km at 7 km/h in horse cart. Find his average speed for the entire distance travelled.

Solution:
We know,
Average speed = Total distance/Total time

Total distance = 96 + 124 + 105 = 325 km

Now,
Bike: 96 km at 16 km/h ∴ Time = 96/16 = 6 hours
Car: 124 km at 31 km/h ∴ Time = 124/31 = 4 hours
Horse cart: 105 km at 7 km/h ∴ Time = 105/7 = 15 hours

∴ Total time = 6 + 4 + 15 = 25 hours

∴ Average speed = Total distance/Total time
= 325/25
= 13 km/h

So Pavel's average speed for the entire journey is 13 km/h.

Question: The area of the triangle whose vertices are given by the coordinates (1, 2), (- 4, -3) and (4, 1) is:

Solution:
Given that, 
Vertices of triangle = (1, 2), (- 4, -3), (4, 1)

We know,
Area of triangle = (1/2) × |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| [whose vertices are (x₁, y₁), (x₂, y₂) and (x₃, y₃)]
= (1/2) × |1(- 3 - 1) + (- 4) (1 - 2) + 4{2 - (- 3)}|
= (1/2) × |(- 4) + 4 + 20|
= 20/2
= 10 sq. units

So the area of the triangle is 10 sq. units.