Bank Math

ধরি,
AB হচ্ছে দেয়াল এবং BC হচ্ছে মই।
এখানে ∠ACB = 60° এবং AC = 4.6 মিটার
তাহলে, AC/BC = cos 60° = 1/2 [ যেহেতু, cosΘ = ভূমি/অতিভুজ]
⇒ BC = 2 × AC
∴ BC = 2 × 4.6
= 9.2m

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

Solution:
4টা 20 মিনিট = 4 + (20/60) ঘন্টা
= 4 + 1/3
= 13/3 ঘন্টা

আমরা জানি,
ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 13/3 ঘণ্টায় ঘোরে = (30° × 13)/3
= 390°/3
= 130°

আবার,
মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 20 মিনিটে ঘোরে = 20 × 6° = 120°

∴ ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = | 130° - 120° |
= 10°

যেহেতু ধারাটি একটি সমান্তর ধারা তাই সব পদের ব্যবধান সমান
(2a + 1) - (a + 1) = (4a - 1) - (2a + 1)
⇒ 2a + 1 - a - 1 = 4a - 1 - 2a - 1
⇒ a = 2a - 2
∴ a = 2

Question: Six friends are sitting in a circle facing inwards. Priti and Ashwini are exactly opposite to each other. Sachin is in between Priti and Dipen. Dipen is immediately to the left of Ashwini. Rishi is not exactly opposite to Dipen. Who is just right to Ashwini?

Solution:

So, Rishi is just right to Ashwini.

Question: 2/5 part of the tank is full of water. When 24 liters of water is taken out, the tank becomes empty. The capacity of the tank is -

Solution: 
Let the total capacity of the tank be 5x liters.
Then, water in the tank = 2/5 × 5x = 2x liters.

When 24 liters of water is taken out, the tank becomes empty.
It means the water taken out = water present in the tank.
∴ 2x = 24 litres
⇒ x = 12 litres

∴ Capacity of the tank = 5x
= 5 × 12
= 60 liters.

Question: If x/y = 1/3,  then (x² + y²)/(x² - y²) = ?

Solution:
(x² + y²)/(x² - y²) 
= {(x² + y²)/y²}/{x² - y²)/y² [Dividing the numerator and denominator by y²]
= {(x²/y²) + 1}/{(x²/y²) - 1}
= {(x/y)² + 1}/{(x/y)² - 1)}
= {(1/3)² + 1}/{(1/3)² - 1} [given, x/y = 1/3]
= {(1/9) + 1}/{(1/9} - 1}
= (10/9)/(-8/9)
= (10/9) × (9/-8)
= - 5/4

Shortcut:
Take x = 1, y = 3 (because 1/3 = 1/3)
∴ (x² + y²)/(x² - y²) = (1² + 3² )/(1² - 3²)
= (1 + 9)/(1 - 9) = - 5/4

Question: At what rate of compound interest per annum will a sum of Tk. 4000 becomes Tk. 4840 in 2 years?

​Solution:
Principal, P = Tk. 4000
Compound Amount, C = Tk. 4840
Time, n = 2 years
Rate, r = ?

We know,
C = P × (1 + r/100)ⁿ
⇒ 4840 = 4000 × (1 + r/100)²
⇒ (1 + r/100)² = 4840/4000 
⇒ (1 + r/100)² = 484/400
⇒ 1 + r/100 = 22/20 [উভয়পাশে বর্গমূল করে]
⇒ r/100 = (11/10) - 1
⇒ r/100 = (11 - 10)/10
⇒ r/100 = 1/10
⇒ r = (1 × 100)/10
∴ r = 10

∴ Interest Rate = 10%

Question: cos45° . sin0° . tan45° . cosec60°

Solution:
Here,  
cos45° . sin0° . tan45° . cosec60°
= (1/√2) × 0 × 1 × (2/√3)
= 0

Cost price = 80000 + 5000 + 1000 = 86000
Given, profit = 25%
∴ Selling price = 86000 + 86000×(25/100)
                       = Tk. 107500

Question: A rectangular field has a perimeter of 110 meters. The length of the field is 5 meters less than three times its width. Find the area of the field in square meters.

Solution:
ধরি, আয়তাকার ক্ষেত্রটির প্রস্থ = x মিটার
সুতরাং, ক্ষেত্রটির দৈর্ঘ্য = 3x - 5 মিটার

আয়তক্ষেত্রের পরিসীমা = 2(দৈর্ঘ্য + প্রস্থ)
প্রশ্নমতে,
2((3x - 5) + x) = 110
⇒ 2(4x - 5) = 110
⇒ 4x - 5 = 55
⇒ 4x = 60
⇒ x = 15 মিটার

সুতরাং,
প্রস্থ = 15 মিটার।
দৈর্ঘ্য = 3x - 5
= (3 × 15) - 5 
= 45 - 5
= 40 মিটার।

আয়তক্ষেত্রের ক্ষেত্রফল = দৈর্ঘ্য × প্রস্থ
∴ ক্ষেত্রফল = 40 × 15 
= 600 বর্গ মিটার।

সুতরাং, ক্ষেত্রটির ক্ষেত্রফল হলো 600 বর্গ মিটার।

Question: (a% of 10b) + (b% of a) is equal to:

Solution:
According to question
(a% of 10b) + (b% of a)
= 10b × (a/100) +  a × (b/100)
= (ab/10) + (ab/100)
= [(10ab + ab)]/100
= 11ab/100
= 11ab%

Or, we can write 11% of ab.

প্রশ্ন: Find the ratio of total surface area to lateral surface area of a cylinder whose radius is 20 cm and height 80 cm.

সমাধান:

দেয়া আছে,
সিলিন্ডারের ব্যাসার্ধ (r) = 20 সেমি এবং উচ্চতা (h) = 80 সেমি।

আমরা জানি,
সিলিন্ডারের সমগ্র পৃষ্ঠতলের ক্ষেত্রফল = 2πr(r + h)
সিলিন্ডারের পার্শ্ব পৃষ্ঠতলের ক্ষেত্রফল = 2πrh

সুতরাং, সমগ্র পৃষ্ঠতলের ক্ষেত্রফল : পার্শ্ব পৃষ্ঠতলের ক্ষেত্রফল
= 2πr(r + h) : 2πrh
= (r + h) : h
= (20 + 80) : 80
= 100 : 80
= 5 : 4

সুতরাং, নির্ণেয় অনুপাত হল 5 : 4

Question: An accurate clock shows the time as 3.00. After the hour hand has moved 135°, the time would be- 

Solution: 
প্রতি ৫ ঘরের মান ৩০° 

১৩৫° তে ঘন্টার কাঁটা যায় ৪.৫ ঘর 
∴ ৪ ঘন্টা ৩০ মিনিট অতিবাহিত হয়েছে। 

নতুন সময় = ৭ : ৩০ মিনিট 

Question: 0.15, 0.3, ____, 1.2, 2.4. What number should fill the blank?
 
Solution:
Each term is being multiplied by 2,

Now,
0.15 × 2 = 0.3  
0.3 × 2 = 0.6  
0.6 × 2 = 1.2  
1.2 × 2 = 2.4

So the complete series is-
0.15, 0.3, 0.6, 1.2, 2.4

∴ Blank should be filled with 0.6

Question: If 8 men or 12 boys can make 60 tables in 6 days, then how many tables will be made by 4 men and 6 boys in 8 days?

Solution:
Here, 8 men = 12 boys
∴ 4 men = (12/2) boys = 6 boys

∴ 4 men and 6 boys = (6 + 6) boys = 12 boys

Now,
12 boys can make 60 tables in 6 days
In 1 day, 12 boys can make (60/6) = 10 tables
∴ In 8 days, 12 boys can make = (10 × 8) tables
= 80 tables

Question: 20% of 45 + 45% of 120 = ?

Solution:
 20% of 45 + 45% of 120
= {(20/100) × 45} + {(45/100) × 120}
= 9 + 54
= 63

Question: If X + Y = 174, and X is half of Y, then find the value of X.

Solution:
Given that,
X + Y = 174 …… (i)
Y = 2X …… (ii)
On solving (i) and (ii), we get,
⇒ X + 2X = 174
⇒ 3X = 174
⇒ X = 174/3
∴ X = 58

Let speed of the water in still water = x kmph
Given that speed of the stream = 3 kmph

Speed downstream = (x + 3) kmph
Speed upstream = (x - 3) kmph

He travels a certain distance downstream in 1 hour and comes back in 1 (1/ 2) hour. That is, (distance travelled downstream in 1 hour = distance travelled upstream in 1 (1 /2) hour).

Since, distance = speed × time ; we have,
(x + 3) × 1 = (x − 3) × (3/2)
⇒ 2 (x + 3) = 3 (x − 3)
⇒ 2x + 6 = 3x − 9
⇒ x = 6 + 9 = 15

Let N be the no. of persons in the group.
Required number of person is given by;
Member in group × aged increased = difference of replacement
N × 5 = 38 - 18
Or, 5N = 20
Or, N = 4

Let Tap A fill the cistern completely in A hours.
So in 1 hour, it fills 1/A amount of the cistern

Also in 1 hour in Tap B fills in 1/4 amount of the cistern
Together they fill the cistern in 2.4 hours
So, Also in 1-hour together they fill in (1/2.4)amount of the cistern
So, Also in 1-hour cistern filled by both is given by 1/A + 1/4 = 1/2.4
∴ 1/A = 1/2.4 - 1/4 = 1/6

∴ Pipe A can fill 1/6^th tank in 1 hour
∴ Pipe a fills the tank completely in 6 hours.
It has a rate of 100-litre water per hour,
So, in 6 hours it gives out 6 x 100 = 600 litres
In 6 hours cistern is full, so capacity = 600 litres.

Let,
Pencil's price x
And eraser's price = x/2
ATQ, 5x + 6x/2 = 80 [As there is an extra eraser] 
Or, 16x = 160
Or, x = 10
∴ eraser costs = x/2 = 10/2 = 5

বিকল্প পদ্ধতি:
5 pencils + 6 erasers = ৳80
The five pencils have the same value as ten erasers, so if we include the extra (non-paired) eraser, we have a total value of 16 erasers:
80 ÷ 16 = 5

Question: What is the distance between the points A (- 1, 3) and B (5, - 5)?

Solution:

Question: What yearly income can be earned by investing Tk. 18000 in 12% stock at Tk. 90?

Solution:
12% stock at Tk. 90 বলতে বুঝায় স্টকটির ফেস ভ্যালু 100 টাকায় 12 টাকা লাভ হয় এবং স্টকটির বাজার মূল্য 90 টাকা।

এখন, 
90 টাকা বিনিয়োগ করে আয় হয় 12 টাকা
∴ 1 টাকা বিনিয়োগ করে আয় হয় 12/90 টাকা
∴ 18000 টাকা বিনিয়োগ করে আয় হয় (18000 × 12)/90 টাকা
= 216000/90 টাকা = 2400 টাকা

Question: sin2A = √3/2, then find the value of A = ?

Solution:
Given that,
sin 2A = √3/2

We know,
sin 60° = √3/2
⇒ sin2A = sin60°
⇒ 2A = 60°
⇒ A = 60°⁄2
∴ A = 30°

Question: 

Solution: 

10 থেকে 100 পর্যন্ত 5 এর গুণিতক সংখ্যাগুলো হলোঃ
10, 15, 20, 25, 30 ,35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95 এবং 100 = 19টি ।

এখন, 10 থেকে 100 পর্যন্ত 9 এর গুণিতক সংখ্যাগুলো হলোঃ
18, 27, 36, 45, 54, 63, 72, 81, 90 এবং 99 = 10 টি

10 থেকে 100 পর্যন্ত মোট সংখ্যা আছে = 100 - 10 + 1 = 91টি ।

এর মধ্যে, 45 এবং 90 দুইটা দুই জায়গাতেই আছে তাই একবার কাউন্ট হবে।

∴ নির্ণেয় সম্ভাব্যতা = (19+10-2)/91
= 27/91

Let the original price be 100
Increase of 10% means the price now = 100 + (10% of 100) = Tk. 110
Now decrease of 10% means price now = Tk. 110 - (10% of 110)
= 110 - 11
= Tk. 99
So change in price = 100 - 99 = Tk. 1
1 is 1 percent of 100.
It is 1%
So change in price is 1%.

Question: What is the compound amount of Tk. 3200 for 2 years at a rate of interest 5% per annum?

Solution:
Given,
Principal, P = 3200
Rate, r = 5% = 5/100 = 1/20
Time, n = 2 years

We know,
A = P(1 + r)ⁿ
  = 3200 × (1 + 1/20)²
  = 3200 × (21/20)²
  = (3200 × 21 × 21) / (20 × 20)
  = (3200 × 441) / 400
  = 1411200 / 400
  = 3528

∴ The compound amount is Tk. 3528.

Question: A cricketer’s average after 20 innings is 45 runs. If he scores 108 runs in the next innings, what is his new average?

Solution:
Average after 20 innings = 45 
Total runs after 20 innings:
= 20 × 45
= 900 runs.

Runs scored in the 21st innings = 108.
Total runs after 21 innings:
= 900 + 108
= 1008 runs.

New average = Total runs / Number of innings
= 1008 / 21
= 48 runs.

Total sum of remaining two
= (8 × 14 – 6 × 16) = 16
∴ Average of these two numbers is = 16 / 2 = 8

Question: The radius of a wheel is 14 cm. How many revolutions will it make in travelling 66 kilometers?

Solution:
আমরা জানি,
চাকার পরিধি = 2πr
= 2 × (22/7) × 14
= 88 সে. মি.

∴ মোট দূরত্ব = 66 কি. মি.
= 66 × 1000 × 100
= 6600000 সে. মি.

∴ ঘূর্ণন সংখ্যা = 6600000 / 88
= 75000 টি

Question: Rakib can do a job in 15 minutes and his friend takes twice as long to do the same job. If they work together, how long will it take to complete the job?

Solution:
রাকিবের সময় = 15 মিনিট, তার বন্ধুর সময় = 30 মিনিট।

1 মিনিটে রাকিব কাজ করে = 1/15 অংশ।
1 মিনিটে তার বন্ধু কাজ করে = 1/30 অংশ।
তাই, 1 মিনিটে তারা একসাথে করে = 1/15 + 1/30
= (2 + 1)/30
= 3/30 = 1/10 অংশ।

1/10 অংশ করতে সময় = 1 মিনিট
∴ সম্পূর্ণ কাজ করতে সময় = 1 ÷ (1/10) = 10 মিনিট।

∴ একসাথে কাজ করলে তারা 10 মিনিটে কাজ শেষ করবে।

Question: (sin⁴θ - cos⁴θ +1) cosec²θ = ?

Solution:
Given that,
(sin⁴θ - cos⁴θ + 1) cosec²θ
= [(sin²θ - cos²θ) (sin²θ + cos²θ) + 1] cosec²θ
= (sin²θ - cos²θ + 1) cosec²θ     ; [sin²A + cos²A = 1]
= [sin²θ - (1 - sin²θ) + 1] cosec²θ
= [2sin²θ - 1 + 1] cosec²θ
= 2sin²θ cosec²θ
= 2sin²θ (1/sin²θ)
= 2

Question: B's income is 60% of A's, and the ratio of their expenditures is 9 : 5. If each saves Tk. 4,000, find A’s income.

Solution:
Suppose,
A's income = 100,
B's income 60% of 100 = 60
A : B = 100 : 60 = 5 : 3

So,
A’s income = 5x, B’s income = 3x
A’s expense = 9y, B’s expense = 5y

Then their savings are:
A’s savings = Income - Expense = 5x - 9y.......(1)
B’s savings = Income - Expense = 3x - 5y........(2)

Given that each saves Tk. 4000:
5x - 9y = 4000
3x - 5y = 4000

Subtract equation (2) from (1):
(5x - 9y) - (3x - 5y) = 0 
 ⇒ 2x - 4y = 0 
⇒ x = 2y

Substitute x = 2y into equation (2):
3(2y) - 5y = 4000
⇒ 6y - 5y = 4000
⇒  y = 4000

Then x = 2y = 8000

Finally, A’s income = 5x = 5 × 8000 = 40,000 Taka

∴ A's income = 40,000 Taka