ব্যাংক নিয়োগ বিষয়ভিত্তিক প্রস্তুতি

Question: If two pens and three notebooks cost Tk. 1,200. and three pens and two notebooks cost Tk. 1,300. Than how much does one pen cost?

Solution:
Let x represent the cost of one pen and y represent the cost of one notebook.

Now,
2x + 3y = 1200 .....(1)
3x + 2y = 1300 .....(2)

Multiply equation (1) by 2,
⇒ 4x + 6y = 2400

Multiply equation (2) by 3,
⇒ 9x + 6y = 3900

Now subtract,
(4x + 6y) - (9x + 6y) = 2400 - 3900
⇒ -5x = -1500
⇒ x = 300

So the cost of one pen is Tk. 300

Question: If x = y = 3z and xyz = 576 than x =?

Solution:
Given that,
x = y = 3z
Now,
xyz = 576
⇒ x × x × (x/3) = 576
⇒ x³/3 = 576
⇒ x³ = 576 × 3
⇒ x³ = 1728
⇒ x³ = 12³
∴ x = 12

Question: One-fifth of Rahim’s investment in Mutual Funds is equal to one-third of his investment in Gold. If his total investment is Tk. 80,000. How much did he invest in Mutual Funds?

Solution:
Let,
m = Investment in Mutual Funds
g = Investment in Gold

Given that,
(1​/5)m = (1/3​)g
g = (3/5)m ...... (1)

And
⇒ m + g = 80,000
⇒ m +  (3/5)m = 80000
⇒ (5m + 3m)/5 = 80000
⇒ 8m/5 = 80000
⇒ m = (80000 × 5)/8
∴ m = 50000

So Rahim invested Tk. 50,000 in Mutual Funds.

Question: A farmer harvested a crop of apples. 15% of them were damaged during transportation. He sold 70% of the remaining good apples to a market. If he has 255 apples left, how many apples did he originally harvest?

Solution:
Let the original number of apples be x

ATQ,
⇒ x × 85% × 30% = 255
⇒ (x × 85 × 30)/(100 × 100) = 255
⇒ x = (255 × 100 × 100)/(85 × 30)
⇒ x = (255 × 1000)/255
∴ x = 1000

So the farmer originally harvested 1,000 apples.

Question: The sum of the squares of three number is 83, while the sum of their products taken two at a time is 71. Their sum is-

Solution:
Given that,
Sum of squares, a² + b² + c² = 83
Sum of products two at a time, ab + bc + ca =71

We know that,
(a + b + c)² = a² + b² +c² + 2(ab + bc + ca)
⇒ (a + b + c)²= 83 + (2 × 71)
⇒ (a + b + c)²= 83 + 142
⇒ (a + b + c)²= 225
⇒ (a + b + c)² = 225
⇒ a + b + c = √225
∴ a + b + c = 15

Question: The sum of the ages of a mother and daughter is 56 years. Four years ago, the mother was five times as old as her daughter. Find the present age of the daughter is?

Solution:
Let daughter's present age = x years
Then mother's present age = (56 - x) years

Four years ago:
Daughter's age = x - 4
Mother's age = (56 - x) - 4 = 52 - x

ATQ,
⇒ 52 - x = 5(x - 4)
⇒ 52 - x = 5x - 20
⇒ 52 + 20 = 5x + x
⇒ 72 = 6x
∴ x = 12

So the daughter's present age is 12 years.

Question: The monthly incomes of two workers are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves TK. 100 per month, find their monthly incomes.

Solution:
Let,
Incomes be 3x and 4x
Expenditures be 5y and 7y

Savings Equations,
3x - 5y = 100 .......(1)
4x - 7y = 100 ........(2)
Now,
(1) × 4 ⇒ 12x - 20y = 400
(2) × 3 ⇒ 12x - 21y = 300

Now subtract ⇒ (12x - 20y) - (12x - 21y) = 400 - 300
⇒ 12x - 20y - 12x + 21y = 100
⇒ y = 100

From (1),
3x - 5(100) = 100
⇒ 3x = 600
∴ x = 200

∴ The monthly incomes are,
First worker = 3x = (3 × 200) = 600
Second worker = 4x = (4 × 200) = 800

Question: Simple interest on a certain sum at the rate of 5.5% p.a. for 4 years and 6 years differs by BDT 220. The sum is?

Solution:
Given that,
Rate of interest, r = 5.5%
Difference in simple interest for 6 years and 4 years = 220
Time difference, n = 6 - 4 = 2 years

We know that,
I = (P × r × n)/100
⇒ 220 = (P × 5.5 × 2)/100
⇒ 11 × P = 22000
⇒ P = 22000/11
∴ P = 2000
∴ The sum is Tk. 2000.

Question: The product of three consecutive even integers is 480. Find their sum.

Solution:
Let the three consecutive even integers are,
x - 2, x, x + 2

Now their product is,
⇒ (x - 2)(x)(x + 2) = 480
⇒ x(x² - 4) = 480
⇒ x³ - 4x = 480

Now try even integer values,
x = 6 than (6)³ - (4 × 6) = 216 - 24 = 192 not valid
x = 8 than (8)3 - (4 × 8) = 512 - 32 = 480 is valid
So,
x = 8
Then the three even integers are,
x - 2 = 6, x = 8, x + 2 = 10

Now their sum is = 6 + 8 + 10 = 24

Question: The average daily wage of 10 workers is Tk. 500. If the lowest wage is Tk. 400, then what is the possible maximum wage?

Solution:
The average wage of 10 workers is Tk. 500.
So total wages = Average × Number of Workers = 500 × 10 = 5000
To find the maximum possible wage of one worker, we must minimize the wages of the other 9 workers.
Let the 10th worker earn the maximum possible wage (M).

Now,
Total Wages = Wages of 9 Workers + Maximum Wage (M)
⇒ 5000 = (9 × 400) + M
⇒ 5000 = 3600 + M
⇒ M = 5000 - 3600
∴ M = 1400

So the maximum possible wage among the workers is Tk. 1400.

Question: After fillings the car's fuel tank, a driver drove from A to B and then to C. He used 2/3 portion of the fuel driving from A to B. If he used another 7 liters to drive from B to C and still had 1/4 of the tank left, how many liters does the tank hold?

Solution:
Let full capacity x liters
Fuel used from B to C = x - {(2x/3) + (1x/4)}
= (12x - 8x - 3x)/12
= x/12

Now,
x/12 of capacity = 7 liters
∴ x of capacity = 7 × 12 = 84 liters

Question: Two trains are running in the same direction on parallel tracks. The first train is 200 meters long and is moving at a speed of 15 meters per second. The second train is 250 meters long and is moving at 10 meters per second. How much time will the faster train take to completely overtake the slower one?

Solution:
Given that,
Length of faster train = 200 m
Length of slower train = 250 m
Speed of faster train = 15 m/s
Speed of slower train = 10 m/s

Since both trains are moving in the same direction, their relative speed
= (15 - 10)m/s
= 5 m/s

To completely overtake, the faster train must cover the length of both trains
= 200 + 250 = 450 m

We know that,
Time = Distance/Speed = 450​/5 = 90 seconds
So the faster train will take 90 seconds to overtake the slower one.

Question: A shopkeeper sells two items for Tk. 6,000 each, neither gaining nor losing in the overall transaction. If he sold one item at a gain of 20%, what is the cost price of the other item?

Solution:
প্রথম পণ্যের ক্রয়মূল্য-
ধরি, ক্রয়মূল্য = ১০০ টাকা
২০% লাভে বিক্রয়মূল্য = ১০০ + ২০ = ১২০ টাকা

বিক্রয়মূল্য ১২০ টাকা হলে ক্রয়মূল্য = ১০০ টাকা
বিক্রয়মূল্য ১ টাকা হলে ক্রয়মূল্য = ১০০/১২০ টাকা
বিক্রয়মূল্য ৬০০০ টাকা হলে ক্রয়মূল্য = (১০০ × ৬০০০)/১২০ = ৫০০০ টাকা

আবার,
মোট বিক্রয়মূল্য = ৬০০০ + ৬০০০ = ১২০০০
মোট ক্রয়মূল্য = মোট বিক্রয়মূল্য = ১২০০০ (কারণ কোনো লাভ বা ক্ষতি হয়নি)

∴ অন্য পণ্যের ক্রয়মূল্য = ১২০০০ - ৫০০০ = ৭০০০ টাকা

Question: The wheel of scooter has diameter 70 cm. How many revolutions per minute must the wheel make so that the speed of the scooter is kept at 26.4 km per hour?

Solution:
Distance travelled by wheel in one revolution = circumference of wheel = (22/7) × 70 = 220cm

Speed of scooter = 26.4km/hr = (26.4 × 1000 × 100)/60 = 44000cm/min

∴ The wheel has therefore got to travel 44000 cm in 1 min i.e. it has to perform 44000/220 revolution in 1min = 200 revolutions.

Question:
 

Solution:

Question: If 2log₄(x) = 1 + log₄(x - 1) find the value of x?

Solution:
⇒ 2log₄(x) = 1 + log₄(x - 1)
⇒ log₄(x²) = log₄4 + log₄(x - 1)
⇒ x² = 4(x - 1)
⇒ x² - 4x + 4 = 0
⇒ (x - 2)² = 0
⇒ x - 2 = 0
∴ x = 2

Question: M takes 10 days more than N to complete a task. If they work together, they finish it in 12 days. How long does N take alone?

Solution:
Let N's time = x days.
Then, M's time = x + 10 days.

ATQ,
⇒ (1/x) + {1/(x + 10)} = 1/12
⇒ (x + 10 + x)/x(x + 10) = 1/12
⇒ 12(2x + 10) = x(x + 10)
⇒ 24x + 120 = x² + 10x
⇒ x² - 14x - 120 = 0
⇒ x² - 20x + 6x - 120 = 0
⇒ x(x - 20) + 6(x - 20) = 0
⇒ (x - 20)(x + 6) = 0
Now,
x - 20 = 0
∴ x = 20
And
x + 6 = 0
∴ x = - 6  ;[since time cannot be negative]

∴ N takes 20 days to complete the task alone.

Question: A man rows downstream 54 km and upstream 42 km, taking 6 hours each time. The speed of the man is?

Solution:
Speed of upstream = 42/6 = 7 km/hr
Speed of downstream = 54/6 = 9 km/hr

Speed of man in still water = (7 + 9)/2 = 16/2 = 8 km/hr

Question: Tamim has a certain average for 12 innings. In the 13th inning, he scores 120 runs, increasing his average by 5 runs. What is his new average?

Solution:
Let Tamim's average be x for 12 innings.
So, Tamim scored 12x run in 12 innings.
In the 13th inning, he scored 120 runs then the average became (x + 5) .
And he scored (x + 5) × 13 runs in 13 innings.
Now,
⇒ 12x + 120 = 13(x + 5)
⇒ 12x + 120 = 13x + 65
⇒ x = 120 - 65
∴ x = 55

∴ New average = (x + 5) = 55 + 5 = 60 runs

Question: In a 50-liter mixture of milk and water, the ratio of milk to water is 4 : 1. How much more water must be added to change the ratio to 2 : 3?

Solution:
Given that,
Milk : Water = 4 : 1
Total mixture = 50 liters

Milk = (50 of 4/5) = 40 liters
Water = (50 of 1/5) = 10 liters
Let x = additional water to be added.
New water = 10 + x liters

ATQ,
⇒ 40 : 10 + x = 2 : 3
⇒ 40/(10 + x) = 2/3
⇒ 20 + 2x = 120
⇒ 2x = 120 - 20
⇒ 2x = 100
⇒ x = 100/2
∴ x = 50
∴ 50 liters of water must be added to achieve the 2 : 3 ratio.

Question: Find the number of divisors of 360.

Solution:
Prime Factorization,
360 = 36 × 10 = (9 × 4) × (2 × 5) = 3² × 2² × 2 × 5 = 2³ × 3² × 5¹

Apply Divisor Formula:
(3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24

∴ 360 has 24 divisors.

Question: In triangle ABC, sides AB = AC, and angle ∠C = 35^∘. Find the measure of angle ∠A.

Solution:
Since AB = AC, triangle ABC is isosceles with the equal sides being AB and AC. In an isosceles triangle, the angles opposite the equal sides are equal. Therefore ∠B =∠C
Given that ∠C = 35^∘, it follows that, ∠B = 35^∘
The sum of the interior angles in any triangle is 180^∘. Therefore,
⇒ ∠A + ∠B + ∠C = 180^∘
⇒ ∠A + 35^∘ + 35^∘ = 180^∘
⇒ ∠A = (180 - 70)°
∴ ∠A = 110°

Question: If (16)^{2x + 3} = (4)^{3x + 6}, than x = ?

Sulotion:
Given that,
⇒ (16)^{2x + 3} = (4)^{3x + 6}
⇒ (2⁴)^{2x + 3} = (2²)^{3x + 6}
⇒ (2)^{8x + 12} = (2)^{6x + 12}
⇒ 8x + 12 = 6x + 12
⇒ 8x - 6x = 12 - 12
⇒ 2x = 0
∴ x = 0

Question: A Company employs 20 persons, each working 42 hours a week. If 5 persons are absent, how many hours a week would the rest of the persons have to work to make up the time lost?

Solution:
একটি কোম্পানি ২০ জন কর্মচারী নিয়োগ দেয়। প্রত্যেকে ৪২ ঘণ্টা কাজ করে।

∴ মোট কাজ হয় = (২০ × ৪২)
= ৮৪০ ঘণ্টা
৫ জন অনুপস্থিত থাকলে, বাকি থাকে = ২০ - ৫ = ১৫ জন

∴ প্রত্যেকের কাজ করতে হবে ৮৪০/১৫ ঘণ্টা
= ৫৬ ঘণ্টা

Question: A craft store has ribbons of lengths 36 cm, 48 cm, and 60 cm. What is the longest possible length that can evenly cut all ribbons without waste?

Solution:
Prime factorization of
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
60 = 2 × 2 × 3 × 5

∴ HCF = 2 × 2 × 3 = 12

So, the longest ribbon length that can evenly cut all three without waste is 12 cm.

Question: A cylindrical water tank has a radius of 35 inches and a height of 120 inches. Calculate the total surface area.

Solution:
Water tank is cylindrical in nature.
Total Surface Area of a cylinder is given by, 2πr(h + r)

∴ Total Surface Area = 2 × (22/7) × 35(120 + 35)
= 2 × 22 × 5 × 155
= 34100

∴ Total Surface Area = 34100 sq. inches.

Question: Solve the following equation:

Solution:

Question: A shopkeeper buys 100 pencils for 500 taka. How many pencils should he sell for 400 taka to make a 25% profit?

Solution:
Cost price of 100 pencils = Tk. 500
Cost price per pencil = 500/100 = Tk. 5

Since the seller wants 25% profit,
The selling price per pencil will be = 5 × (1 + 25/100) = (5 × 1.25) = 6.25 tk

Now, the number of pencils the seller can sell = 400/6.25 = 64

So the shopkeeper should sell 64 pencils for 400 taka to make a 25% profit.

Question: Dhaka and Khulna apart from each other 760 km. A train starts from Dhaka at 9 am and travels towards Khulna at speed 100 km/h. Another train starts from Khulna at 11 am and travels towards Dhaka at speed 40 km/h. At what time both will meet?

Solution:
Total distance between Dhaka and Khulna = 760km
A travels 2 hour before B so it travels = 100 × 2 = 200km

Now the remaining distance Dhaka and Khulna = 760 - 200 = 560km

∴ Relative speed = (100 + 40)km/h = 140km / h

∴ Time = Distance/Speed = 560/140 = 4 hour

Therefore, the trains will meet at = 11 am + 4 hour = 3 : 00 pm

Question: The sum of ages of 5 siblings born at 2 years intervals is 60 years. What is the age of the elder sibling?

Solution:
Let youngest age = x
Ages = x, x + 2, x + 4, x + 6, x + 8

ATQ,
⇒ x + x + 2 + x + 4 + x + 6 + x + 8 = 60
⇒ 5x + 20 = 60
⇒ 5x = 60 - 20
⇒ 5x = 40
⇒ x = 40/5
∴ x = 8

∴ elder sibling = 8 + 8 = 16 years.

Question: Find the difference between compound and simple interest for Tk. 8,000 invested at 10% per annum for 3 years.

Solution:
Given that,
Principal, P = Tk. 8,000
Rate, r = 10% = 10/100 = 0.1 
Time, n = 3 years

We know that,
Simple Interest = Prn/100
= (8000 × 10 × 3)/100
= 2400
And
Compound Interest = P(1 + r)ⁿ - P
= 8000(1 + 0.1)³ - 8000
= 8000(1.1)³ - 8000
= 10648 - 8000
= 2648

∴ Difference = 2648 - 2400 = 248
The difference between compound and simple profit is Tk. 248

Question: In an arithmetic sequence, the 4th term is 20 and the 10th term is 44. Find the first term is-

Solution:
Given that,
4th term = 20
10th term = 44

We use general formula for the nth term of an arithmetic sequence, T_n = a + (n - 1)d
Now,
From 4th term, T₄ = a + 3d =20 .........(1)
From10th term, T₁₀ = a + 9d = 44.......(2)

Subtract (1) from (2),
(a + 9d) - (a + 3d) = 44 - 20
⇒ 6d = 24
⇒d = 24/6
∴ d = 4
Put value the of d = 4 into equation (1)
⇒ a + 3 × 4 = 20
⇒ a + 12 = 20
⇒ a = 20 - 12
∴ a = 8
So the first term is 8

Question: If x and y are both odd numbers, which of the following must be ab even number?

Solution:
ধরি,
x = 3 এবং y = 5
ক) xy + 4 = (3 × 5) + 4 = 19 ; যা বিজোড়
খ) x + y = 3 + 5 = 8  ; যা জোড়
গ) x + y + 1 = 3 + 5 + 1 = 9  ; যা বিজোড়

∴ x + y সব সময় জোড়।

Question: Pipe X can fill a cistern in 2 hours, and Pipe Y can empty it in 3 hours. If both pipes are opened at 8 : 15 A. M., when will the cistern be full?

Solution:
Given that,
Pipe X can fill the cistern in 2 hours
Pipe Y can empty the cistern in 3 hours
Both pipes are opened at 8:15 A.M.

Let the total capacity of the cistern = 1 unit
Pipe X fills 1/2 unit/hour
Pipe Y empties 1/3 unit/hour

∴ Net rate = (1/2) - (1/3) = (3 - 2)/6 = 1/6 unit/hour

∴ Time to fill the cistern = 1/Net rate = 1/(1/6) = 6 hour

If both pipes are opened at 8:15 A.M., then the cistern will be full at = 8 : 15 A.M. + 6 hours
= 2 : 15 P. M.

Question: A train running at a speed of 72 km/hr crosses a platform double its length in 1 minute. What is the length of the platform in metres?

Solution:
Speed of the train = 72 km/h
The platform is double the length of the train
Time to cross the platform = 1 minute = 60 seconds

∴ Speed = (72 × 1000)/(60 × 60) = 20 m/s

Let, the length of the train is x
Then, the length of the platform = 2x

∴ Total distance to cross the platform = x + 2x = 3x

We know,
Distance = Speed × Time
⇒ 3x = 20 × 60 =1200
⇒ 3x = 1200
⇒ x = 1200/3
∴ x = 400

So, length of the platform = 2x = 2 × 400 = 800 metres.

Question: Find the value of x, if log(x + 1) + log(x - 1) = 3log2

Solution:
Given,
⇒ log(x + 1) + log(x - 1) = 3log2
⇒ log{(x + 1)(x - 1)} = log2³
⇒ log(x² - 1) = log8
⇒ x² - 1 = 8
⇒ x² = 8 + 1
⇒ x² = 9
⇒ x = ± √9
⇒ x = ± 3
∴ x = 3 ; [Only positive value]

Question: If 3x + 2y = 8 and 2x - y = 3. Find the value of 3x - 4.

Solution:
Given that,
3x + 2y = 8 .......... (1)
2x - y = 3 ............(2)

Now,
(1) + (2) × 2 ⇒ 3x + 2y + 2(2x - y) = 8 + 6
⇒ 3x + 2y + 4x - 2y = 14
⇒ 7x = 14
⇒ x = 14/7
∴ x = 2
Now puting the value of x = 2 into equation = 3x - 4
= 3(2) - 4
= 6 - 4
= 2

Question: A and B invest in a business in the ratio 3 : 2. If 5% of the total profit goes to charity and A's
share is Taka 855, the total profit is?

Solution:
Given that,
A and B invest in a business in the ratio = 3 : 2
5% of the total profit goes to charity
A’s share of the remaining profit = Tk. 855
Now,
Let the total profit = Tk. x
5% of x goes to charity, so the remaining profit = 95% of x = 95x/100
A’s share = (3/5)(95x/100) = 57x/100

ATQ,
⇒ 57x/100 = 855
⇒ x = (855 × 100)/57
⇒ x = 15 × 100
∴ x = 1500

∴ Total profit = Tk. 1500