Bank Math

Question: A worker earns Tk. 250 on the first day and spends Tk. 200 on the second day, earns Tk. 250 on the third day and again spends Tk. 200 on the fourth day and so on. On which day would he have had Tk. 1000?

Solution:
১ম দিনে আয় করে = ২৫০ টাকা
২য় দিনে ব্যয় করে = ২০০ টাকা
∴ প্রতি ২ দিনে জমা হয় = ২৫০ - ২০০ = ৫০ টাকা

শুধু ১ম দিনে আয় করে তার হাতে থাকে ২৫০ টাকা
তাহলে, ১০০০ - ২৫০ = ৭৫০ টাকা আরও জমা করতে হবে।

৫০ টাকা জমা হয় ২ দিনে
∴ ৭৫০ টাকা জমা হয় = (২ × ৭৫০)/৫০ = ৩০ দিনে

অর্থাৎ, ৩০ দিনের শেষে তার হাতে জমা থাকে = ৭৫০ টাকা
৩১তম দিনে সে আবার আয় করে = ২৫০ টাকা
⇒ মোট = ৭৫০ + ২৫০ = ১০০০ টাকা

∴ ৩১তম দিনে তার কাছে ১০০০ টাকা ছিল।

12 men can do a piece of work in 24 days
⇒ M₁ = 12 and D₂ = 24
8 men can do this work in D₂ days
⇒ M₂ = 8
M₁D₁ = M₂D₂
⇒ 12 × 24 = 8 × D₂
⇒ D₂ = (12 × 24)/8
= 36 days.

Question: If A and B are in the ratio 5 : 7 and B and C are in the ratio 14 : 15 then what is the ratio of A to C?

Solution: 
Given that, 
A : B = 5 : 7 and B : C = 14 : 15 

Now, 
(A/B) × (B/C) = (5/7) × (14/15)
⇒ A/C = (2/3)
∴ A : C = 2 : 3

Say both egg and apple cost $10 each per unit
An increase of 10% for the eggs would bring the new total to $11
And an increase of 2% for the apples would bring the new total to $10.2
The original total was $20 and the new total is $21.2.
Change in price = 21.2 - 20 = 1.2
∴ Percentage increased in price  = 1.2/20 × 100 = 6%.

 We know, Area of rhombus = 1/2 × x × y [Here, x and y are two diagonals of the rhombus]
Or, x = (54 × 2) / 6 = 18 cm

Let the length of the tunnel = x meter
Then, distance = (800 + x) meter.

Given,
Time = 1 minute = 60 seconds.
Speed = 78 km/hr
= 78 × (5/18)
= (65/3) m/s

According to the question,
800 + x = 60 × (65/3)
⇒ 800 + x = 1300
⇒ x = 500 meter.

So, the length of the tunnel = 500 meter

A : B : C = 7 : 8 : 9
Size of step, 4A = 5B = 6C
the ratio of speeds = 7/4 : 8/5 : 9/6
= 35 : 32 : 30.

ধরি, ঝুড়িতে কলা ছিলো 3x টি এবং আপেল ছিলো 2x টি
প্রশ্নমতে , 3x-5 / 2x = 1/1
⇒ 3x - 5 = 2x
⇒ x = 5
∴ আপেল ছিল = 2x = 2 × 5 = 10 টি

Question: Time required by two pipes A and B working separately to fill a tank is 36 seconds and 45 seconds respectively. Another pipe C can empty the tank in 30 seconds. Initially, A and B are opened and after 7 seconds, C is also opened. In how much more time the tank would be completely filled?

Solution:
Let the capacity of the tank be LCM (36, 45, 30) = 180 units
∴ Efficiency of pipe A = 180/36 = 5 units/second
Efficiency of pipe B = 180/45 = 4 units/second
Efficiency of pipe C = - 180 / 30 = - 6 units/second

Now,
for the first 7 seconds, A and B were open. 
Combined efficiency of A and B = 5 + 4 = 9 units/second 
∴ Part of the tank filled in 7 seconds = 7 × 9 = 63 units

Part of tank empty = 180 - 63 = 117 units

Now, all pipes are opened.
Combined efficiency of all pipes = 5 + 4 - 6 = 3 units/second
Therefore, more time required = 117/3 = 39 seconds.

Here,
(r × 100)/(100 + r)
= (20 × 100)/(100 + 20)
= 16.67

Question: In how many ways can a group of 3 teachers and 4 students be formed from 6 teachers and 10 students?

Solution:
We have 6 teachers and 10 students.
We need to choose 3 teachers from 6 and 4 students from 10.

∴ Number of ways = ⁶C₃ × ¹⁰C₄
= 6!/{3!(6 - 3)!)} × 10!/{4!(10 - 4)!)}
= 6!/(3! × 3!) × 10!/(4! × 6!)
= {(6 × 5 × 4)/(3 × 2 × 1)} × {(10 × 9 × 8 × 7)/(4 × 3 × 2 × 1)}
= 20 × 210
= 4200 ways

Let the number be x

ATQ,
60x/100 - 40 = 50
Or, (60x - 4000) /100 = 50
Or, 60x - 4000 = 50 ×100
Or, 60x -4000 = 5000
Or, 60x = 5000 + 4000
Or, 60x = 9000
Or, x = 9000/60
Or, x = 150
Hence, the number is 150

Question: A shopkeeper marks an article 40% above the cost price and allows a discount of 10%. What is his profit percentage?

Answer:
Cost Price 100 Taka
Then Mark Price = 140, (40% above the cost price)

10% discount
If marked price is 100, then discount is 10 taka
If marked price is 1, then discount is 10/100 taka 
If marked price is 140, then discount is 140/10 = 14 taka

Selling Price = 140 - 14 = 126,
As the Cost Price = 100
Then, the Profit percentage is = 26%

Question: A car travels a distance of 60 km in 1 hour and 40 minutes. How much faster, in kilometers per hour, on an average, must it travel to make the same trip in 20 minutes less time?

সমাধান:
দেওয়া আছে,
মোট অতিক্রান্ত দূরত্ব = 60 কিমি।
প্রাথমিক সময় = 1 ঘন্টা 40 মিনিট = 1 + 40/60 ঘন্টা = 1 + 2/3 = 5/3 ঘন্টা।

প্রাথমিক গতিবেগ = দূরত্ব/সময়
= 60 / (5/3) কিমি/ঘন্টা
= (60 × 3/5) কিমি/ঘন্টা
= 36 কিমি/ঘন্টা।

20 মিনিট কম সময় = 20/60 = 1/3 ঘন্টা কম।
∴ নতুন সময় = (প্রাথমিক সময় - কমানো সময়)
= 5/3 - 1/3 ঘন্টা
= 4/3 ঘন্টা।

∴ নতুন গতিবেগ = 60/(4/3) কিমি/ঘন্টা
= (60 × 3/4) কিমি/ঘন্টা
= 45 কিমি/ঘন্টা।

গতি বৃদ্ধি করতে হবে = (নতুন গতিবেগ - প্রাথমিক গতিবেগ)
= (45 - 36) কিমি/ঘন্টা
= 9 কিমি/ঘন্টা।

∴ গাড়িটিকে গড়ে 9 কিমি/ঘন্টা বেশি গতিতে যেতে হবে।

First 4L orange juice was removed
So now in 40L mixture, there is 40 - 4 = 36 litres orange juice and 4 litres water
Now, remove 4L mixture. While doing this proportionate amount of juice and water gets removed.
Amount of juice removed = 4 × (Juice quantity/Mixture quantity)
= 4 × (36/40)
= 3.6 Litres.

Orange Juice remaining = 36-3.6 = 32.4 Litres
Again 4L mixture removed
Amount of juice removed = 4 × (Juice quantity/Mixture quantity)
= 4 × (32.4/40)
= 3.24 Litres.

Juice remaining = 32.4 - 3.24
= 29.16 Litres.

Question: A person's present age is two-fifth of the age of his father. After 10 years, he will be one half of the age of his father. What is the present age of the father?

Solution:
Let the age of the person's father be 'x' years
∴ Age of the person = (2x/5)

After 10 years, age of the person's father = 'x + 10' years
Age of the person after 10 years = (2x/5) + 10 years

According to question,
(2x/5) + 10 = (1/2)(x + 10)
⇒ (2x/5) + 10 = (x/2) + 5
⇒ (x/2) - (2x/5) = 10 - 5
⇒ (5x - 4x)/10 = 5
⇒ x/10 = 5
⇒ x = 50

So, the present age of his father is 50 years

Question: Out of 6 persons working on a project, 2 are graduated. If 2 are selected, what is the probability that there is at least one graduate among them?

Solution:
এখানে মোট ব্যক্তি = 6 জন, গ্র্যাজুয়েট = 2 জন এবং নন-গ্র্যাজুয়েট = (6 - 2) = 4 জন।

6 জন থেকে 2 জনকে বাছাই করার মোট উপায় = ⁶C₂
= (6 × 5)/(2 × 1)
= 15 

এখন, 2 জনের মধ্যে একজনও গ্র্যাজুয়েট না থাকার (অর্থাৎ 2 জনই নন-গ্র্যাজুয়েট হওয়ার) উপায় = ⁴C₂
= (4 × 3)/(2 × 1) = 6 

∴ একজনও গ্র্যাজুয়েট না থাকার সম্ভাবনা = 6/15 = 2/5

আমরা জানি, কমপক্ষে একজন গ্র্যাজুয়েট থাকার সম্ভাবনা = 1 - (একজনও গ্র্যাজুয়েট না থাকার সম্ভাবনা)
= 1 - (2/5)
= 3/5

If the second box has each dimension three times that of the first box, then its volume is 3 × 3 × 3 = 27 times.
So, the second box contains 27 times as much the first box.

Bubli's age = 14 years
⇒ Sakib's age = (14 - 9) years
= 5 years
Let the present age of Zayed Khan be x years.
∴ (x - 10)/14 = 5
⇒ x - 10 = 70
⇒ x = 80 years.
Answer: Zayed Khan age is 80 years.

Speed of the first train = 48 km/hr.
Let the length of the first train = 2x meter.
Speed of the second train = 42 km/hr.
Let the length of the second train = x meter.
Distance = (2x + x) = 3x meter.
Time = 12 seconds
Relative speed = 48 + 42 = 90 km/hr.
= 90 × (5/18) = 25 m/s.
3x = 25 × 12
⇒ x = 100 meter.
Length of the first train = 200
Time is taken to cross the platform = 45 seconds.
Speed of first train = 48 km/hr
= 48 × (5/18)
= 40/3 m/s.
Let the length of the platform = y meter.
Distance = 200 + y meter.
⇒ 200 + y = 45 × (40/3)
⇒ 200 + y = 600
⇒ y = 400 meter.

Question: In how many ways can a team of 4 people be selected from 8 people?

Solution:
Total number of people, n = 8
Number of team members, r = 4

The number of ways to choose the team = ⁿC_r =
⁸C₄ = 8!/[4! × (8 - 4)!]
= (8 × 7 × 6 × 5 × 4!)/(4 × 3 × 2 × 1 × 4!)
= 70

Question: The ratio of two numbers is 5 : 6 and their H.C.F is 7. Their L.C.M is -

Solution:
ধরি, সংখ্যা দুটি হলো 5x এবং 6x
∴গসাগু (H.C.F) = x = 7

∴ সংখ্যা দুটি: হলো 5 × 7 = 35 এবং 6 × 7 = 42

∴ সংখ্যাদ্বয়ের গুণফল = 35 × 42 = 1470
এবং H.C.F = 7

আমরা জানি,
L.C.M = (Product of two numbers)/H.C.F
= 1470/7
= 210

∴ সংখ্যা দুটির লসাগু (L.C.M) = 210

Question: The mean weight of three club members is 42 kg. If none of them weighs less than 40 kg, what is the maximum possible weight of one of the members?

Solution:
Given,
the mean weight of three members is 42 kg
Total weight of three members = (42 × 3) kg = 126 kg

According to the question,
Minimum weight of any member = 40 kg
 So, Minimum weight of 2 members = (40 × 2) = 80 kg

∴ Maximum weight of any of three members = (126 - 80) kg = 46 kg 

Question: There are 5 red, 4 white, and 3 blue balls in a box. If 3 balls are drawn at random from the box, what is the probability that all three are red?

Solution:
Total number of balls = 5 + 4 + 3 = 12 balls
We are drawing 3 balls at random without replacement.
Total number of ways to choose 3 balls from 12
= ¹²C₃ 
= 12!/(3! × 9!)
= (12 × 11 × 10)/(3 × 2 × 1)
= 220

And
Number of favorable ways (all 3 balls are red)
= Number of ways to choose 3 red balls from 5 red balls
= ⁵C₃
= 5!/(3! × 2!)
= (5 × 4 × 3)/(3 × 2 × 1)
= 10

∴ Probability = (favorable outcomes)/(total outcomes)
= 10/220
= 1/22
So the probability that all three balls are red is 1/22.

Alternatively, using sequential probability (without replacement):
P(1st Red) = 5/12
P(2nd Red) = 4/11 (4 red left out of 11 total)
P(3rd Red) = 3/10 (3 red left out of 10 total)

∴ Total Probability = (5/12) × (4/11) × (3/10) 
= 60/1320
= 1/22. 

Question: In your wallet, there are Tk 1000, Tk 500, and Tk 200 notes in the ratio 3 : 7 : 5. The total amount of money in the wallet is Tk 22,500. Find the number of each note respectively.

Solution:
Let,
The number of Tk 1000 notes = 3x
The number of Tk 500 notes = 7x
The number of Tk 200 notes = 5x

According to the question,
(1000 × 3x) + (500 × 7x) + (200 × 5x) = 22500
⇒ 3000x + 3500x + 1000x = 22500
⇒ 7500x = 22500
⇒ x = 22500/7500
∴ x = 3

∴ Number of Tk 1000 notes = 3 × 3 = 9
∴ Number of Tk 500 notes = 7 × 3 = 21
∴ Number of Tk 200 notes = 5 × 3 = 15

Given, 624A
= 13 × 8 × 6 × A
= 13 × 2³ × 2 × 3 × A
= 13 × 2⁴ × 3 × A

So, to make the number square of an integer, required value of A is 13 × 3 = 39

Question: Find the simple interest on BDT 15,000 at 8% per annum for 5 months.

Solution:
Principal, P = 15,000 Taka
Time, n = 5 months = 5/12 years
Rate of interest, r = 8% = 8/100

Simple Interest, I = P × n × r
= 15,000 × (5/12) × (8/100)
= (15,000 × 5 × 8)/(12 × 100)
= 600,000/1200
= 500

∴ The simple interest is Tk. 500.

Let the number of hens be x and the number of cows be y.
Then, x + y = 48 .... (i)
and 2x + 4y = 140
x + 2y = 70 .... (ii)
Solving (i) and (ii) we get:
x = 26, y = 22.
The required answer = 26.

দেওয়া আছে,
sinA + sin²A = 1
⇒ sinA = 1 - sin²A
⇒ sinA = cos²A
এখন,
cos²A + Cos⁴A
= cos²A + cos²A.cos²A
= cos²A + sinA.sinA [sina = cos²A]
= Cos²A + sin²A
= 1[sin²A + cos²A = 1]

Question: A rhombus has diagonals of 10 cm and 20 cm. Find the side length of a square that has the same area as the rhombus.

Solution: 
Given that,
d₁ = 10 cm and d₂ = 20 cm
where d₁ and d₂ are the lengths of the diagonals.

We know,
The area of a rhombus is = (d₁ × d₂)/2
= (10 × 20)/2
= 200/2
= 100 cm²

Now, let the side length of the square be a cm.
So the area of the square is a².
Since the square has the same area as the rhombus.
⇒ a² = 100
⇒ a = √100
∴ a = 10 cm

So the side length of the square is 10 cm.

মোট খরচ = (80 × 40 + 40 × 50) টাকা
= (3200 + 2000) = 5200 টাকা
10% লাভে বিক্রয়মূল্য (5200 + 5200×10/100) = 5720 টাকা
∴ প্রতি কেজির বিক্রয়মূল্য (5720 ÷ 120) টাকা = 47.7 টাকা

Question: State the order of the matrix is-

Solution:
ম্যাট্রিক্সের মাত্রা বা ক্রম(Order of Matrix): একটি ম্যাট্রিক্সের সারি ও কলামের সংখ্যা যথাক্রমে m ও n হলে, ঐ ম্যাট্রিক্সকে m × n ক্রমের বা আকারের ম্যাট্রিক্স বলা হয়।
অর্থাৎ ম্যাট্রিক্সের আকার বা মাত্রা বোঝাতে প্রথমে সারি এবং পরে কলাম উল্লেখ করা হয়।
প্রদত্ত ম্যাট্রিক্সটি একটি আয়তাকার ম্যাট্রিক্স কারণ এর সারি ও কলাম অসমান।
এখানে,
সারি m = 2 এবং কলাম n = 3
∴ প্রদত্ত ম্যাট্রিক্সটি একটি 2 × 3 আকারের ম্যাট্রিক্স।

Question: What is the slope of a line perpendicular to the line whose equation is 8x + 3y = 14?

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,
8x + 3y = 14
3y = - 8x + 14
y = - (8/3)x + 14/3
(1) নং এর সাথে তুলনা করে পাই,
m = - (8/3)

∴ লম্ব (perpendicular) রেখার ঢাল হবে, m' = - {1/- (8/3)} = 3/8

x is odd
⇒ x² is odd [square of an odd number is always odd]
⇒ 5x² is odd 
⇒ (5x² + 2) is odd. [The sum of an odd number and even number is always odd]