Bank Math

Question: X, Y, and Z are hired to complete a project for Tk. 6900. X and Y together complete 17/25 of the work, and Y and Z together complete 13/25 of the work. What is the wage of Y in Taka?

Solution:
X + Y = 17/25
Y + Z = 13/25

অতএব,
X + Y + Y + Z = (17/25) + (13/25)
= 30/25

যেখানে, X + Y + Z = 1 (সম্পূর্ণ কাজ)

এখন,
(X + Y + Z + Y) - (X + Y + Z) = 30/25 - 1
∴ Y = 5/25 = 1/5

∴ Y এর বেতন = (1/5) × 6900 = Tk. 1380

Question: A factory produces 180 items in 2 days working 9 hours per day. How many items would it produce in 9 days working 10 hours per day?

Solution: 
180 items in 2 days, working 9 hours per day
Total hours worked = 2 × 9 = 18 hours
So, production rate = (180/18) = 10 items/hour

Total hours for 9 days working 10 hours/day
= 9 × 10 = 90 hours

So, total items = 90 × 10 = 900 items

Question: A woman borrowed a certain amount of money for 8 months. She paid Tk. 1,200 as simple interest at a rate of 12% per year. What was the original amount (principal) she borrowed?

Solution:
Simple Interest (SI) formula:
SI = [P (principal) × R (rate) × T (time)] / 100

Here,
SI = 1200
R = 12% per annum
T = 8 months = 8/12 = 2/3 year

Now,
1200 = [P × 12 × (2/3)] / 100
⇒ 1200 = (P × 8) / 100
⇒ P = (1200 × 100) / 8
⇒ P = 150000 / 8
⇒ P = Tk. 15000

Question: For what values of m are the roots of the quadratic equation mx (x - 2√5) + 10 = 0 real and equal?

Solution:
mx (x - 2√5) + 10 = 0
⇒ mx² - 2√5 mx + 10=0

Compare given equation with the general form of quadratic equation, which ax² + bx + c=0
a = m, b = - 2√5m, c = 10

Since roots are real and equal, discriminant, D = 0
b² - 4ac = 0
⇒ (- 2√5m)² - 4 × m × 10 = 0
⇒ 20m² - 40m = 0
⇒ 20(m² - 2m) = 0
⇒ m² - 2m = 0
⇒ m(m - 2) = 0

Either, m = 0 

Or, m - 2 = 0
∴ m = 2

Question: A company's profits have doubled for each of the last 5 years. If the total profits for the last 5 years were Tk. 62 lacs, what were the profits in the first year?

Solution:
ধরা যাক, প্রথম বছরের লাভ P টাকা। যেহেতু লাভ প্রতি বছর দ্বিগুণ হয়েছে,
সুতরাং, গত 5 বছরের লাভের পরিমাণ ছিল: P, 2P, 4P, 8P, এবং 16P।
প্রশ্ন অনুযায়ী, গত 5 বছরের মোট লাভ ছিল 62 লক্ষ টাকা।

∴ P + 2P + 4P + 8P + 16P = 62,00,000
⇒ 31P = 62,00,000
⇒ P = 62,00,000 / 31
⇒ P = 2,00,000
সুতরাং, প্রথম বছরে লাভ ছিল 2,00,000 টাকা।

এখানে দিন বাকি = 30 - 20 = 10 দিন এবং কাজ বাকি = 1 - 1/2 = 1/2অংশ
20 দিনে একটি কাজের 1/2 অংশ সম্পন্ন করে 50 জন
শ্রমিক
1 দিনে একটি কাজের 1/2 অংশ সম্পন্ন করে (50×20) = 1000 জন শ্রমিক
10 দিনে একটি কাজের 1/2 অংশ সম্পন্ন করে 1000/10 জন = 100 জন
সুতরাং অতিরিক্ত শ্রমিক লাগবে = 100 - 50 = 50 জন।

The perimeter is = 25 + 40 + 55 = 120 ft
Distance between pillars is 5 feet.
Required pillars = 120/5 + 1 = 25 [If it was a straight single line]
But as it is a triangular plot, one pillar overlaps 
So required pillar number is 25 - 1 = 24

Question: Rafsan and Karim entered into a business partnership by investing Tk. 20,000 and Tk. 30,000 respectively. If the total profit earned is Tk. 7,000, what is Rafsan’s share of the profit?

Solution:
Given,
Investment ratio = 20000 : 30000
= 2 : 3
Sum of the ratio's = 2 + 3 = 5

∴ Rafsan's share = 7000 × (2/5)
= Tk. 2800

Question: A student scored 30% marks and failed by 15 marks. Another student scored 50% marks and secured 25 marks more than the pass marks. What is the pass percentage?

Solution:
Let total marks = x

According to the question,
30% of x +15 = 50% of x - 25
⇒ 0.30x + 15 = 0.50x - 25
⇒ 0.50x - 0.30x = 25 + 15
⇒ 0.20x = 40
⇒ x = 40/0.20
∴ x = 200

∴ Pass marks = 30% of x + 15
= 0.30 × 200 + 15
= 60 + 15 
=75

∴ Pass percentage = (75/200) × 100
= 37.5%

Question: A boat can cover 'r' km upstream in 6 hours. It can cover 'r + 18' km downstream in 4 hours. Find the time taken by boat to cover 81 km upstream and 90 km downstream if the speed of boat in still water and speed of stream are in the ratio 4 : 1, respectively.

Solution:
Given that,
Boat covers r km upstream in 6 hours
Boat covers (r + 18) km downstream in 4 hours
Ratio of speed of boat in still water : speed of stream = 4 : 1

Let speed of boat in still water = 4k km/hr
Let speed of stream = 1k km/hr

∴ Upstream speed = 4k − 1k = 3k
∴  Downstream speed = 4k + 1k = 5k

Now,
r/6 = 3k
∴ r = 18k …… (1)

And, (r + 18)/4 = 5k
(18k + 18)/4 = 5k ; [From 1]
⇒ 18k + 18 = 20k
⇒ 20k - 18k = 18
⇒ 2k = 18
⇒ k = 18/2 = 9
∴ k = 9

Now, upstream speed = 3k = 27 km/hr
Downstream speed = 5k = 45 km/hr

Required Time,
Time for 81 km upstream = 81/27 = 3 hours
Time for 90 km downstream = 90/45 = 2 hours

∴ Total time = 3 + 2 = 5 hours

∴ The boat takes 5 hours to cover 81 km upstream and 90 km downstream.

Question: Rupa and Puja started from the opposite direction of a 12km racing track. Their speed is 10kmph and 8kmph respectively. After which point they will meet if they start simultaneously?

Solution: 
let, they will meet after t hour at point X.
in t hour,
Rupa will cross = 10t km
Puja will cross = 8t km

ATQ,
10t + 8t = 12
18t = 12
t = 12/18 hour

in 12/18 hours,
Rupa will cross = (12/18)10 km = 6667m
puja will cross = (12/18)8 km = 5333m

Question: A bag contains 7 green balls, 8 blue balls, and 5 yellow balls. One ball is drawn at random. What is the probability that the ball drawn is neither green nor yellow?
​
​Solution:
​Total balls = 7 + 8 + 5 = 20
​
​Favorable outcomes = balls that are neither green nor yellow, that mean blue balls = 8
​
​∴ P(blue) = Favorable outcomes​/total outcomes​ = 8/20 = 2/5

Given,
x = y^a
⇒ x = (z^b)^a
⇒ x = (x^c)^ab
⇒ (x)^abc = x¹
∴ abc = 1

Question: The H.C.F. of two number is 11 and their L.C.M. is 693. If one of the numbers is 77, find the other.

Solution:
Given that,
H.C.F. of two numbers = 11
L.C.M. of two numbers = 693
One number = 77
Let the other number = x


We know the relation between H.C.F., L.C.M., and two numbers.
⇒ Product of the two numbers = H.C.F. × L.C.M.
⇒ 77 × x = 11 × 693 
⇒ x = (11 × 693)/77
⇒ x = 693/7
∴ x = 99

So the other number is 99.

Question: Two dice are thrown simultaneously. What is the probability that the total score is an even number?

Solution:
Clearly, n(S) = (6 × 6) = 36

Let E = Event that the sum is an even number.
E = {(1, 1), (1, 3), (1, 5),
     (2, 2), (2, 4), (2, 6),
     (3, 1), (3, 3), (3, 5),
     (4, 2), (4, 4), (4, 6),
     (5, 1), (5, 3), (5, 5),
     (6, 2), (6, 4), (6, 6)}

∴ n(E) = 18

∴ P(E) = n(E)/n(S)
= 18/36
= 1/2

প্রশ্ন: What will be the number in the question mark? 3, 7, 15, 27, 43, ?

সমাধান:
প্রদত্ত ধারাটি হলো: 3, 7, 15, 27, 43, ?

ধারার সংখ্যাগুলোর মধ্যে পার্থক্য নির্ণয় করি:
7 - 3 = 4
15 - 7 = 8
27 - 15 = 12
43 - 27 = 16

এখানে, প্রতিবার পার্থক্য 4 করে বৃদ্ধি পাচ্ছে।

∴ পরবর্তী পার্থক্য হবে = 16 + 4 = 20
∴ পরবর্তী সংখ্যাটি হবে = 43 + 20 = 63

অতএব, প্রশ্নবোধক স্থানে 63 বসবে।

Shortcut: 3 (+4)→ 7 (+8)→ 15 (+12)→ 27 (+16)→ 43 (+20)→ 63

According to the question,
Retail price 40 Tk. (per pen).
20% discount = 40 × (25/100)
= 10 Tk.
অর্থ্যাৎ,
প্রতি pen এ save হয় = 10 Tk.
Total savings = 240 Tk.
∴ Number of pen = 240/10
= 24 pens.

Question: The average of six numbers is A and the average of three of these is B. If the average of the remaining three is C, then which one is correct?

Solution:
Total sum of six numbers = 6A
Total sum of three numbers = 3B
Total sum of the other numbers = 3C

Now,
6A = 3B + 3C
or, A = 3(B + C)/6
or. A = (B + C)/2
∴ 2A = B + C

Question: A boat travels 18 km downstream in 45 minutes. If the speed of the stream is 3 km/h, what is the speed of the boat in still water?

Solution:
স্রোতের অনুকূলে 45 মিনিটে যায় 18 কিমি
স্রোতের অনুকূলে 1 মিনিটে যায় 18/45 কিমি
স্রোতের অনুকূলে 1 ঘণ্টা বা 60 মিনিটে যায় (18 × 60)/45 কিমি
= 24 কিমি

∴ স্রোতের অনুকূলে বেগ = 24 কিমি/ঘণ্টা

দেওয়া আছে,
স্রোতের বেগ = 3 কিমি/ঘণ্টা।

∴ স্থির পানিতে নৌকার বেগ = স্রোতের অনুকূলে বেগ - স্রোতের বেগ
= 24 - 3 = 21 কিমি/ঘণ্টা।

Question: If n is an even integer, which of the following must be an odd integer?

Solution:
Let,
n = 2

ক) n + 2 = 2 + 2 = 4  ; even
খ) n² = 2² = 4   ; even
গ) 2n +1 = (2 × 2) + 1 = 5   ; odd
ঘ) n² + n = 2² + 2 = 6   ; even

Question:

Solution:

Question: The sum of the present ages of a son and his father is 70 years. After 5 years, the age of the father will be three times that of the son. At present, their ages are?

Solution:
Let,
The son's age is x years.
Then, father's age = (70 - x) years.

ATQ,
(70 - x) + 5 = 3(x + 5)
⇒ 75 - x = 3x + 15
⇒ 4x = 60
∴ x = 15

∴ Son's age = 15 years
And the father's age = (70 - 15) = 55 years.

From the figure, ∠DOC = 180°
Or, ∠DOB + ∠BOC = 180°
Or, ∠BOC = 180° - 63° = 117°

Question: In how many different ways can the letters of the word 'MEETING' be arranged so that the vowels always come together?

Solution:
In the word 'MEETING' we treat the three vowels 'E, E' and 'I' as one letter. Thus, we have MTNG(EEI) this has 5 letters.
As we know that number of ways of arranging n letters out of which r are of same type = n!/r!

∴ Number of ways of arranging these letters = 5! = 120
And 3 vowels (E, E,I) can be arranged in 3!/2! = 3 ways

∴  Required number of ways = 120 × 3 = 360

Question: An F‑7 BGI fighter jet covers a certain distance at a speed of 1200 km/h in 5 hours. What speed must it maintain to cover the same distance in 250 minutes?

Solution:
Total distance = Speed × Time
= (1200 × 5) km
= 6000 km

Given time = 250 minutes = (250/60) hours
= 25/6 hours

∴ Required speed = Distance/Time
= {6000/(25/6)} km/h
= {6000 × (6/25)} km/h
= (240 × 6) km/h
= 1440 km/h

To gain 20% on whole he must sell all good for,
Tk. 400 + 20% of 400 = 480
As he get 5% gain on half of the goods i.e. 200 + 5% of 200 = 210
So required balance = 480 - 210 = 270
He must gain Tk. 70 on rest Tk. 200
% gain on remainder goods = (70×100)/200
= 35%

Question: P is five years younger than Q, who is twice as old as R. If the sum of their ages is 55, how old is Q?

Solution:
Let,
the age of R be = x years
Then,
the age of Q = 2x years
and age of P = (2x – 5) years

According to the question,
(2x – 5) + 2x + x = 55
⇒ 5x – 5 = 55
⇒ 5x = 60
⇒ x = 12

Hence, age of Q = 2x = (2 × 12) years = 24 years

Question: If the volume of a sphere is 972π cm³, what is the surface area of the sphere?

Solution:
দেওয়া আছে,
গোলকের আয়তন, V = 972π cm³

আমরা জানি,
গোলকের আয়তন, V = (4/3)πr³
⇒ (4/3)πr³ = 972π
⇒ r³ = (972π × 3)/(4π)
⇒ r³ = (972 × 3)/4 
⇒ r³ = 243 × 3
⇒ r³ = 729 
∴ r = 9 cm

গোলকের সমগ্র পৃষ্ঠতলের ক্ষেত্রফল = 4πr²
= 4π(9)²
= 4π × 81
= 324π cm².

অতএব, গোলকের সমগ্র পৃষ্ঠতলের ক্ষেত্রফল = 324π cm²

Question: Four times the first of three consecutive even integers is 4 more than twice the third. The second integer is -

Solution:
Let the three integers be x, (x + 2) and (x + 4)

ATQ,
4x = 2(x + 4) + 4
⇒ 4x = 2x + 12
⇒ 2x = 12
∴ x = 6

∴ Second integer = x + 2
= 6 + 2
= 8

Question: If x and y are odd numbers, which number is even?

Solution:
Let x = 1 and y = 3 (both are odd numbers)

a) x × y = (1 × 3) = 3 ............. Odd

b) x + y + 1 = (1 + 3 + 1) = 5 ......... Odd

c) 3x + 4 = (3 × 1 + 4) = 3 + 4 = 7 ......... Odd 

d) 2x + 2y = (2 × 1 + 2 × 3) = 2 + 6 = 8 .......... Even 

Average speed of Joy = 2xy/(x + y)
= (2 × 25 × 4)/(25 + 4)
= 200/29 km/h

Distance traveled = Speed × Time
= 200/29 × 29/5
= 40 Km

Distance between city and town = 40/2 = 20 km.

Question: Rihan can write 120 pages in 30 hours. Zayan and Rihan together can write 240 pages in 40 hours. In what time can Zayan write 60 pages?

Solution:
Given,
In 30 hours Rihan can write 120 pages
∴ In 1 hour Rihan can write 120 ÷ 30 = 4 pages

Rihan and Zayan together can write 240 ÷ 40 = 6 pages per hour

∴ Zayan's 1 hour work = (Rihan + Zayan)'s 1 hour work - Rihan's 1 hour work
= 6 - 4 = 2 pages/hour

Zayan's time:
2 pages in 1 hour
∴ 1 page in 1/2 hour
∴ 60 pages in (1 × 60) ÷ 2
= 30 hours

Question: In how many different ways can the letters of the word 'LEARN' be arranged so that the vowels are never together? 

Solution:
Total letters = 5 (L, E, A, R, N)
Vowels = E, A (2 vowels)
Consonants = L, R, N (3 consonants)

Total arrangements of all 5 letters = 5! = 5 × 4 × 3 × 2 × 1 = 120

Treating the vowels (EA) as one single letter,
Now we have: (EA), L, R, N → 4 units.
These 4 units can be arranged in = 4! = 24 ways.

The two vowels EA can be arranged in their place in 2! = 2 ways
∴ Number of arrangements where vowels are together = 24 × 2 = 48 ways

∴ Number of arrangements where vowels are never together
= Total - Together
= 120 - 48
= 72

To earn Tk. 135, investment = Tk. 1620.
To earn Tk. 8, investment = Tk.(1620/135 × 8)
= Tk 96
So, Market value of Tk. 100 stock = Tk. 96.

Question: A clock shows 7 O’clock in the morning. By how much angle will the hour’s hand rotate when the clock shows 9 O’clock in the morning?

Solution:
We know, 
The hour hand takes 12 hours to rotate 360°
The difference of time = 9 O'clock - 7 O'clock = 2 hours

Now,
in 12 hours the hour hand rotates 360°
In 1 hour the hour hand rotates = (360/12)°
In 2 hours the hour hand rotates = 30° × 2 = 60°

Question: In a class, 30 students play basketball, 20 students play volleyball, and 8 students play both. 12 students play neither basketball nor volleyball. What is the total number of students in the class?

Solution:
Let the number of students who play basketball = 30
Number of students who play volleyball = 20
Number of students who play both basketball and volleyball = 8
Number of students who play neither = 12

First, calculate the number of students who play basketball or volleyball:
n(B ∪ V) = n(B) + n(V) − n(B ∩ V)
n(B ∪ V) = 30 + 20 − 8 = 42

Now, add the students who play neither sport to get total students:
Total students = n(B ∪ V) + neither

Total students = 42 + 12 = 54

Question: A circular garden has a radius of 10 feet. If the radius is increased by 10%, what is the new area of the garden?

Solution:
Given that,
Original radius, r = 10 ft
And Increase radius by 10%

∴ New radius, r' = 10 + (10% of 10) = 10 + 1 = 11 ft.

We know, Area of circle = πr²

∴ New area with radius 11 ft
A′ = π(11²) = π × 121 = 121π ft²