Bank Math

Question: What would be the measure of the perimeter of a square whose area is equal to 225 square cm?

Solution:
দেওয়া আছে,
বর্গক্ষেত্রের ক্ষেত্রফল = 225 বর্গ সেমি
এক বাহুর দৈর্ঘ্য = a

প্রশ্নমতে,
a² = 225
⇒ a² = 15²
∴ a = 15

∴ বর্গক্ষেত্রের পরিসীমা = 4a
= 4 × 15 = 60 সেমি

Question: If θ = 60° , then what is the value of (1 - sec²θ)/(1 + sec²θ)?

Solution:
Here, θ = 60°

Now,
(1 - sec²θ)/(1 + sec2θ)
= {1 - (sec60°)²}/{1 + (sec60°)²}
= (1 - 2²)/(1 + 2²)
= (1 - 4)/(1 + 4)
= - 3/5

Question: A train 400 meters long passes a pole in 16 seconds. How long will it take to pass a platform that is 800 meters long?

Solution:
Train's speed = Distance/Time
= 400/16 = 25m/s

Total distance to pass the platform,
= Length of train + Length of platform
= 400 + 800
= 1200 meters

∴ Required time to pass platform = Distance/Speed
= 1200/25
= 48 seconds

Question: Three numbers are in the ratio 3 : 4 : 6 and their products is 1944. The largest number is - 

Solution:
Let the number be 3x, 4x, 6x.

ATQ,
3x × 4x × 6x = 1944
⇒ 72x³ = 1944
⇒ x³ = 1944/72
⇒ x³ = 27
∴ x = 3

So largest number = 6x = 6 × 3 = 18

As per the question, three girls can’t occupy consecutive seats but two can.

Therefore, if we find the number of ways in which all three girls occupy consecutive seats and subtract this number from the total number of ways in which the five people can be arranged among themselves, we will get the required answer.

5 students can be arranged among themselves in ⁵p₅ ways = 120 ways.

Assume that the 3 girls are one entity. The total number of ways in which they can be arranged among themselves = 3! = 6
Also, the set of three girls and the other students can be arranged among themselves in 3! = 6 ways.
Thus, the total number of ways in which three girls are together = 6 × 6 = 36

Thus, a number of ways in which all 3 girls will not occupy consecutive seats = 120 – 36 = 84.

Question: Two taps, P and Q, can fill a tank in 6 and 8 minutes respectively. A leak (outlet pipe) R can empty 20 liters of water per minute. If all three are opened together, the tank is filled in 12 minutes. What is the capacity of the tank in liters?

সমাধান:
প্রথম নল P এর 1 মিনিটে পূর্ণ করে = 1/6 অংশ
দ্বিতীয় নল Q এর 1 মিনিটে পূর্ণ করে = 1/8 অংশ
তিনটি নল একত্রে 1 মিনিটে পূর্ণ করে = 1/12 অংশ।

ছিদ্র নল R এর 1 মিনিটে খালি করার অংশ = (P + Q এর কাজ - সম্মিলিত কাজ)
= (1/6 + 1/8) - 1/12 অংশ
= (4 + 3)/24 - 1/12 অংশ
= 7/24 - 1/12 অংশ
= (7 - 2)/24 অংশ
= 5/24 অংশ।

অর্থাৎ, ছিদ্র নল R একা 5/24 অংশ খালি করে 1 মিনিটে।

ছিদ্র নল R একা সম্পূর্ণ ট্যাঙ্কটি খালি করতে সময় নেয় = 1/(5/24) মিনিট
= 4.8 মিনিট।

ছিদ্র নল R 1 মিনিটে খালি করে 20 লিটার।
∴ ট্যাঙ্কটির মোট ধারণ ক্ষমতা = (মোট সময় × প্রতি মিনিটের নির্গমনের হার)
= (4.8 × 20) লিটার
= 96 লিটার।

Let, length = x and Width = 3x - 6
ATQ, 
2(x + 3x - 6) = 104
Or, 4x - 6 = 104/2 = 52
Or, 4x = 58
Or, x = 14.5
So, width = 3 × 14.5 - 6= 37.5

Question: A blend consists of an equal amount of lemon juice and sugar syrup. When mixed with extra sugar syrup in a 1 : 3 ratio, what is the final ratio of lemon juice to sugar syrup?

Solution: 
Let, the new mixture is 12 litres

The old mixture = 12 × (1/4) = 3 liters 
The sugar syrup = 9 liters 

The new sugar syrup = 9 + (3/2) = 10.5 

∴ The final ratio of lemon juice and sugar syrup = 1.5 : 10.5 
= 15 : 105 
= 1 : 7

[The statement "Let, the new mixture is 12 liters" is used as an assumption to make the calculations easier. By assuming the total amount of the new mixture is 12 liters, it simplifies the process of determining the amounts of lemon juice and sugar syrup in the mixture. This assumption is just a way to set up the problem in a manageable way.

12 liters is a convenient number because it is divisible by 4 (the ratio of lemon juice and sugar syrup in the original mixture), and it helps make the calculation easier for the final ratio.]

x adults = 2x children
to complete in d days 2x children is required.
∴ to complete in (d + 2) days (2x × d)/(d + 2) children is required.

Question: A and B are partners in a business. A invests Taka 25,000 for 8 months and B invests Taka 30,000 for 6 months. If the profit at the end of the year is Taka 15,200, what is B's share in the profit?

Solution: 
A’s Contribution = 25000×8 = 200000 Taka
B’s Contribution = 30000×6 = 180000 Taka

The ratio of A's and B's investment = 200000 : 180000
= 10 : 9

So, B's share in profit = {9/(10+9)} × 15200
= 7200 Taka

Question: Find the number of factors of 360.

Solution:
Factorize 360 into prime factors:
360 = 2³ × 3² × 5¹
The formula for the number of factors of a number n =p^aq^br^c…n is:
Number of factors = (a + 1) (b + 1)(c + 1)
Apply the formula:
(3 + 1) (2 + 1) (1 + 1) = 4 × 3 × 2 = 24

Question: In how many different ways can be letters of the word 'CYCLE' be arranged?

Solution:
CYCLE whereas total 5 letters and C comes two times.

So, arrangements are = 5!/2! 
= 60 ways

Question: What should be the value of "P" so that the expression (16 − 24x + Px²) becomes a perfect square?

Solution:
(16 − 24x + Px²)
= (4)² − 2 × 4 × 3x + (3x)²+ Px² − (3x)²
= (4 − 3x)² + Px² − 9x²

∴ The expression becomes a perfect square if,
Px² − 9x² = 0
⇒ Px² = 9x²
∴ P = 9

Question: A machine is sold at a profit of 10%. Had it been sold for Tk. 40 less, there would have been a loss of 10%. What was the cost price?

Solution: 
Let, cost price of the machine is x taka 

Selling price = 1.1x taka 

ATQ, 
1.1x - 40 = 0.9x 
⇒ 1.1x - 0.9x = 40 
⇒ 0.2x = 40
⇒ x = 40/.2
= Tk. 200 

Question: Which number replaces the question mark?

Solution:
এই পিরামিডের ক্ষেত্রে,
উপরে থাকা প্রতিটি সংখ্যা তার সরাসরি নিচের দুটি সংখ্যার পার্থক্যের সমান।
অর্থাৎ, প্রতিটি সংখ্যা = নিচের বাম সংখ্যা - নিচের ডান সংখ্যা
এখানে,
65 = 110 - 45
27 = 45 - 18
38 = 65 - 27
15 = 27 - 12
23 = 38 - 15

যেহেতু, 12 = 18 - ?
∴ ? = 18 - 12 = 6
সুতরাং, প্রশ্নবোধক চিহ্নের স্থানে 6 বসবে।

Question: A person make a profit of 10% on 25% of the quantity and a loss of 20% on the rest of the quantity. What is the gain or loss in percentage on the whole?

Solution: 
Let Cost Price = 100
Profit = 25 x 10%
= 25/10
= 2.5

Loss = 75 x 20%
= 75/5
= 15

Net loss = 15 - 2.5 
= 12.5

∴ Net Loss as % = 12.5/100 x 100%
= 12.5%

Let the mother's present age be x year.
Then, the person's present age = 2/5 x year.
(2x/5 + 8) = 1/2 (x + 8)
2(2x + 40) = 5(x + 8) 
=> x = 40.

Question: 120% of 45 + 45% of 120 = ?

Solution:
 120% of 45 + 45% of 120
= {(120/100) × 45} + {(45/100) × 120}
= 54 + 54
= 108

Question: Three metal wires of lengths 90 cm, 126 cm, and 162 cm are provided. Calculate the maximum length of wire segments that can be cut, ensuring no waste.

Solution:
The maximum length of wire segments that can be cut (ensuring no waste) = (H.C.F. of 90, 126, 162) cm = 18 cm.

Elaborately,

The prime factorization of each number:
 90 = 2 × 3² × 5
126 = 2 × 3² × 7
162 = 2 × 3⁴

Common factors of all three numbers are: 2 (common), 3² = 9 (common)

∴ HCF = 2 × 3² = 18

Question: A pipe can fill an empty tank in 15 minutes. Another pipe drains water at a rate of 10 liters per minute. If both pipes are opened simultaneously, the tank gets filled in 60 minutes. What is the capacity of the tank (in liters)?

Solution:
মনে করি, ট্যাঙ্কটির মোট ধারণক্ষমতা V লিটার।

প্রথম পাইপটি 15 মিনিটে খালি ট্যাঙ্কটি পূর্ণ করে।
∴ প্রথম পাইপ দিয়ে প্রতি মিনিটে পানি পূর্ণ হয় = V/15 লিটার।

দ্বিতীয় পাইপ দিয়ে প্রতি মিনিটে পানি বের হয়ে যায় = 10 লিটার।

দুটি পাইপ একসাথে খোলা থাকলে ট্যাঙ্কটি 60 মিনিটে পূর্ণ হয়।

∴ প্রতি মিনিটে ট্যাঙ্কটি কার্যকরভাবে পূর্ণ হয় = V/60 লিটার।

প্রশ্নমতে,
(V/15) - 10 = V/60
⇒ (V/15) - (V/60) = 10
⇒ (4V - V)/60 = 10
⇒ 3V/60 = 10
⇒ 3V = 10 × 60
⇒ 3V = 600
⇒ V = 600/3
∴ V = 200

সুতরাং, ট্যাঙ্কটিতে 200 লিটার পানি ধরে।

Question: If the least common multiple of two numbers is twelve times their highest common factor, and their sum (HCF + LCM) equals 403, then what is the other number when one number is 93?

Solution:
Let HCF be h and LCM be l
Then l = 12h and
l + h = 403

∴12h + h = 403
⇒ h = 31

So, l = (403 - 31) = 372

Hence, the other number = (31 × 372)/93 = 124

April has 30 days. So Protik takes 30 days to build the pavement.
Mahmud is 25% faster than Protik

25% = 25/100 = .25
This means, if Protik is 1, then Mahmud is (1 + 0.25) = 1.25
Protik takes 30 days to do the work.
Mahmud will take = 30/1.25 = 24 days to get the work done.

Let small number is x, so large number is (3/2)x
ATQ, x + 14 = (3/2)x
Or, 2x + 28 = 3x
So, x = 28

Question: The angle of elevation of a ladder leaning against a wall is 60º and the foot of the ladder is 5.5 m away from the wall. The length of the ladder is-

Solution: 

Let AB be the wall and BC be the ladder.
Then, ∠ACB = 60° = AC = 5.5 m
AC/BC = cos⁡60^∘= 1/2
⇒ BC = 2 × AC = 2 × 5.5 = 11 m

Question: The sum of 7 consecutive natural numbers is 77. Find how many of these are prime numbers?

Solution:
Given that,
The sum of seven consecutive natural numbers = 77

Now,
Let the numbers be n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6 respectively
∴ 7n + 21 = 77
⇒ 7n = 77 - 21
⇒ 7n = 56
⇒ n = 56/7
∴ n = 8

So the numbers is 8, 9, 10, 11, 12, 13, 14
Out of these 11, 13 are prime numbers

∴ Required prime numbers is 2

Let,
The pipe Z alone takes A minutes to fill the tank.
Given that,
Y is thrice as fast as Z.
Then, Y takes A/3 minutes to fill the tank.
And, X is thrice as fast as Y.
X takes (A/3)/3 = A/9 minutes to fill the tank.
Now,
Part filled by X in 1 minute = 9/A
Part filled by Y in 1 minute = 3/A
Part filled by Z in 1 minute = 1/A
Net part filled by (X+Y+Z) in 1 minute = 9/A + 3/A + 1/A
= 13/A
(X+Y+Z) take 10 minutes to fill the cistern.
Part filled by (X+Y+Z) in 1 minute = 1/10
Thus,
We have,
1/10 = 13/A
⇒ A = 13 × 10
⇒ A = 130
Therefore, Z alone takes 130 minutes
So, X can fill the cistern in 130/9 minutes.
Hence, the correct answer is - ঘ) none of these

Question: The perimeter of a circle measures 16πcm, what is the area of the circle in sq.cm?

Solution:
মনেকরি
বৃত্তের ব্যাসার্ধ r
বৃত্তের পরিধি = 2πr
বৃত্তের ক্ষেত্রফল = πr²

প্রশ্নমতে
2πr = 16π
2r = 16
r = 8

বৃত্তের ক্ষেত্রফল = πr²
=π8²
=64π

Question: The population of a city grows from 175,000 to 262,500. What is the percentage growth in the city's population?

Solution:
জনসংখ্যা বৃদ্ধি পেয়েছে 
 = (262500 - 175000) = 87500

175000 জনে বৃদ্ধি পায় = 87500 জন
∴ 1 জনে বৃদ্ধি পায় = 87500/175000 জন
∴ 100 জনে বৃদ্ধি পায় = (87500 × 100)/175000 = 50 জন

∴ জনসংখ্যা বৃদ্ধির শতকরা হার = 50%

Question: The greatest number will divide 3026 and 5053 leaving remainders 11 and 13 respectively?

Solution: 
৩০২৬ - ১১ = ৩০১৫ ও ৫০৫৩ - ১৩ = ৫০৪০ এর গ.সা.গু 

৩০১৫ = ৩ × ৩ × ৫ × ৬৭ 
৫০৪০ = ২ × ২ × ২ × ২ × ৩ × ৩ × ৫ × ৭ 

৩০১৫ ও ৫০৪০ এর গ সা গু = ৩ × ৩ × ৫
= ৪৫ 

Let the speed of the slower train = X m/sec
Then, the speed of the faster train will be = 2X m/sec

Relative Speed = X + 2X = 3X m/sec
Relative Speed is also = Sum of the length of the trains/Time taken to cross each other
= (200 + 200)/10
= 400/10
= 40.

So, 3X = 40
X = 40/3
X (speed) in Km/hr = (40/3) × (18/5)
= 720/15
= 48 km/hr.

Question: If CODE is 31545 and BOOK is 2151511, then LIVE will be equal to-

Solution:
CODE শব্দটির জন্য অক্ষরগুলোর অবস্থান হলো:
C = 3
O = 15
D = 4
E = 5

∴ CODE = 31545

BOOK শব্দটির জন্য অক্ষরগুলোর অবস্থান হলো:
B = 2
O = 15
O = 15
K = 11

∴ BOOK = 2151511।

LIVE শব্দটির জন্য প্রয়োগ করি:
L = 12
I = 9
V = 22
E = 5

∴ LIVE = 129225

সুতরাং, LIVE এর কোড হবে 129225

Question: A sum of money is to be divided among P, Q, R, S in the ratio 7 : 3 : 5 : 2. If R gets Tk. 2000 more than S, what is Q's share?

Solution:
Let their shares be 7x, 3x, 5x, and 2x respectively.

ATC,
5x - 2x = 2000
⇒ 3x = 2000
⇒ x = 2000/3

Therefore,
Q's share = 3x
= 3 × (2000/3)
= 2000 taka.

Let C.P.=Tk. 100 
Then, 
 M.P. = Tk. 135,
S.P. =Tk.108
∴Discount %  =(27/135 ×100)% =20%

Question: If sec²θ + tan²θ = 7/12, then sec⁴θ - tan⁴θ = ?
 
Solution: 
Given that, 
sec²θ + tan²θ = 7/12

Now, 
sec⁴θ - tan⁴θ
= (sec²θ)² - (tan²θ)²
= (sec²θ + tan²θ)(sec²θ - tan²θ) ; [sec²θ - tan²θ = 1]
= (7/12) × 1
= 7/12

Total number of balls = (4 + 5 + 6)
= 15.
P(drawing a red ball or a green ball)
= P(red) + P(green)
= (4/15 + 5/15)
= 9/15
= 3/5.