Bank Math

Compound interest for 1 year.
2{3%(P)} + 3%{3%(P)}
= 6%(P) + 0.09%(P)
= 6.09%(P)
That is an effective interest rate for 1 year = 6.09%.

Question: Two ships, Alpha and Beta, start towards each other from two ports, 160 km apart. The speeds of Ship Alpha and Ship Beta in still water are 16 km/h and 24 km/h respectively. If Ship Alpha proceeds downstream and Ship Beta proceeds upstream, they will meet after how many hours?

সমাধান:
ধরি, স্রোতের গতিবেগ হলো x কিমি/ঘন্টা।
ধরি, জাহাজ দুটি t ঘন্টা পর মিলিত হবে।

জাহাজ Alpha (স্রোতের অনুকূলে) এর গতিবেগ = (16 + x) কিমি/ঘন্টা।
জাহাজ Beta (স্রোতের প্রতিকূলে) এর গতিবেগ = (24 - x) কিমি/ঘন্টা।

তারা একে অপরের দিকে আসছে, তাই তাদের আপেক্ষিক গতিবেগ হলো তাদের গতির যোগফল।
আপেক্ষিক গতিবেগ = (16 + x) + (24 - x) কিমি/ঘন্টা
= (16 + 24 + x - x) কিমি/ঘন্টা
= 40 কিমি/ঘন্টা।

প্রশ্নমতে,
দূরত্ব = আপেক্ষিক গতিবেগ × সময়।
160 = 40 × t
⇒ t = 160/40
⇒ t = 4 ঘন্টা।
∴ জাহাজ দুটি 4 ঘন্টা পর মিলিত হবে।

Question: An observer who is 1.8 meters tall is standing 20 meters away from a tower. If the angle of elevation from his eye to the top of the tower is 45°, what is the height of the tower?

Solution:

পর্যবেক্ষকের উচ্চতা, CD = 1.8 মিটার
এখানে, CD = EB
টাওয়ারের উচ্চতা = AB

এখন,
tan∠C = AE/CE
⇒ tan45° = AE/20
⇒ 1 = AE/20
∴ AE = 20

∴ AB = AE + BE
= 20 + 1.8
= 21.8 m

∴ টাওয়ারটির উচ্চতা 21.8 meters.

Market Value of Company X (his selling price) = Tk. 30
Total shares sold = 4000

The amount he gets = Tk. (4000 × 30)
He invests this amount in ordinary shares of Company Y
Market Value of Company Y(His purchasing price) = Tk. 15

Number of shares of company Y which he purchases = (4000 × 30)/15
= Tk. 8000.

Question: A train 300 meters long passes a pole in 20 seconds. How long will it take to pass a platform that is 450 meters long?

Solution:
Train's speed = Distance/Time
= 300/20
= 15 m/s

Total distance to pass the platform = Length of train + Length of platform
= 300 m + 450 m
= 750 m

∴ Required time = Distance/Speed
= 750/15
= 50 seconds

∴ The train will take 50 seconds to pass the platform.

Question: Find the greatest number which on dividing 1567 and 1979, leaves remainders 3 and 7 respectively.

Solution:
1567 - 3 = 1564
1979 - 7 = 1972

বৃহত্তম সংখ্যাটি হবে 1564 এবং 1972 এর গসাগু।

ইউক্লিডীয় পদ্ধতিতে গসাগু বের করি,

∴ গসাগু = 68

∴ নির্ণেয় বৃহত্তম সংখ্যা = 68

Question: The side of a square is increased by 10%, by what percent will the area be increased?

Solution:
আমরা জানি, 
বর্গক্ষেত্রের ক্ষেত্রফল = (বাহু × বাহু) বর্গ একক

ধরি, 
বর্গক্ষেত্রের এক বাহুর দৈর্ঘ্য = 10 মিটার 
∴ বর্গক্ষেত্রের ক্ষেত্রফল = (10 × 10) বর্গমিটার 
= 100 বর্গমিটার

আবার, 
10% বৃদ্ধিতে বর্গক্ষেত্রের বাহুর সংখ্যা = 10 + (10 এর 10%) মিটার 
= 10 + 1 = 11 মিটার

∴ 10% বৃদ্ধিতে বর্গক্ষেত্রের ক্ষেত্রফল = (11 × 11) = 121 বর্গমিটার

∴ বর্গক্ষেত্রের ক্ষেত্রফল বৃদ্ধি পেয়েছে = (121 - 100) = 21 বর্গমিটার

∴ বর্গক্ষেত্রের ক্ষেত্রফল বৃদ্ধি পাবে = 21%

Question: X can complete a task in 12 days and Y in 18 days. If they work together for 3 days, what fraction of work is left?

Solution:
The work rate of X is 1/12 (since X can complete the work in 12 days).
The work rate of Y is 1/18 (since Y can complete the work in 18 days).
In 3 days they complete = 3{(1/12) + (1/18)}
= 3 × (5/36)
= 5/12

Then the fraction of work that is left is = 1 - (5/12)
= (12 - 5)/12
= 7/12

Question: The average age of A, B, C, D and E is 48 years. The average age of A and B is 40 years and the average of C and D is 50 years. Age of E is :

Solution:
A + B + C + D + E = 48 × 5 = 240
A + B = 40 × 2 = 80
C + D = 50 × 2 = 100
Therefore,
E = (A + B + C + D + E) - (A + B + C + D)
E = 240 - 80 - 100
E = 60 years

Question: The average of nine numbers is 60, that of the first five numbers is 55 and the next three is 65. The ninth number is 10 less than the tenth number. Then, tenth number is-

Solution:
Average of nine numbers = 60
Average of first five numbers = 55 and
average of next three numbers = 65
Tenth number = Ninth number + 10 

The sum of nine numbers = 60 × 9 = 540
The sum of the first five numbers = 55 × 5 = 275
The sum of the next three numbers = 65 × 3 = 195
Ninth number = (540 - 275 - 195) = (540 - 470) = 70

∴ Tenth number = 70 + 10 = 80

Question: A pipe can fill a tank in x hours, and another can empty it in y hours. In how many hours do they together fill it in (y > x)?

Solution:
Pipe fills the tank in x hours
∴ filling rate = 1/x tank/hour
Pipe empties the tank in y hours
∴ emptying rate = 1/y tank/hour
We are told y > x, so the filling pipe is faster than the emptying pipe

When both pipes are open, the net rate = (1/x) - (1/y)

∴ Time to fill the tank = Total work/Net rate
= 1/[(1/x) - (1/y)]
= 1/[(y - x)/xy]
= xy/(y - x) hours

Question: A pole 120 m long breaks at a point and the broken part bends such that it makes an angle of 30° with the ground (without getting separated). The length of the broken part is:

Solution:
 
খুঁটির মোট দৈর্ঘ্য = 120 মিটার
ধরি,ভাঙা অংশটির দৈর্ঘ্য = x মিটার
∴ অবশিষ্ট অংশটির দৈর্ঘ্য = (120 - x) মিটার
মই ভূমির সাথে কোণ তৈরি করে, θ = 30°

আমরা জানি,
sinθ = লম্ব/অতিভুজ
⇒ sin 30° =(120 - x)/x
⇒ 1/2 = (120 - x)/x
⇒ x = 2(120 - x)
⇒ x = 240 - 2x
⇒ 3x = 240
∴ x = 80 মিটার

অতএব, খুঁটির ভাঙা অংশটির দৈর্ঘ্য = 80 মিটার।

Question: If the radius of a right cylinder is tripled and the height is reduced by 40%, then what would the percentage change in volume?

Solution: 
We know, 
Volume of a cylinder, V = πr²h .....(1)

Given that, 
New radius = 3r
New height = h - 40% of h
= (1 - 0.40)h
= 0.6h

∴ New volume, V' = π(3r)²(0.6h)
= π × 9r² × 0.6h
= π × 5.4 × r²h
= 5.4 × (πr²h)
= 5.4 × V ; [From 1]
So, new volume = 5.4 times the original volume

∴ Percentage change in volume = {(V' - V)/V} × 100%
= {(5.4V - V)/V} × 100%
= (4.4V/V) × 100%
= 4.4 × 100%
= 440% increase

So the volume increases by 440%.

Question: At what rate of compound interest per annum will a sum of Tk. 1800 become Tk. 3042 in two years?

Solution:
Given that,
Principal, P = Tk. 1800 
Amount after 2 years, C = Tk. 3042
Time, n = 2 years
Rate of interest per annum, r = ?


According to the compound interest formula,
C = P{1 + (r/100)}ⁿ

Question: City B is 5 miles east of city A. City C is 10 miles southeast of city B. Which of the following is the closest to the distance from city A to City C?

Solution:

BD এবং DC দুটো সমান যেহেতু বিপরীত কোন দুইটাই 45°
অর্থাৎ, BDC ত্রিভুজ থেকে আমরা পাই,
BC² = BD² + DC²
⇒ 10² = x² + x²
⇒ 2x² = 100
⇒ x² = 50
∴ x = 5√2

অনুরূপে, ADC থেকে পাই,
AC² = AD² + DC²
⇒ AC² = (5 + x)² + x²
= 5² + 2 · 5 · x + x² + x²
= 25 + 10 · 5√2 + (5√2)² + (5√2)²
= 25 + 50√2 + 50 + 50
= 125 + 50√2
= 125 + 70.71
= 195.71
∴ AC = √195.71 = 13.99
≈ 14 miles

অর্থাৎ, A থেকে C এর নিকটবর্তী দূরত্ব 14 মাইল।

Question: Two pipes, A and B, can fill a cistern together in 3 hours. If each pipe were opened separately, pipe B would take 8 hours more than pipe A to fill the cistern. How long would it take pipe A to fill the cistern on its own?

Solution: Let the time taken by A alone be x hours.
Then time taken by B alone = x + 8 hours.
Rate of A = 1/x cistern/hour. Rate of B = 1/(x+8) cistern/hour.
Combined rate = 1/x + 1/(x+8) = 1/3 (since together they fill in 3 hours).

Now,
1/x + 1/(x+8) = 1/3
⇒ (x+8 + x) / [x(x+8)] = 1/3
⇒ (2x + 8) / [x(x+8)] = 1/3
⇒ 3(2x + 8) = x(x+8) [Cross multiply]
⇒ 6x + 24 = x² + 8x
⇒ x² + 2x - 24 = 0
⇒ (x + 6)(x - 4) = 0
So, x = 4 (positive value).

(Other root is negative and discarded.)
Therefore, A will take 4 hours alone. 

3 year ago the age was = 27×3 = 81
Currently sum of their age is  = 81 + 9 = 90
5 year ago = 20×2 = 40
Now, the sum is = 40 + 10 = 50

So, Present age of the husband = 90 - 50 = 40

Question: A certain sum of money becomes 2.5 times itself in 10 years at simple interest. What is the rate of interest per annum? 

Solution: 
Let the principal amount be P.
The sum becomes 2.5 times itself in 10 years.

So, A = 2.5P
∴ Simple Interest, I = 2.5P - P = 1.5P

We know, I = Pnr/100
1.5P = P × 10 × r/100
10r = 150
∴ r = 15%

Numbers can be divided by 7 are: 7, 14, 21, 28, 35
Among them, 14, 28 are divided by 14 with no remainder
But when 7, 21, 35 these numbers are divided by 14, the remainder is 7

যেহেতু,
N ঋনাত্মক পূর্ণসংখ্যা।
ধরি, N = -7

∴ (N)² = (-7)² = 49
6 - N = 6 - (-7) = 13
- N = -(-7) = 7
6 + N = 6 + (-7) = -1

তবে অপশন D তে N এর মান -6 বা তার কম হলে এটার মানও ধণাত্মক হতো। এজন্যই প্রশ্নে 'Could have a negative value' টার্ম ব্যবহার করা হয়েছে

∴ 6 + N এর মান ঋনাত্মক হতে পারে।

y% of x = 29
Or, y/100 × x = 29
So, x = 29×100/y = 2900/y

Question: In the sequence (1/√2), 1, √2............... which term is 8√2?

Solution:
দেওয়া আছে,
অনুক্রমটির প্রথম পদ, a = 1/√2
সাধারন অনুপাত, r = 1/(1/√2) = √2
n তম পদ = ar^{n - 1}

প্রশ্নমতে,
ar^{n - 1} = 8√2
⇒ (1/√2) × (√2)^{n - 1} = 8√2
⇒ (√2)^{n - 1} = 8√2 × √2
⇒ (√2)^{n - 1} = 16
⇒ (√2)^{n - 1} = (√2)⁸
⇒ n - 1 = 8 
⇒ n = 8 + 1 = 9 

অর্থাৎ অনুক্রমটির 9-তম পদ হলো 8√2

Question: A train, 150 m long, passes a pole in 15 seconds and another train of the same length travelling in the opposite direction in 12 seconds. The speed of the second train is-

Solution:
Given that,
Length of both trains = 150 m
First train passes a pole in 15 s
First train passes the second train (opposite direction) in 12 s

Now,
Speed of the first train,
= 150/15
 = 10 m/s
Time taken by trains to cross each other = 12 sec
And, relative speed of two trains :
= (150 + 150)/12
= 25 m/s

∴ Speed of the second train is 
= (25 - 10) × 18/5
= 15 × 18/5
= 54 km/hr

So the speed of the second train is 54 km/hr.

Question: How many ways can the word QUESTION be arranged with 2 letters each time?

Solution:
'QUESTION' শব্দটিতে মোট 8 টি ভিন্ন ভিন্ন বর্ণ আছে (Q, U, E, S, T, I, O, N)

8টি ভিন্ন বর্ণ থেকে প্রতিবার 2টি বর্ণ নিয়ে সাজাতে হবে।

n টি ভিন্ন বস্তু থেকে r টি বস্তু নিয়ে সাজানোর উপায় = ⁿP_r = n!/(n - r)!

এখানে n = 8, r = 2
∴ ⁸P₂ = 8!/(8 - 2)!
= 8!/6!
= (8 × 7 × 6!)/(6!)
= 8 × 7
= 56

∴ 'QUESTION' শব্দের বর্ণগুলো থেকে প্রতিবার 2টি বর্ণ নিয়ে মোট 56 উপায়ে সাজানো যায়।

Let, John's present age = 6x,
and, Mary’s present age = 4x
Therefore,
(6x - 5)/(4x - 5) = 5/3
Or, 20x - 25 = 18x - 15
Or,  x = 5
∴ John’s age = 6 × 5 = 30 years

Question: The mean of 8 numbers is 22. Five of them are 20, 18, 24, 16, and 26. Find the mean of the remaining three numbers.

Solution:
Mean of 8 numbers = 22 
Total sum of 8 numbers,
= 8 × 22 
= 176

Sum of the given 5 numbers,
= 20 + 18 + 24 + 16 + 26
= 104

Sum of the remaining 3 numbers:
= 176 - 104
= 72

∴ Mean of the remaining 3 numbers,
= 72/3
= 24

Question: If A is a zero matrix, then A + B = ?

Solution:
যদি A একটি শূন্য ম্যাট্রিক্স (zero matrix) হয়, তবে এর সব উপাদান শূন্য।
ম্যাট্রিক্স যোগ করার সময় প্রতিটি অবস্থানের উপাদানগুলি যথাক্রমে যোগ করা হয়।

তাই A + B মানে প্রতিটি অবস্থানে A-এর উপাদান এবং B-এর উপাদান যোগ করা।
যেহেতু A-এর সব উপাদান শূন্য, প্রতিটি অবস্থানে যোগফল শুধু B-এর উপাদানই থাকবে। তাই A + B = B.

এটি একটি মৌলিক বৈশিষ্ট্য যা শূন্য ম্যাট্রিক্সের সাথে যে কোনো ম্যাট্রিক্স যোগ করলে মূল ম্যাট্রিক্স অপরিবর্তিত থাকে।
সুতরাং সঠিক উত্তর হলো Matrix B।

- উত্তর: খ) Matrix B

Let father's age be x year and son's age be y year.
According to question,
2(x+y) = 8y _______(I)
and (x+y)/2 = 30
=> x+y = 60 year_______(II)
From equation (I) and (II)
8y = 120
y = 15 year,
Hence x = 45 year.

Let the number be x and y, such that x > y.
then, 3x - 4y = 5 ..........(i)
And (x + y) - 6 (x - y) = 6 ⇔ -5x + 7y = 6 ........(ii)
Solving (i) and (ii), we get : x = 59 and y = 43
Answer : 59, 43.

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: A reduction of 10% in the price of tea enables a dealer to purchase 25 kg more tea for Tk 22500. What is the reduced price per kg of tea?

Solution:
Let 10% of 22500 = 2250
Now,
25 kg = 2250
⇒ 1kg = 2250/25
∴ 1kg = 90

Question: It was Friday on January 1, 2016. What was the day of the week on Jan 1, 2017?

Solution:
• অধিবর্ষ নির্ণয়:
2016 সালটি 4 দ্বারা বিভাজ্য (2016 ÷ 4 = 504), তাই এটি একটি অধিবর্ষ (Leap Year)।

• অতিরিক্ত দিন নির্ণয়:
একটি অধিবর্ষে 366 দিন থাকে। 366 ÷ 7 = 52 সপ্তাহ এবং 2 দিন অতিরিক্ত (Odd Days) হয়।

এখন,
1লা জানুয়ারি 2016 দিনটি ছিল Friday।
যেহেতু 2016 সালটি অধিবর্ষ, তাই 1লা জানুয়ারি 2017 দিনটি হবে Friday এর থেকে 2 দিন বেশি।
⇒ Friday + 1 দিন = Saturday
⇒ Saturday + 1 দিন = Sunday

∴ 1লা জানুয়ারি 2017 দিনটি হবে Sunday।

Given that,
A dog takes 3 leaps for every 5 leaps of a hare
Therefore,
Dog : Hare = 3 : 5
One leap of a dog is equal to 3 leap of the hare
Therefore,
1 leap of dog = 3 leap of hare
Now the ratio becomes,
Dog : Hare = 3 (3) : 5
Dog : Hare = 9 : 5
Thus the ratio of the speed of the dog to that of the hare is 9 : 5

Question: The length of a room is 5.5 m and the width is 3.75 m. Find the cost of paving the floor with slabs at the rate of Tk. 800 per square metre.
(একটি ঘরের দৈর্ঘ্য ৫.৫ মিটার এবং প্রস্থ ৩.৭৫ মিটার। প্রতি বর্গমিটার ৮০০ টাকা হারে মেঝেতে স্ল্যাব বসানোর খরচ কত হবে?)

Solution:
ফ্লোরের ক্ষেত্রফল
= (5.5 × 3.75) m²
= 20.625 m²

∴ মেঝে বাঁধানোর খরচ
= (800 × 20.625) Tk
=16500 Tk

(7^a)(7^b) = 7^c/7^d
Or, 7^{a + b} = 7^c-d
Or, a + b = c - d
Or, d = c - a - b

Question: Two pipes A and B can fill the tank in 24 and 36 minutes, respectively. Both the pipes are opened together. After how many minutes should the pipe B be turned off, so that the tank be fill in 18 minutes?

Solution:
Given that,
Pipe A fills the tank in 24 minutes.
Pipe B fills the tank in 36 minutes.
Total time to fill the tank = 18 minutes.
Now,
LCM of 24 and 36 = 72 (Total capacity of the tank).
Efficiency of pipe A = 72/24 = 3 units/minute.
Efficiency of pipe B = 72/36 = 2 units/minute.

Let,
pipe B be turned off after x minutes.
Pipe A works for 18 minutes.
Pipe B works for x minutes.
Work done by A in 18 minutes = 3 × 18 = 54 units.
​Work done by B in x minutes = 2x = 2x units.

Total work done = 54 + 2x = 72
⇒ 2x = 72 - 54
⇒ 2x = 18
⇒ x = 18/2
∴ x = 9

∴ Pipe B should be turned off after 9 minutes.

If the remainder is same in each case and remainder is not given,
HCF of the differences of the numbers is the required largest number.

9997 - 7654 = 2343
9997 - 8506 = 1491
8506 - 7654 = 852
Hence, the greatest number which divides 7654, 8506 and 9997 and leaves same remainder
= HCF of 2343, 1491, 852
= 213
Now we need to find out the common remainder.
Take any of the given numbers from 7654, 8506 and 9997, say 7654
7654/213 = 35,
remainder = 199.

(a - b)² = a² - 2ab + b²
⇒ 6² = (a² + b²) - 2ab
⇒ 6² = 116 - 2ab
⇒ 36 = 116 - 2ab
⇒ 2ab = 116 - 36
⇒ 2ab = 80
⇒ ab = 40.

Let, first number be x and second number be y
Here,
x + y = 36 .... (i)
and, 5x + 3y = 142 ...(ii)

by multiplying the first equation by 3
3x + 3y = 108
5x + 3y = 142

by subtraction the second equation from the first
(3–5)x + (3–3)y = 108–142
Or, -2x = -34
Or, x = 17

by substituting x in the first equation
17 + y = 36
Or, y = 19

As we know,
a³ + b³ + c³ - 3abc = a² + b² + c² - ab - bc - ca(a + b + c)
when (a + b + c) = 0
Then a³ + b³ + c³ - 3abc = 0
When x + y + z = 0
⇒ x³ + y³ + z³ = 3xyz
⇒ x³ + y³ + z³ + 3xyz = 3xyz +3xyz
= 6xyz.

Question: A fruit shop has 12 types of fruits. You don’t like Mango and Papaya. How many ways can you select 5 different fruits from the ones you like?

Solution:
Given that, 
Total fruits = 12
Fruits you don’t like = 2
∴ Fruits you can choose = 12 - 2 = 10
Number of fruits to choose = 5

∴ Number of ways = ¹⁰C₅ = 10!/5!(10 - 5)!
= (10 × 9 × 8 × 7 × 6 × 5!)/(5! × 5!)
= (10 × 9 × 8 × 7 × 6)/(5 × 4 × 3 × 2)
= 252

So, there are 252 ways to select 5 different fruits from the ones you like.

In the original 125 gallons of mixture, 20% is water.
Hence, no. of gallons of other materials in the mixture: 125 x 80% = 100 gallons
In the new mixture, water makes up 25%, thus 75% is other materials.
As no. of gallons of others is unchanged, 100 gallons = 75% in the new mixture volume.
The total volume of the new mixture is : 100 / 75% = 100/ 0.75 = 133.33 gallons.
∴ Required additional amount of water = 133.3 – 125 = 8.33 = 8(1/3) gallons