Bank Math

Let, The numbers are x & y,
therefore,
x - y = 11 ------ (1) and
1/5(x + y) = 9 or, x + y = 45 ------ (2)
adding two equation we got,
2x = 56 or, x = 28,
putting the value of x in equation 1,
we get, y = 17

Question: B’s age after 12 years would be equal to 4 times his age 6 years ago. Find his age 5 years hence?
 
Solution:
Let B’s present age be ‘x’ years.

According to the question,
x + 12 = 4(x - 6)
⇒ x + 12 = 4x - 24
⇒ 3x = 36
⇒ x = 12

∴ B’s present age = 12 years

Therefore, B’s age 5 years hence = 12 + 5 = 17 years

The average age of five members is 27
Total age = 27× 5 = 135
After excluding one person, the new average = 27 - 2 = 25
New total age = 25× 4 = 100
Then the age of excluded person = Total age - New total age
= 135 - 100
= 35
Hence the required answer is 35.

Question: A cube has a total surface area of 486 square meters. What is the volume of the cube?

Solution:
ধরি, ঘনকের বাহুর দৈর্ঘ্য = a মিটার।

আমরা জানি, ঘনকের সম্পূর্ণ পৃষ্ঠের ক্ষেত্রফল = 6a²

প্রশ্নমতে,
6a² = 486
⇒ a² = 486/6
⇒ a² = 81
∴ a = 9 মিটার

এখন, ঘনকের আয়তন = a³ = 9³ = 729 ঘন মিটার

অতএব, ঘনকটির আয়তন = 729 ঘন মিটার

Question: What is the total surface area of a right circular cone with a base radius of 7 cm and a height of 24 cm?

Solution:
দেওয়া আছে,
ভূমির ব্যাসার্ধ r = 7 cm, উচ্চতা h = 24 cm
∴ হেলানো উচ্চতা, l = √(r² + h²)
= √(7² + 24²)
= √(49 + 576)
= √(625)
= 25 cm.

সমগ্র পৃষ্ঠতলের ক্ষেত্রফল (total surface area), = πr(l + r)
= (22/7) × 7 × (25 + 7)
= 22 × 32
= 704 cm².

∴ সমগ্র পৃষ্ঠতলের ক্ষেত্রফল (total surface area) = 704 cm².

Question: 3 pumps working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?

Solution: 
3 pumps need 2 × 8 hours = 16 hours
1 pump needs  16 × 3  hours
4 pumps need (16 × 3)/4 hours
= 12 hours

Question: In covering a distance of 48 km, Robin takes 2 hours more than Karim. If Robin triples his speed, he would take 2 hours less than Karim. What is Robin's original speed in km/h?

সমাধান:
ধরি, Robin-এর মূল গতিবেগ = x কিমি/ঘন্টা
∴ Robin-এর 48 কিমি অতিক্রম করতে সময় লাগে = 48/x ঘন্টা

ধরি, Karim-এর 48 কিমি অতিক্রম করতে সময় লাগে = t ঘন্টা

প্রথম শর্ত অনুযায়ী:
Robin, Karim-এর চেয়ে 2 ঘন্টা বেশি সময় নেয়,
⇒ 48/x = t + 2 ........ (i)

দ্বিতীয় শর্ত অনুযায়ী:
Robin যদি তার গতিবেগ তিনগুণ করে (3x কিমি/ঘন্টা), তাহলে সে Karim-এর চেয়ে 2 ঘন্টা কম সময় নেয়,
⇒ 48/(3x) = t - 2 ........ (ii)

সমীকরণ (i) থেকে: t = 48/x - 2

সমীকরণ (ii)-তে বসিয়ে পাই,
48/(3x) = (48/x - 2) - 2
⇒ 16/x = 48/x - 4
⇒ 48/x - 16/x = 4
⇒ 32/x = 4
⇒ x = 32/4
⇒ x = 8 কিমি/ঘন্টা

সুতরাং, Robin-এর মূল গতিবেগ হলো 8 কিমি/ঘন্টা।

Question: For a circle where the diameter is 4π, calculate the radius-to-circumference ratio.

Solution:
Here
The diameter of the circle is d = 4π
So the radius of the circle r = 2π

∴ Circumference of circle = 2. π. 2π
= 4π²

So the ratio between radius and Circumference of circle = 2π : 4π²

= 2π/4π²

= 1 : 2π

Question: A box contains 20 electric bulbs, out of which 4 are defective. Two balls are chosen at random from this box. The probability that at least one of them is defective, is

Solution:
Given that,
Total bulbs = 20
Defective bulbs = 4
Non-defective bulbs = 20 - 4 = 16
Two bulbs are chosen at random (without replacement)

Now,
P(both non-defective) = (16/20) × (15/19) = 240/380 = 12/19

And,
∴ P(at least one defective) = 1 - P(both non-defective)
= 1 - (12/19)
= (19 - 12)/19
= 7/19
∴ The probability that at least one of them is defective is 7/19

Let Mira paid x,
so, Akhi paid 2x/3, and
Lamia paid 2x,
So total bill paid is given by,
x + (2x/3) + 2x = 1;
we get,
x = 3/11

So, Mira paid 3/11 fraction of the total bill.

Suppose he buys each share for Tk. x.
Then, Tk.(25 × 9/100) = (x × 10/100)
or x = Tk. 22.50.
Cost of each share = Tk. 22.50.

Question: The price of a phone is Tk. 8000. Its price is first increased by 25% and then decreased by 20%. What is the present price of the phone?

Solution:
Initial Cost = Tk. 8000
After 25% increase in the cost, it becomes,
(8000 + 25% of 8000)
= 8000 + 2000
= Tk. 10000

Now, cost is decreased by 20%, so cost will become,
(10000 - 20% of 10000)
= 10000 - 2000
= Tk. 8000

So, present cost is Tk. 8000.

Question: Sanzida ate 3/4 of a pizza. Her brother Babu ate 1/2 of what was left. Then their friend Pavel ate 2/3 of what was still left. What fraction of the pizza remains uneaten?

Solution:
Sanzida ate = 3/4
∴ Remaining = 1 - 3/4 = 1/4

Babu ate = 1/2 × 1/4 = 1/8
∴ Remaining = 1/4 - 1/8 = (2 - 1)/8 = 1/8

Pavel ate = 2/3 × 1/8 = 2/24 = 1/12
∴ Remaining = 1/8 - 1/12 = (3 - 2)/24 = 1/24

∴ Fraction of pizza remaining uneaten = 1/24.

Question: If the workforce is doubled, how much longer will it take to finish the task?
​
Solution:
ধরি,
শ্রমিক সংখ্যা = x, এর দিগুণ = 2x,
সময় = n
x জন কাজটি করে n সময়ে

১ জন কাজটি করে = xn সময়ে
২x জন কাজটি করে = xn/২x
= n/২ সময়ে বা ১/২ সময়ে।

Question: The difference between the ages of a father and his son is 40% of the father’s age. If the son is 18 years old, find the father’s age.

Solution:
Let
Let father’s age = x

Then,
x - 18 = 40% of x
⇒ x - 18 = 40x/100
⇒ x - 18 = 2x/5
⇒ x - 2x/5 = 18
⇒ (5x - 2x)/5 = 18
⇒ 3x = 90
∴ x = 30

So the father’s age is 30 years.

Question: Tickets numbered 1 to 30 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 4 or 5?

Solution:
Here, S = {1, 2, 3, 4, ...., 29, 30}
n(S) = 30

Let E = event of getting a multiple of 4 or 5
∴ n(E) = {4, 5, 8, 10, 12, 15, 16, 20, 24, 25, 28, 30}

∴ P(E) = n(E)/n(S)
= 12/30
= 2/5

P/S.I.=10/3
Let Principal = 10
S.I. for 5 year = 3
S.I. for 1 year = 0.6
Rate = S.I./Principal×100
Rate = 0.6/10×100
=6%

Distance = X
Distance covered at 5km and 4 km = x/3 + 2x/5
= (5x + 6x)/15
= 11x/15
Rest of the distance = 1 - 11x/15 = 4x/15
ATQ,
4x/15 = 12
or, x = (12 × 15)/4
or, x = 45.

Product of two different irrational numbers is sometimes irrational and sometimes rational. 
For example, product of √2 and √3 is √6, which is irrational
but product of √3 and √12 is √36, which is rational number 6. 

Two parts are = A and (40800 - A)
Time period and Simple interest are same for both

Simple Interest = (P × R × T)/100

∴ (A × 6.25 × 8)/100 = {(40800 - A) × 7 × 5}/100
∴ 50A = 35 × 40800 - 35A
∴ 85A = 35 × 40800
∴ A = 8400

Smaller amount is Tk. 16800
By direct observation, we can say that 16800 is smaller - as it is smaller than half of 40800.

Question: In a badminton tournament, every player plays against every other player exactly once. If there were 66 matches played in total, how many players participated in the tournament?

Solution:
ধরি, টুর্নামেন্টে মোট খেলোয়াড়ের সংখ্যা = x

যেহেতু প্রতিটি খেলোয়াড় অন্য সকল খেলোয়াড়ের সাথে একবার করে ম্যাচ খেলে, তাই মোট ম্যাচের সংখ্যা হবে:
^xC₂ = 66
⇒ x!/{2!(x - 2)!} = 66
⇒ x(x - 1)/2 = 66
⇒ x(x - 1) = 132
⇒ x² - x - 132 = 0
⇒ x² - 12x + 11x - 132 = 0
⇒ x(x - 12) + 11(x - 12) = 0
⇒ (x - 12)(x + 11) = 0
⇒ x = 12 অথবা x = -11

যেহেতু খেলোয়াড়ের সংখ্যা ঋণাত্মক হতে পারে না, তাই x = 12

∴ টুর্নামেন্টে মোট খেলোয়াড়ের সংখ্যা = 12 জন।

Given that,
Two thirds of the 6 km was covered at 4 km/hr
i.e. 4 km distance was covered at 4 km/hr.
Time taken to cover 4 km = 4 km/(4 km/hr)
= 1 hr
= 60 minutes
Time left = (84 – 60) minutes.
= 24 minutes
Now,
The man has to cover remaining 2 km in 24 minutes
or 24/60
= 2/5 hours
Speed required for remaining 2 km
= 2 km/(2/5)hr
= 5 km/hr.

Interest for the first month = p×(10000/1000) = 10p
Interest for the 2nd month will be =  q×(10000/1000) = 10q
Interest for the 3rd month will be = q×(10000/1000) = 10q

∴ Total Interest = 10p + 20q

যে কোন করমর্দন অথবা কোলাকুলির অংকে শুধু কত জন করমর্দন (Handshake), বা কোলাকুলি করল তা দেয়া থাকবে।
এক্ষেত্রে মনে রাখতে হবে যে প্রত্যেক বার করমর্দন বা কোলাকুলি করার সময় মোট ২ জন লোকের প্রয়োজন।
তাই এক্ষেত্রে সূত্রটি হবে ⁿC₂ = ^{মোট লোক}C_{২ জন সব সময়}
¹⁰C₂ = 10!/2!(10 - 2)!
= 10!/2!8!
= (10 × 9)/2
= 5 × 9
= 45.

Question: If 3 jackets and 5 sweaters cost Tk. 12,000, and 5 jackets and 3 sweaters cost Tk. 13,600, what is the cost of one jacket?

Solution:
ধরি, একটি জ্যাকেটের মূল্য x টাকা এবং একটি সোয়েটারের মূল্য y টাকা।

প্রশ্নমতে,
3x + 5y = 12000 ............... (i) 
5x + 3y = 13600 .............. (ii)

(ii) × 5 - (i) × 3 ⇒
(25x + 15y) - (9x + 15y) = 68000 - 36000
⇒ 25x - 9x = 32000
⇒ 16x = 32000
⇒ x = 32000/16
⇒ x = 2000

সুতরাং, একটি জ্যাকেটের মূল্য 2000 টাকা।

Question: A train 180 meters long passes a pole in 12 seconds. How long will it take to pass a platform that is 420 meters long?

Solution:
Train's speed = Distance/Time
= 180/12 = 15 m/s

Total distance to pass the platform,
= Length of train + Length of platform
= 180 m + 420 m
= 600 m

∴ Required time = Distance/Speed
= 600/15
= 40 seconds

∴ The train will take 40 seconds to pass platform.

Cost price of one doughnut = 35/100 = 0.35
Selling price of one doughnuts = 7.20/12 = 0.6 tk
Profit in one doughnuts = 0.6 - 0.35 = 0.25 tk
So, Total doughnuts bought = 30 / 0.25 = 120

Question: A sum of money is borrowed and paid back in two annual instalments of Tk. 882 each allowing 5% compound interest. The sum borrowed was -

Solution:
Principle,
= (P.W. of Tk. 882 due 1 year hence) + (P.W of Tk. 882 due 2 years hence)
= [{882/(1 + 5/100)} + {882/(1 + 5/100)²}]
= [{(882 × 20)/21} + {(882 × 400)/441}]
= 840 + 800
= Tk. 1640

The sum borrowed was Tk. 1640.

Question:

Solution:

Question: What is the angle between the hour and minute hand of a clock when it is 4 : 40 pm?

সমাধান:
4টা 40 মিনিট = 4 + (40/60) ঘন্টা
= 4 + 2/3 = 14/3 ঘন্টা

আমরা জানি,
ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 14/3 ঘণ্টায় ঘোরে = (30° × 14)/3
= 140°

আবার,
মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 40 মিনিটে ঘোরে = 40 × 6° = 240°

ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = |240° - 140°| = 100°

Question: Calculate the area of a rhombus if the length of its side is 4 cm and one of its angles A is 120 degrees.

Solution:
Given that,
Side of rhombus, a = 4 cm
And One angle, A = 120°

We know,
Area of a rhombus = a² × sin⁡A [Where a = side of rhombus, A = any interior angle.]
= 4²  × sin⁡120°
= 16 × (√3/2)
= 8√3

So the area of the rhombus is 8√3 cm²

Note:
sin(180∘ - θ) = sinθ,
So sin120° = sin⁡(180° - 60°) = sin⁡60° = √3/2

Question: If tan(θ + 30°) = √3, then what is the value of cosθ?

Solution:
Given that,
tan(θ + 30°) = √3
⇒ tan(θ + 30°) = tan60°
⇒ (θ + 30°) = 60°
∴  θ = 30°

Now,
cosθ
= cos30°
= √3/2

Question: A cuboid has dimensions in the ratio 1 : 2 : 4 and a total surface area of 112 cm². What is its volume?

Solution:
দেয়া আছে,
 আয়তাকার ঘনবস্তুর মাত্রাগুলির অনুপাত = 1 : 2 : 4
এবং সমগ্র পৃষ্ঠতলের ক্ষেত্রফল = 112 cm²

ধরি , আয়তাকার ঘনবস্তুর মাত্রাগুলির অনুপাত যথাক্রমে x, 2x এবং 4x

আমরা জানি,
আয়তাকার ঘনবস্তুর সমগ্র পৃষ্ঠতলের ক্ষেত্রফল = 2(lb + bh + lh)
⇒ সমগ্র পৃষ্ঠতলের ক্ষেত্রফল = 2(x)(2x) + (2x)(4x) + (4x)(x)
⇒ 112 = 2(2x² + 8x² + 4x²)
⇒ 112 = 2(14x²)
⇒ 112 = 28x²
⇒ x² = 112/28
⇒ x² = 4
⇒ x = 2

সুতরাং, ঘনবস্তুটির মাত্রাগুলি হল,
দৈর্ঘ্য (l) = x = 2 cm
প্রস্থ (b) = 2x = 2 × 2 = 4 cm
উচ্চতা (h) = 4x = 4 × 2 = 8 cm

এখন, আয়তাকার ঘনবস্তুর আয়তন, V = l × b × h
⇒ V = 2 × 4 × 8
⇒ V = 64 ঘন সেমি।

সুতরাং, নির্ণেয় আয়তন হল 64 cm³।

Area to be plastered = [2(l + b)×h] + (l×b)
= [2(25 + 12)×6] + (25×12)
= 744 m² 
Cost of plastering = 744 × 75/100 = 558 tk

Question: The supplement of an angle exceeds twice the angle by 30°. Then the angle is equal to-

Solution:
Let the angle be x
Then, its supplement = 180 - x

According to the question,
180 - x = 2x + 30
⇒ 180 - 30 = 3x
⇒ 150 = 3x
⇒ x = 50°

Question: What is the ratio of 5 inches to 9 feet?

Solution:
We know, 1 foot = 12 inches
So, 9 feet = 9 × 12 = 108 inches

Now,
5 inches : 9 feet = 5 : 108

∴ The ratio = 5 : 108

Question: A tap can completely fill a water tank in 8 hours. The water tank has a hole in it through which the water leaks out. The leakage will cause the full water tank to empty in 10 hours. How much time will it take for the tap to fill the tank completely with the hole?

Solution: 
Tap alone fills the tank in 8 hours
⇒ Filling rate = 1/8 tank/hour
Leakage alone empties the full tank in 10 hours
⇒ Emptying rate = 1/10 tank/hour

∴ Net rate = Filling rate - Emptying rate
= (1/8) - (1/10)
= (5 - 4)/40
= 1/40

Time to fill the tank with the hole = 1 full tank/Net rate
= 1/(1/40) hours
= 40 hours 

x + y + xy/100
= 10 - 10 - (100/100)
= -1%
so, 1% = 10
and 100% = 10 × 100
= 1000

Question: A candidate has to obtain a minimum of 40% of the total marks to pass. He got 30% of the total marks and failed by 50 marks. What are the maximum marks?

Solution: 
Let the maximum marks be x.

Then,
40% of x - 30% of x = 50
⇒ 10% of x = 50
⇒ 10x/100 = 50
⇒ x= (50 × 100)/10
∴  x = 500

Question: A contractor employs 45 persons to do a job in 40 days. After 10 days, it was found that only one-sixth of the work was finished. How many more persons are to be employed to finish the job as per schedule?

Solution:
দেওয়া আছে:
মোট লোক = 45 জন
নির্ধারিত সময় = 40 দিন
10 দিনে সম্পন্ন কাজ = 1/6 অংশ

ধরি, সম্পূর্ণ কাজ = 1 একক

45 জন লোক 10 দিনে করে = 1/6 অংশ কাজ
∴ 45 জন লোক 1 দিনে করে = (1/6) ÷ 10 = 1/60 অংশ
∴ 1 জন লোক 1 দিনে করে = (1/60) ÷ 45 = 1/2700 অংশ

অবশিষ্ট কাজ = 1 - 1/6 = 5/6 অংশ
অবশিষ্ট সময় = 40 - 10 = 30 দিন

∴ অবশিষ্ট 5/6 অংশ কাজ 30 দিনে করতে হবে

∴ প্রতিদিনের প্রয়োজনীয় কাজের হার = (5/6) ÷ 30 অংশ
= 5/180 = 1/36 অংশ

এখন,
প্রতিদিন 1/2700 অংশ কাজ করে 1 জন
∴ 1 অংশ কাজ করে = 1 ÷ (1/2700) জন
∴ 1/36 অংশ কাজ করে = (2700/36) জন
= 75 জন

∴ অতিরিক্ত লোকের প্রয়োজন = 75 - 45 = 30 জন

Question: If cosA = 8/17 than, what is the value of tanA = ?

Solution:
Given that, 
cosA = 8/17

We know, 
sin²A = 1 - cos²A = 1 - (8/17)²
= 1 - (64/289)
= (289 - 64)/289
= 225/289
∴ sinA = √(225/289) = 15/17

Now, 
tanA = sinA/cosA = (15/17)/(8/17) = 15/8
∴ tanA = 15/8

y = -5x + 9
⇒ y + 5x = 9 .....(i)
সুতরাং (i) নং রেখাটির লম্বরেখার সমীকরণ 5y - x = k
⇒ y = 1/5x + k
∴ লম্ব রেখাটির ঢাল = 1/5

Length of the wire fencing
= perimeter
= 2 (90+ 50) = 280 metre

Two poles are kept 5 metre apart. Note that the poles are placed along the perimeter of the rectangular plot, not in a single straight line.

Hence, number of poles required
= 280/5
= 56 metre

Question: The bus fare was recently raised from Tk. 3.70 to Tk. 4.00 per kilometer. What is the approximate percentage increase?

Solution: 
বাস ভাড়া বাড়ে = (4.00 - 3.70) টাকা
= 0.30 টাকা

বাস ভাড়া শতকরা বাড়ে = (0.30/3.70) × 100%
= 8.1 %
≈ 8%

Question: A student scored 30% marks and failed by 12 marks. Another student scored 55% marks and secured 38 marks more than the pass marks. What is the pass percentage?

Solution:
Let total marks = x

According to the question,
30% of x + 12 = 55% of x - 38
⇒ 0.3x + 12 = 0.55x - 38
⇒ 0.55x - 0.3x = 12 + 38
⇒ 0.25x = 50
⇒ x = 50/0.25
∴ x = 200

∴ Pass marks = 30% of x + 12
= 0.3 × 200 + 12
= 60 + 12
= 72

∴ Pass percentage = (72/200) × 100
= 36%

Question: If the day before yesterday was Thursday, when will Sunday be?

Solution:
Day before yesterday = Thursday
Yesterday = Friday
Today = Saturday
Tomorrow = Sunday

Therefore, Sunday will be tomorrow.

Question: A man invested TK 4500 in a stock at 108 to obtain an income of TK 250. What is the dividend from the stock?

Solution:
By investing TK 4500, income  = TK 250
By investing TK 108 = (108 × 250)/4500 = 6

Hence, the dividend is 6%.

We have to find the speed of a current.
t (downstream) = 3 h 45 min = 3.75 h
t (upstream) = 2 h 30 min = 2.5 h
Speed downstream = 22.5 km / 3.75 h = 6 km/h
Speed upstream = 10 km / 2.5 h = 4 km/h

So, Speed of the current = (Speed downstream - speed upstream) / 2
= (6 - 4) / 2
= (2 / 2) km/h
= 1 km/h

Ratio of rates of working of A and B = 2:1.
So, ratio of times taken = 1:2
Therefore, A's 1 day's work = 1/9
B's 1 day's work = 1/18
(A + B)'s 1 day's work = 1/9 + 1/18 = 1/6
So, A and B together can finish the work in 6 days.