Bank Math

Cost of machine 10000/-
Depreciation 10% each year.
1st year = 10000 - 1000 = 9000/-
2nd year = 9000 - 900 = 8100/-
3rs year = 8100 - 810 = 7290/-
4th year = 7290 - 729 = 6561/-
Depreciation will be deducted from the amount left after subtracting previous depreciation value.

Let,
He has TK. x
ATQ, x - x/2 - x/2×1/4 = 3600
or, x/2 - x/8 = 3600
or, (4x - x)/8 = 3600
∴ x = (3600×8)/3 = 9600

Question: The length, breadth, and height of a rectangular box are in the ratio 3 : 2 : 1. If its total surface area is 352 cm², what is its volume in cm³?

Solution:
দেওয়া আছে,
আয়তাকার বাক্সের দৈর্ঘ্য, প্রস্থ ও উচ্চতার অনুপাত 3 : 2 : 1।
সমগ্রতলের ক্ষেত্রফল 352 বর্গ সে. মি.

মনে করি,
আয়তাকার বাক্সের দৈর্ঘ্য, a = 3x সে. মি.
আয়তাকার বাক্সের প্রস্থ, b = 2x সে. মি.
আয়তাকার বাক্সের উচ্চতা, c = x সে. মি.

আমরা জানি,
আয়তাকার বাক্সের সমগ্রতলের ক্ষেত্রফল = 2(ab + bc + ca) বর্গ একক
= 2(3x × 2x + 2x × x + x × 3x) বর্গ একক
= 2(6x²+ 2x²+ 3x²) বর্গ একক
= 2 × 11x² 
= 22x² বর্গ একক

প্রশ্নমতে,
22x²= 352
⇒x²= 352/22
⇒x²= 16
⇒x = √16
∴ x = 4

আয়তাকার বাক্সের আয়তন = abc ঘন একক
= 3x × 2x × x ঘন একক
= 6x³ ঘন একক
= 6 × 4³ ঘন সে. মি.
= 384 ঘন সে. মি.

∴ আয়তাকার বাক্সের আয়তন 384 ঘন সে. মি. ।

A : B : C
Ratio of efficiency - 3 : 1 : 2
Ratio of No. of days - 1/3 : 1/1 : 1/2
or 2 : 6 : 3

Question: A cone’s slant height is 21 cm, and the curved surface area is 264 cm². Determine the diameter of the cone’s base.

Solution:
দেওয়া আছে,
তির্যক উচ্চতা, l = 21 সে.মি
বক্রপৃষ্ঠের ক্ষেত্রফল = 264 cm²

মনে করি,
কোণকটির ভূমির ব্যাসার্ধ = r 

∴ কোণকটির বক্রপৃষ্ঠের ক্ষেত্রফল = πrl

প্রশ্নমতে,
πrl = 264
⇒ r × (22/7) × 21 = 264
⇒ 66r = 264
⇒ r = 264/66
⇒ r = 4

∴ কোণকটির ভূমির ব্যাস = 2r = 2 × 4 = 8 সে.মি

Question: Out of 60 books in a school library, 18 belong to the science fiction genre. If a student chooses one book randomly, find the probability it isn’t a science fiction book.

Solution:
Probability of picking a book that is a science fiction book = 18/60
= 3/10

Probability of picking a book that is not a science fiction book = (1 - 3/10)
= (10 - 3)/10
= 7/10

Let the smaller number be x
Then larger number = (x + 1365)
∴ x + 1365 = 6x + 15
⇒ 5x = 1350
⇒ x = 270
∴ Smaller number = 270

Question: P scored 30% marks and failed by 15 marks. Q scored 45% marks and obtained 30 marks more than the pass marks. What is the pass percentage?

Solution:
Let the total marks be x.

Given,
P scored 30% marks and failed by 15 marks:
0.30x + 15 = Pass marks
Q scored 45% marks and obtained 30 marks more than the pass marks:
0.45x - 30 = Pass marks

Now,
0.30x + 15 = 0.45x - 30
⇒ 0.45x - 0.30x = 15 + 30
⇒ 0.15x = 45
⇒ x = 45/0.15
∴ x = 300

Pass marks = 0.30 × 300 + 15
= 90 + 15 = 105

∴ Pass percentage = (105/300) × 100% = 35%

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

Number = (12 x 35)
Correct Quotient = 420 /21 = 20
Answer : 20

Question: A tank is 12 m long, 8 m wide and 5 m deep. The cost of plastering its walls and bottom at 75 paisa per sq. m is-

Solution:
Let, l = 12 m, b = 8 m and, h = 5 m

∴ Area to be plastered = [2(l + b) × h] + (l × b) 
= [2(12 + 8) × 5] + (12 × 8) sq. m
= (200 + 96) sq. m
= 296 sq. m

∴ Cost of plastering = 296 × (75/100) Tk
= 296 × (3/4) Tk
= (74 × 3) Tk
= 222 Tk

Question: If (5 - 2x) ≤ 13, then which one is correct?

Solution: 

Given, 
⇒ 5 - 2x ≤ 13
⇒ 5 - 2x - 5 ≤ 13 - 5
⇒ - 2x ≤ 8
∴ x ≥ - 4

Assume, distance traveled by P in x hrs = 25 x km -----(1)
distance traveled by Q in (x-1) hrs = 20 (x-1) km -----(2)

Adding (1) & (2),
25 x + 20 (x -1) = 275
x = 6.5 hrs
(x -1) = (6.5 -1) = 5.5 hrs

Time at which they cross each other = 9 a.m. + 5.5hrs = 2.30 p.m.
The two motorcycle riders cross each other at 2.30 p.m.

Question: If x² - 3x + 1 = 0, and x > 1, then what is the value of x - 1/x?

Solution:
Given, x² - 3x + 1 = 0
⇒ (x²/x) - (3x/x) + (1/x) = 0 [উভয়পক্ষকে x দ্বারা ভাগ করে]
⇒ x - 3 + 1/x = 0
⇒ x + 1/x = 3

এখন, (x - 1/x)² = (x + 1/x)² - 4 . x . 1/x
⇒ (x - 1/x)² = (3)² - 4
⇒ (x - 1/x)² = 9 - 4
⇒ (x - 1/x)² = 5
∴ x - 1/x = ±√5

যেহেতু x > 1 দেওয়া আছে, তাই x - 1/x > 0
সুতরাং, x - 1/x = √5

When the owner of the ball to be included always, we have to select 10 players out of 14.

The required no. of ways
¹⁴C₁₀ = 14!/(10!4!) 
= 14.13.12.11 / 4.3.2.1
= 7.13.11
= 1001

Question: Six identical machines can produce 540 articles in 12 hours. How many articles would 8 such machines produce in 15 hours?

Solution: 
Total articles produced by 6 machines in 12 hours = 540.
Articles produced by 1 machine in 12 hours = 540/6
Articles produced by 1 machine in 1 hour = 540/(6×12) = 7.5 articles

So, Articles produced by 8 machines in 15 hours = 7.5 × 8 × 15 
= 900 articles

Question: A man has Tk. 1050 in the denominations of five-taka notes, ten-taka notes and twenty-taka notes. The number of notes of each denomination is equal. What is the total number of notes that he has?

Solution:
ধরি, 5 টাকা, 10 টাকা ও 20 টাকার নোটের সংখ্যা প্রত্যেকটি = x

∴ মোট টাকা = 5x + 10x + 20x
= 35x

প্রশ্নমতে,
35x = 1050
∴ x = 1050/35
= 30

অতএব,
5 টাকার নোট = 30
10 টাকার নোট = 30
20 টাকার নোট = 30

∴ মোট নোটের সংখ্যা = 30 + 30 + 30
= 90

Question: The price of a mobile set is Tk 8,000 and that of a tab is 50% more than the price of a mobile set. If a total of 18 mobiles and tabs were sold for a total of Tk 188,000, how many tabs were sold?

Solution:
Given that,
Price of mobile = Tk. 8,000
Price of tab = 50% more than mobile
= 8,000 + (50/100) × 8,000
= 8,000 + 4000 = Tk. 12,000
Total items sold = 18 (mobiles + tabs)
Total sales = Tk 188,000

Now, Let, Number of mobiles = m,Number of tabs = t 
Then we get,
m + t = 18
∴ m = 18 - t .......(1)

And, 
8000m + 12000t = 188,000
⇒ 2m + 3t = 47   ; [Dividing by 4,000]
⇒ 2(18 - t) + 3t = 47   ; [From (1)]
⇒ 36 - 2t + 3t = 47
⇒ t = 47 - 36
∴ t = 11

∴ The number of tabs sold is 11.

টসে প্রথমবারে হেড আসার সম্ভাবনা 1/2
এবং পরেরবারে টেল আসার সম্ভাবনা 1/2
সুতরাং, প্রথম টসে হেড এবং পরের টসে টেল আসার সম্ভাবনা = 1/2 × 1/2 = 1/4

Let the required no of hours be x. Then
Less men , More hours (Indirect Proportion)
Therefore, 15:36 :: 25:x
=> (15 × x) = (36 × 25)
=> x = 60
Hence, 15 men can do it in 60 hours.

Question: A swimming pool maintenance service charges a fixed fee of Tk. 200 plus Tk. 150 per hour for cleaning. If a customer's total budget is Tk. 1,400, what is the maximum number of full hours the technician can work?

Solution:
Given,
Fixed service fee = 200 Tk
Charge per hour = 150 Tk
Total budget = 1,400 Tk

Let h = number of full hours the technician works.

According to the condition,
200 + 150h ≤ 1,400
⇒ 150h ≤ 1,400 - 200
⇒ 150h ≤ 1,200
⇒ h ≤ 1,200/150
∴ h ≤ 8

Therefore, the technician can work a maximum of 8 full hours.

We know,
Distance(D) = Speed(S) × Time(T)
∴ S = D/T ; T = D/S

Let distance traveled by cat before dog catches it be D
We know, time for which Dog and Cat ran is the same
∴ T = T
∴ D/5 = (D + 80)/7
⇒ 7D = 5D + 400
⇒ 7D - 5D = 400
⇒ 2D = 400
⇒ D = 200 m.

Question: What is the value of the expression?
(√3 + √12)² = ?

Solution:
(√3 + √12)²
= {√3 + √(3 × 4)}²
= (√3 + 2√3)²
= (3√3)²
= 3² × (√3)²
= 9 × 3
= 27

Question: A man buys an article for 20% more than its value and sells it for 20% less than its value. His gain or loss percentage is –

Solution:
Let the original value of the article = 100 টাকা
∴ Cost Price (CP) = 100 + 20% of 100 টাকা
= 100 + 20 = 120 টাকা

∴ Selling Price (SP) = 100 - 20% of 100 টাকা
= 100 - 20 = 80 টাকা

Since SP (80 টাকা) is less than CP (120 টাকা), there is a Loss.

∴ Loss = CP - SP = 120 - 80 = 40 টাকা
∴ Loss percentage = (Loss/CP) × 100%
= (40/120) × 100%
= (1/3) × 100%
= 33.33% loss

Let the numbers be x and x + 2.
Then, (x + 2)² - x² = 84
⇒ 4x + 4 = 84
⇒ 4x = 80
⇒ x = 20.
∴ The required sum
= x + (x + 2)
= 2x + 2
= 42

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

সমাধান:
ধরি, m = 3 (একটি বিজোড় সংখ্যা)

ক) m² + m = 3³ + 3 = 12 → জোড় 
খ) 2m + 1 = 2(3) + 1 = 6 + 1 = 7 → বিজোড়
গ) 5m - 2 = 5(3) - 2 = 15 - 2 = 13 → বিজোড় 
ঘ) m³ + 2 = 3³ + 2 = 27 + 2 = 29 → বিজোড়

Question: Solve the inequality: 3(2x - 5) + 1 > 4(x - 3)

Solution:
Given inequality,
3(2x - 5) + 1 > 4(x - 3)
⇒ 6x - 15 + 1 > 4x - 12
⇒ 6x - 14 > 4x - 12
⇒ 6x - 4x > −12 + 14
⇒ 2x > 2
∴ x > 1

So, the solution of the inequality is x > 1.

Question: The ratio of the number of boys and girls in a school is 7 : 4. If the percentage increase in the number of boys and girls be 25% and 15% respectively, what will be the new ratio?

Solution: 
Let,
The number of boys and girls in a school be 7X and 4X respectively
their increased number number is (125% of 7X) and (115% of 4X)
⇒ (125/100) of 7X and (115/100) of 4X
⇒ 35X/4 and 23X/5

∴ required ratio = 35X/4 : 23X/5
= 175X : 92X        [multiply by 20]
= 175 : 92

Question: In a set of 3 numbers, the average of first two numbers is 2, the average of the last two numbers is 3, and the average of the first and the last numbers is 4. What is the average of three numbers?

Solution: 
let, the numbers are x, y, z 

x + y = 2 × 2 = 4
y + z = 2 × 3 = 6 
z + x = 2 × 4 = 8

2 (x + y + z) = 4 + 6 + 8 = 18 
⇒ (x + y + z) = 9 

∴ the average of three numbers is = 9/3 = 3 

Let,
the cost of the table and chair be Tk. 5x and Tk. 7x respectively.
New cost of chair = 120% of Tk. 7x = Tk (6/5 × 7x)
= Tk. 42x/5.
New cost of table = 110% of Tk. 5x = Tk.(11/10 × 5x)
= Tk. 55x/10.
∴ New ratio = 55x/10 : 42x/5
= 55 : 84

এছাড়াও,
5 : 7
নতুন অনুপাতঃ
5 এর 110% : 7 এর 120%
= 5.5 : 8.4

Question: A train crosses two bridges that are 600 meters and 200 meters long in 60 seconds and 40 seconds respectively. What is the length of the train?

Solution:
We know,
To cross a bridge, a train must cover the length of the bridge along with its own length.

Let the length of the train = x meters

Then,
For the first bridge, the distance covered by the train = (x + 600) meters
And,
For the second bridge, the distance covered by the train = (x + 200) meters

According to the question,
(x + 600)/60 = (x + 200)/40
⇒ 40(x + 600) = 60(x + 200)

⇒ 40x + 24000 = 60x + 12000
⇒ 60x - 40x = 24000 - 12000
⇒ 20x = 12000
⇒ x = 12000/20 
⇒ x = 600

∴ The length of the train is 600 meters.

Question: One pipe can fill a tank four times as fast as another pipe. If together the two pipes can fill the tank in 24 minutes, then the slower pipe alone will be able to fill the tank in-

Solution:
Let,
the slower pipe alone fill the tank in x minutes.
Then, Faster pipe alone will fill it in x/4 minutes.

ATQ,
(1/x) + (4/x) = 1/24
⇒ 5/x = 1/24
∴ x = 120

∴ The slower pipe alone fill the tank in 120 minutes
= (120/60) hours
= 2 hours

Question: A wholesaler buys a television for Tk. 18,000 and sells it to a retailer at a profit of 30%. The retailer then sells it to a customer at a profit of 25%. How much does the customer pay to the retailer?

সমাধান:
পাইকারের 30% লাভে বিক্রয়মূল্য = 18,000 + 18,000 এর 30%
= 18,000 + (18,000 × 30/100)
= 18,000 + 5,400 = 23,400 টাকা

পাইকারের বিক্রয়মূল্য = খুচরা বিক্রেতার ক্রয়মূল্য = 23,400 টাকা

খুচরা বিক্রেতার 25% লাভে বিক্রয়মূল্য = 23,400 + 23,400 এর 25%
= 23,400 + (23,400 × 25/100)
= 23,400 + 5,850 = 29,250 টাকা

সুতরাং, খুচরা বিক্রেতার বিক্রয়মূল্য = ক্রেতার ক্রয়মূল্য = Tk. 29,250

Question: A and B are two positive integers such that AB = 72. Which of the following cannot be the value of A + B?

Solution:
Factor pairs of 72:
(1, 72) → A + B = 73
(2, 36) → A + B = 38
(3, 24) → A + B = 27
(4, 18) → A + B = 22
(6, 12) → A + B = 18
(8, 9) → A + B = 17

So, possible values of A + B are: 73, 38, 27, 22, 18, 17.

Among the options, 25 cannot be the value of A + B.