Bank Math

Question: What is the value of 1 + {tan²A/(1 + secA)} ?

Solution:
1 + {tan²A/(1 + secA)}
= 1 + {(sce²A - 1)/(1 + secA)}
= {(1 + secA) + (sce²A - 1)}/(1 + secA)
= (1 + secA + sce²A - 1)/(1 + secA)
= (secA + sce²A)/(1 + secA)
= secA(1 + secA)/(1 + secA)
= secA

Question: A and B can do a piece of work in 20 days and 30 days respectively. They began to work together but A leaves after some time, and B completed the remaining work in 10 days. After how many days did A leave?

 
Solution:
Work done by B in 10 days = (1/30 × 10) part
= 10/30
= 1/3 part

Remaining work = 1 - 1/3 = 2/3 part

(A + B)'s 1 day's work = (1/20 + 1/30) part
= (3/60 + 2/60) = 5/60
= 1/12 part

Now, 1/12 part work is done by (A + B) in 1 day
∴ 1 part work is done by (A + B) in 12 days

∴ 2/3 part work is done by (A + B) in = 12 × (2/3) = 8 days

∴

Question: The difference in Taka between simple and compound interest at 10% annually on a sum of Tk. 8,000 after 2 years is:

Solution:
Given that,
Principal, P = Tk. 8,000
Rate, r = 10%
Time, n = 2 years

We know that,
Simple Interest = Pnr/100
= (8000 × 2 × 10)/100
= 1600 Tk.

And Compound Interest = P(1 + r/100)ⁿ - P
= 8000(1 + 10/100)² - 8000
= 8000(11/10)² - 8000
= 8000(1.21) - 8000
= 9680 - 8000
= 1680 Tk.

∴ Difference = 1680 - 1600 = 80 Tk.
The difference between compound and simple interest is Tk. 80.

Total annual income
= TK (10000 X 4 + 12000 X 8 + 26000)
= TK. 162000
Average monthly income = TK 16200012 = TK. 13500

Pages typed by Anik in 1 hour = 32/6 = 16/3
Pages typed by Shuvo in 1 hour = 40/5 = 8

Pages typed by Anik and Shuvo in 1 hour = 16/3 + 8 = 40/3
Time taken to type 110 pages when Anik and Shuvo work together = 110 × 3 /40 = 33/4
= 8 (1/4) hours = 8 hour 15 minutes.

At 5% profit, selling price = 100 + 5 = 105 tk
At 5% loss, selling price = 100 - 5 = 95 tk
Difference between selling price = 105 - 95 = 10 tk
When the difference 10, buying price is 100 tk
∴ When the difference is 15, buying price is (100 × 15)/10 tk
= 150 tk

30% of 10 = 10% of z
⇒(30/100)×10=(10/100)×z
⇒3=z/10
So,z=30

Given, 2x - 3y = 6 .... (i)
Lets (i) × (-2),
So, - 4x + 6y = - 12
Or, 6y - 4x = - 12

Discount = 80 - 68 = 12
Discount Rate = (12/80) × 100
= 15%

Question: How many prime numbers are there between 110 and 120?

Solution:
The numbers from 110 to 120 are,
111, 112, 113, 114, 115, 116, 117, 118, 119

Now let's check which of these are prime numbers (divisible only by 1 and themselves).
111 = 3 × 37 = not prime
112 = even = not prime
113 = prime (only divisible by 1 and 113)
114 = even = not prime
115 = ends with 5 = divisible by 5 = not prime
116 = even = not prime
117 = 9 × 13 = not prime
118 = even = not prime
119 = 7 × 17 = not prime

∴ Only one number is prime 113

Question: To fill a tank, 28 buckets of water are required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to four-fifth of its present?

Solution: 

ধরি,
বালতির ধারণক্ষমতা x
ট্যাঙ্কের ধারণক্ষমতা 28x

আবার,
বালতির নতুন ধারণক্ষমতা (4x/5) 

∴ প্রয়োজনীয় বালতির সংখ্যা = 28x/(4x/5) 
= 28x × (5/4x) 
= (7 × 5) টি
= 35 টি

Question: If x, y and z are the sides of a right angled triangle, where ‘z’ is the hypotenuse, then find the value of 1/(log_{x + z}y) + 1/(log_{x - z}y).

Solution:
Here x, y and z are the sides of a right angled triangle, so z² = x² + y².

ঋণাত্মক সংখ্যার লগারিদম হয় না বিধায়, সঠিক উত্তর: ঙ) None of these

LCM of 30, 36, 80 = 720
Number = 720 × K + 11 (K = 2)
Then the number = 720 × 2 + 11
= 1440 + 11
= 1451

Question: (0.01 × 0.001 × 0.1 × 10⁶) is equal to:

Solution:
Given expression,
(0.01 × 0.001 × 0.1 × 10⁶)
= (1/100) × (1/1000) × (1/10) × 10⁶
= 10⁶/10⁶
= 10^{(6 - 6)}
= 10⁰
= 1

Questtion: A certain amount of money earning simple interest becomes 7/5th of the initial amount in 4 years. What is the interest rate?

Solution:
মনেকরি 
আসল P = 100 টাকা
মুনফা- আসল A = 100 এর 7/5 = 140 টাকা
মুনাফা I = 140 - 100 = 40 টাকা
সময় n = 4 বছর 
মুনাফার হার r = ?

আমরাজানি 
I = Pnr
r = I/Pn
 = (40 × 100)/(100 × 4)
= 10% 

Question: The volume of a sphere is the same as the volume of a right circular cylinder whose radius is 4 cm and height is 18 cm. What is the radius of the sphere?

Solution:
ধরি, গোলকের ব্যাসার্ধ = r₁
এবং বেলনের ব্যাসার্ধ = r₂

দেওয়া আছে,
বেলনের ব্যাসার্ধ, r₂ = 4 সেমি
বেলনের উচ্চতা, h = 18 সেমি

আমরা জানি,
গোলকের আয়তন = (4/3)πr₁³
বেলনের আয়তন = πr₂²h

প্রশ্নমতে,
গোলকের আয়তন = বেলনের আয়তন
(4/3)πr₁³ = πr₂²h
⇒ (4/3)r₁³ = (4)² × 18
⇒ (4/3)r₁³ = 16 × 18
⇒ 4r₁³ = 16 × 18 × 3
⇒ r₁³ = (16 × 18 × 3)/4
⇒ r₁³ = 4 × 18 × 3
⇒ r₁³ = 216
⇒ r₁ = 6

∴ গোলকের ব্যাসার্ধ = 6 সেমি

Question: A rectangular hall measures 10 meters in length and 6 meters in width. If carpeting costs Tk. 15 per square meter, what will be the total cost to carpet the entire hall?

Solution:
Area of the hall = Length × Width
= 10 × 6 = 60 m²

Cost of carpet = Area × Cost per m²
= 60 × 15
= Tk. 900

Let, the numbers be x and y
According to the question, x  +  y = 21 ......(i) and x²  +  y²  =  333 ......(ii) 
∴ (x + y)² = x² + y² + 2xy
⇒ (21)² = 333 + 2xy
⇒ 441 = 333 + 2xy
⇒ 2xy = 108
⇒ xy = 54

ট্যাংকের ধারণক্ষমতা = 12 × 13.5 = 162
দেয়া আছে, প্রতিটি বাকেটের ধারণক্ষমতা = ৯ লিটার
মোট বাকেট লাগবে = 162/9 = 18 টি
উত্তরঃ 18 টি

Question: The volume of a cone is 300π cubic centimeters. If the radius of its base is 6 cm, what is the height of the cone?

solution:
দেওয়া আছে,
কোণকের আয়তন, V = 300π ঘন সে.মি.
ভূমির ব্যাসার্ধ, r = 6 সে.মি.
ধরি, কোণকের উচ্চতা = h সে.মি.

আমরা জানি,
কোণকের আয়তন, V = 1/3 × π × r² × h
∴ 300π = 1/3 × π × 6² × h
⇒ 300 = 1/3 × 36 × h (π উভয় পক্ষ থেকে বাদ দিয়ে)
⇒ 300 = 12h
⇒ h = 300 / 12
∴ h = 25 সে.মি.

অতএব, কোণকটির উচ্চতা = 25 সে.মি.

Question: Which of the following is equivalent to the pair of inequalities 2x - 5 > 3 and x + 4 ≤ 12?

Solution:
প্রথম অসমতাটি হলো:
⇒ 2x - 5 > 3
⇒ 2x > 3 + 5
⇒ 2x > 8
⇒ x > 4

দ্বিতীয় অসমতাটি হলো:
⇒ x + 4 ≤ 12
⇒ x ≤ 12 - 4
⇒ x ≤ 8

এখন প্রাপ্ত ফলাফল দুটিকে একত্রে সাজালে পাই:
4 < x এবং x ≤ 8
∴ 4 < x ≤ 8

Both offered = (75 + 45)% - 100% = 20%
Numbers of student offered both = 7500 × 20% = 1500

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.

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

Solution: 
We know,
The angle between the hour and minute hands is,
Angle = |11M - 60H|/2
= |(11 × 20) - (60 × 4)|/2
= |220 - 240|/2
= |- 20|/2
= 20/2
= 10°

So the angle between the hour and minute hands at 4 : 20 is 10°.

Question: 12 women can complete a piece of work in 10 days. 10 men can complete the same work in 8 days. What is the ratio of the work capacity of one man to one woman?

Solution:
12 women can complete a piece of work in 10 days.
∴ 1 woman can complete a piece of work in 10 × 12 = 120 days.

∴ 1 Woman's 1 day's work = 1/(12 × 10) 
= 1/120
∴ One woman's 1 day's work = 1/120

Similarly, 
Men's 1 day's work = 1/(10 × 8) = 1/80
∴ One man's 1 day's work = 1/80

∴ Ratio of capacity of one man to one woman = (1/80) : (1/120)
= 120 : 80
= 3 : 2

Question: If a pole 12 m high casts a shadow 4√3 m long on the ground, then the elevation of the sun is -

Solution:

ধরি,
AB = 12, BC = 4√3

ABC সমকোণী ত্রিভুজ হতে পাই,
tanθ = AB/BC
⇒ tanθ = 12/4√3
⇒ tanθ = 3/√3
⇒ tanθ = (√3 × √3)/√3
⇒ tanθ = √3
⇒ tanθ = tan60°
∴ θ = 60°

So the elevation of the sun is 60°.

Question: A box contains 9 blue balls, 6 green balls, and 10 yellow balls. One ball is drawn at random. What is the probability that the ball drawn is neither blue nor yellow?

Solution:
Number of blue balls = 9
Number of green balls = 6
Number of yellow balls = 10

∴ Total balls = 9 + 6 + 10 = 25

Let, event E = The ball drawn is neither blue nor yellow, so it must be green.

∴ Number of favorable outcomes = 6

∴ P(E) = 6/25

Sum of the present age of husband, wife and child
= (23 × 2 + 5 × 2) + 1 = 57 years
∴ Required average = (57/3) years
= 19 years

Question: A mechanic repairs 4 cars in 5/3 hours. How many cars can he repair in 50 minutes?

Solution:
Here,
5/3 hours = (5/3) × 60 = 100 minutes

In 100 minutes, he repairs = 4 cars
In 1 minute, he repairs = 4/100 
So, in 50 minutes, he repairs = (4 × 50)/100
= 2 cars