Bank Math

Question: If ⁶P_r = 120, what is the value of r?

Solution:
আমরা জানি, ⁿP_r = n!/(n - r)!
দেওয়া আছে,
⁶P_r = 120
⇒ 6!/(6 - r)! = 120
⇒ 720/(6 - r)! = 120 (কারণ 6! = 720)
⇒ (6 - r)! = 720/120
⇒ (6 - r)! = 6
⇒ (6 - r)! = 3!   (3! = 3 × 2 × 1 = 6)
⇒ 6 - r = 3
⇒ r = 6 - 3
∴ r = 3

Question: What are the roots of the equation √(x + 7) = x - 5 ?

Solution:
দেওয়া আছে,
√(x + 7) = x - 5
⇒ {√(x + 7)}² = (x - 5)² [উভয়পক্ষ বর্গ করে]
⇒ x + 7 = x² - 10x + 25
⇒ x² - 10x - x + 25 - 7 = 0
⇒ x² - 11x + 18 = 0
⇒ x² - 9x - 2x + 18 = 0
⇒ x(x - 9) - 2(x - 9) = 0
⇒ (x - 9)(x - 2) = 0
⇒ x = 9 or x = 2

এখন মূলগুলো যাচাই করি:
যখন x = 2,
বাম পক্ষ: √(2 + 7) = √9 = 3
ডান পক্ষ: 2 - 5 = - 3
(3 ≠ - 3) ⇒ তাই x = 2 অতিরিক্ত মূল (বর্জনীয়)।

যখন x = 9,
বাম পক্ষ: √(9 + 7) = √16 = 4
ডান পক্ষ: 9 - 5 = 4
(4 = 4) ⇒ গ্রহণযোগ্য।

সুতরাং, সমীকরণটির একমাত্র মূল হলো x = 9

Question: Rina ,Ratul and Ayesha started a business. Rina invested 1/3 part, Ratul 1/4 part and rest of the capital was invested by Ayesha. The ratio of their profits will be-

Solution:
Let,
Total capital be TK x
Then,
Rina's share = x/3 TK
Ratul's share = x/4 TK
Ayesha's share = x - {(x/3) + (x/4)}
= x - {(4x + 3x)/12}
= x - (7x/12)
= (12x - 7x)/12
= 5x/12 TK

∴ Required ratio
= (x/3) : (x/4) : (5x/12)
= 4x : 3x : 5x     [multiply by 12]
= 4 : 3 : 5

Question: How much water should be added to 40 liters of pure milk to gain an extra 20% profit when selling the mixture at the price of pure milk?

Solution:
Assume,
Price of pure milk per liter = 100 Taka
So, the price of 40 liters of pure milk = 40 × 100 = 4000 Taka

Let, the amount of water added = x liters
Total mixture = (40 + x) liters

Since the mixture is sold at the price of pure milk,
Selling price of (40 + x) liters = 100(40 + x) Taka

According to the question,
100(40 + x) = 4000 + 4000 of 20%
⇒ 4000 + 100x = 4000 + (4000 × 20/100)
⇒ 4000 + 100x = 4000 + 800
⇒ 100x = 800
⇒ x = 800/100
⇒ x = 8

∴ Required water: 8 liters.

Question: An accurate clock shows 10:00 AM. Through how many degrees will the hour hand rotate when the clock shows 4:00 PM?

Solution: 
10:00 AM থেকে 4:00 PM পর্যন্ত অতিবাহিত সময় = 6 ঘণ্টা।

আমরা জানি,
ঘণ্টার কাঁটা 12 ঘণ্টায় ঘোরে 360°
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 6 ঘণ্টায় ঘোরে = 6 × 30° = 180°

অতএব, 10:00 AM থেকে 4:00 PM পর্যন্ত ঘণ্টার কাঁটাটি 180° ঘুরবে

Let the x, x + 2, x + 4 and x + 6 be the 4 consecutive odd integers.
We have to find the lowest integer x.
It is given that the average of these numbers is 24.
i,e., (x + x + 2 + x + 4 + x + 6)/4 = 24
⇒ 4x + 12 = 24x4
⇒ 4x = 84
⇒ x = 84/4
= 21
Hence, the answer is 21.

Question: When 25% of the first number is added to the second number, the second number becomes 3/2 times the first number. What is the ratio of the first number to the second number?

Solution:
Let the first number = x
and the second number = y.

According to the question,
y + 25% of x = (3/2)x
⇒ y + (25/100)x = (3/2)x
⇒ y + (1/4)x = (3/2)x
⇒ y = (3/2)x - (1/4)x
⇒ y = (6 - 1)x/4
⇒ y = 5x/4
∴ y = 5x/4

Therefore, x : y = 4 : 5 

Question: A cube has a total surface area of 384 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 = 384
⇒ a² = 384/6
⇒ a² = 64
∴ a = 8 মিটার

এখন,
ঘনকের আয়তন = a³
= 8³
= 512 ঘন মিটার

অতএব, ঘনকটির আয়তন = 512 ঘন মিটার।

Question: What is the difference between the simple interest on a principal of Tk. 1000 being calculated at 7% per annum for 2 years and 6% per annum for 3 years?

Solution:
Given that,
Principal, P = Tk. 1000
Rate, r₁ = 7% and Time, n₁ = 2 years
Rate, r₂ = 6% and Time, n₁ = 3 years

We know,
I = Prn/100
1st case:
I₁ = Prn/100 = (1000 × 7 × 2)/100 = 140

2nd case:
I₂ = Prn/100 = (1000 × 6 × 3)/100 = 180

∴ Difference = 180 - 140 = Tk. 40

Question: Yesterday, a shop made Tk. 5,000 in total sales. One-fifth of these sales came from items that were sold at a 20% discount. What would the shop’s total sales have been yesterday if all items had been sold at full price?

Solution:
Total sales = Tk. 5,000
Discounted sales = 1/5 of 5,000 = 1,000

Since these items were sold at 20% discount,
The full price of these items = 1,000 ÷ (1 - 0.20) = 1,000 ÷ 0.8 = Tk. 1,250
Sales from items not discounted = 5,000 - 1,000 = Tk. 4,000

If all items were sold at full price, total sales
= 4,000 + 1250
= Tk. 5250

Let, B invested the money for t months
Then the ratio of investment = (12×11 : 11× t) = 12 : t
So, 12/t = 4/1
⇒ t = 3 months

Question: The ratio of the number of red balls to yellow balls to green balls in an urn is 3 : 4 : 5. What is the probability that a ball chosen at random from the urn is a red ball? 

Solution:
লাল, হলুদ, ও সবুজ বলের সংখ্যার আনুপাতিক মান যথাক্রমে 3, 4, 5
মোট বলের সংখ্যার আনুপাতিক মান = 3 + 4 + 5
 = 12

∴ বলটি লাল হওয়ার সম্ভাব্যতা = 3/12
= 1/4

Question: A pole 120 meters long breaks into two parts without complete separation and makes an angle of 30° with the ground. Find the length of the broken part of the pole.

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: 6 pipes, working 10 hours a day, can empty a cistern in 3 days. How many hours a day must 9 pipes work to empty the cistern in one day?

Solution:
By applying the MDH method,
it can be written as,

6 × 10 × 3 = 9 × x × 1
⇒ x = 20 hours

Required no. of numbers = 5 ×⁵P₄
= 5 × 5!
= 5 ×120
= 600

Let the age of the son before 8 years ago = x
Then, age of Kamal before 8 years age = 4x
After 8 years, Kamal will be twice as old as his son
(4x + 16) = 2(x + 16)
⇒ 4x - 2x = 32 - 16
⇒ 2x = 16
⇒ x = 8 years.
The present age of kamal = 4x + 8
= (4 × 8) + 8
= 32 + 8
= 40 years.

প্রশ্ন: যদি a = 0.202 হয়, তাহলে 

এর মান কত?

সমাধান:

সঠিক উত্তর 1.202 হবে, যেহেতু (+) যোগ চিহ্ন দিয়ে বের করা রাশির উত্তর নেই।

Man's/Boat's Speed = X
Stream/Current/River Speed = Y

∴ Downstream speed = X + Y
Upstream speed = X - Y

X + Y = 18 km/hr and X - Y = 10 km/hr
Adding them we get,
X + Y + X - Y = 28 km/hr

∴ X = 14 km/hr = Speed of Motorist

Y = 18 - 14 = 4 km/hr = Speed of stream

Question: When a discount of 20% is given on a sweater, the profit is 20%. If the discount is 10%, then the profit is

Solution:
Let the C.P. of sweater be Tk. 100 and its marked price be Tk. x.
ATQ,
x × (80/100) = 120
⇒ 4x/5 = 120
⇒ x = 150 Tk

When discount = 10%, then
S.P. of sweater = 150 × (100 - 10)%
= (150 × 90)/100
= 135 Tk

∴ Profit = 135 - 100 = 35

∴ Profit percentage = (35/100) × 100% = 35%

Question: March 20, 1991 was a Wednesday. What day of the week was March 20, 1992?

Solution:
Given that, 
March 20, 1991 was a Wednesday.

1992 was a leap year which is divisible by 4.
A normal year (non-leap year) has 365 days and 365 ÷ 7 = 52 weeks + 1 day remainder ; day advances by + 1.
A leap year has 366 days and 366 ÷ 7 = 52 weeks + 2 days remainder ; day advances by + 2.

Since the period from March 20, 1991 to March 20, 1992 includes February 29, 1992 (the leap day), it is a full 366-day period.
Therefore Wednesday + 2 days
Wednesday ⇒ Thursday ⇒ Friday

So March 20, 1992 is Friday.

30% discount = 30% of 2000 = Tk. 600
Now, if you offer two successive discounts of 15% each, it works out to
First discount of 15% = 15% of 2000
= Tk. 300
After discount value = Tk. 2000 - Tk.300
= Tk. 1700
Second discount of 15% = 15% of Tk. 1700
= {(15/100) × 1700}
= Tk. 255

Difference = 600 - (300+255)
= 600 - 555
= Tk. 45

According to math,
If,
x = y
Then, 1 - q = 2q + 1
⇒ 2q + q = 1 - 1
⇒ 3q = 0
⇒ q = 0.

Question:

Solution:

Solution: 
Let, present ages of three persons 4x, 7x, 9x 

Eight years ago, the sum of their ages was 56 years
At present, the sum of their ages is = 56 + 8 + 8 + 8 
= 56 + 24 years 
= 80 years 

4x + 7x + 9x = 80 
⇒ 20x = 80
⇒ x = 4 

The present age of the eldest person is = 4 × 9 = 36 years 

A's 1 day work = 1/18
B's 1 day work = 1/9 [because B take half time than A]
So, (A + B)'s one day work
= 1/18 + 1/9
= (1+2)/18
= 1/6

Question: If tan(θ + 30°) = 1, find the value of cosθ.

Solution:
Given,
tan(θ + 30°) = 1
⇒ tan(θ + 30°) = tan 45°
⇒ θ + 30° = 45°
⇒ θ = 45° - 30°
⇒ θ = 15°

Now,
cosθ = cos 15°
∴ cos 15° = cos(45° - 30°)
= cos45° cos30° + sin45° sin30°
= (1/√2 × √3/2) + (1/√2 × 1/2)
= (√3/2√2) + (1/2√2)
= (√3 + 1)/(2√2) 

Question: Bimol takes twice as much time as Rakib to complete a work and Alfi does it in the same time as Bimol and Rakib together. If all three working together can finish the work in 9 days, then the time taken by Bimol to finish the work is-

Solution:
Let,
Rakib takes x days to complete a work
Then, Bimol takes 2x days to complete the work

Rakib's 1 day's work = 1/x
Bimol's 1 day's work = 1/2x
Alf's 1 day's work = (1/x) + (1/2x) = 3/2x

(Bimol + Rakib + Alfi)'s 1 day's work = (1/x) + (1/2x) + (3/2x)
= (2 + 1 + 3)/2x
= 3/x

ATQ,
3/x = 1/9
⇒ x = 27
Hence, Bimol takes (2 × 27) = 54 days to complete the work.

Question: If log_x(16/81) = - 4, then what is the value of x?

Solution:
log_x(16/81) = - 4
⇒ x^{- 4} = 16/81 [log_ba = c ⇒ b^c = a]
⇒ x^{- 4} = (2/3)⁴
⇒ x^{- 4} = 1/(3/2)⁴
⇒ x^{- 4} = (3/2)^{- 4}
⇒ x = 3/2

Question: Two unbiased coins are tossed. What is the probability of getting at most one head?

Solution:
Total cases = {HH, HT, TH, TT} = 4
Favorable cases = {HH, HT, TH} = 3

∴ Required Probability = 3/4

Question: Mina has 3 Taka more than Babu has, but 5 Taka less than Shelly has. If Mina has x taka, how much money Shelly and Babu have althogether?

Solution:
মিনার কাছে আছে = x টাকা
বাবুর কাছে আছে = x - 3 টাকা
শেলির কাছে আছে = x + 5 টাকা

বাবু ও শেলির কাছে আছে = x - 3 + x + 5 টাকা
= 2x + 2 টাকা

0.1 + 0.1² + 0.1³
= 0.1 + 0.01 + 0.001
= 0.111

Question: 8 men can complete a piece of work in 20 days. 8 women can complete the same work in 32 days. In how many days will 5 men and 8 women together complete the same work?

Solution:
8 × 20 men = 8 × 32 women
5 men = 8 women
Now, 5 men + 8 women = 8 + 8 = 16 women

ATQ, D₁ × M₁= M₂× D₂
⇒ 8 × 32 women = 16 × D₂
⇒ D₂= (32 × 8)/16
∴ D₂= 16 days

1 pound = 16 ounces
∴ 4.75 pounds = 76 ounces

Question: One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card (Jack, Queen and King) only?

Solution:
A standard deck has 52 cards.
Face cards are Jack, Queen, King
There are 4 suits (hearts, diamonds, clubs, spades)
So number of face cards = 3 × 4 = 12

Total possible outcomes (when drawing one card) = 52
Favorable outcomes (drawing a face card) = 12

∴ Probability = favorable outcomes/total outcomes
= 12/52
= 3/13

So the probability that the card drawn is a face card (Jack, Queen, or King) is 3/13. 

Question: An equilateral triangle has a perimeter of 30 meters. What is its area?

Solution:
দেওয়া আছে,
সমবাহু ত্রিভুজের পরিসীমা = 30 মিটার

সমবাহু ত্রিভুজের এক বাহুর দৈর্ঘ্য, a = পরিসীমা/3
= 30/3 মিটার
∴ a = 10 মিটার

সমবাহু ত্রিভুজের ক্ষেত্রফল = (√3/4) × (বাহুর দৈর্ঘ্য)²
= (√3/4) × 10² বর্গ মিটার
= (√3/4) × 100 বর্গ মিটার
= 100√3/4 বর্গ মিটার
= 25√3 বর্গ মিটার

অতএব, সমবাহু ত্রিভুজের ক্ষেত্রফল = 25√3 বর্গ মিটার।

Question: Two partners invest TK 1,00,000 and TK 60,000 .After 6 months, they admit a new partner with TK 80,000. What is the ratio of their profits after one year?

Solution:
Profit sharing ratio depends on : Capital × Time
Let,
the partners be P, Q and R

P's invest : TK 1,00,000 for 12 months
⇒ 1,00,000 × 12 = 12,00,000

Q's invest : TK 60,000 for 12 months

⇒ 60,000 × 12 =7,20,000

R's invest: TK 80,000 for 6 months

⇒ 80,000 × 6 = 4,80,000

Now, the ratio of profits:
12,00,000 : 7,20,000 : 4,80,000

Simplify = 5 : 3 : 2

∴ Ratio = 5 : 3 : 2

Question: Eight people are planning to share equally the cost of a rental car. If one person withdraws from the arrangement and the others share equally the entire cost of the car, then the share of each of the remaining persons will be increased by-

Solution:
ধরি,
গাড়ির ভাড়া ক টাকা 
৮ জনে ভাড়া দিলে ১ জন দিবে ক/৮ টাকা 

৭ জনে ভাড়া দিলে ১ জন দিবে ক/৭ টাকা 

জন প্রতি ভাড়া বৃদ্ধি পায় = ক/৭ - ক/৮ টাকা 
= (৮ক - ৭ক)/৫৬ টাকা 
= ক/৫৬ টাকা 

∴ জনপ্রতি ভাড়া (ক/৫৬)/(ক/৮) গুণ বৃদ্ধি পায় 
= ৮/৫৬ গুণ বৃদ্ধি পায় 
= ১/৭ গুণ বৃদ্ধি পায়

Given that, time is taken to travel upstream = 2 × times taken to travel downstream
When the distance is constant, speed is inversely proportional to the time
Hence, 2 × speed upstream = speed downstream
Let speed upstream = x
Then speed downstream 2x
we have,
1/2(x + 2x) = speed in still water
⇒ 1/2(3x)=7.5
⇒ 3x = 15
⇒ x = 5
i.e., speed upstream = 5 km/hr
Rate of stream = 1/2(2x - x)
= x/2
= 5/2
= 2.5 km/hr.