Bank Math

Question: A student has three iron rods of lengths 44 cm, 22 cm, and 55 cm. He needs to cut them into rods of the largest possible length such that no iron is wasted. What is the longest rod length he can achieve?
(একজন ছাত্রকে ৩টি বিভিন্ন দৈর্ঘ্যের লোহার টুকরা দেওয়া হয়েছে – যথাক্রমে ৪৪ সেমি, ২২ সেমি ও ৫৫ সেমি। তাকে এমন একটি সর্বোচ্চ দৈর্ঘ্যের রড তৈরি করতে হবে যাতে কোনো লোহার অপচয় না হয়। রডটির সর্বাধিক দৈর্ঘ্য নির্ণয় করো।)

Solution:
H.C.F. = গ.সা.গু
এ ধরনের রডের সর্বাধিক সম্ভাব্য দৈর্ঘ্য = (44, 22, 55 এর গ.সা.গু) cm = 11 cm.

Question: In a meeting, every person shakes hands with every other person exactly once. If the total number of handshakes was 28, how many people were in the meeting?

Solution:
Let,
the number of people be n.

ATQ,
number of total handshakes,
n(n - 1)/2 = 28
⇒ n(n - 1) = 28 × 2
⇒ n² - n = 56
⇒ n² - n - 56 = 0
⇒ n² - 8n + 7n - 56 = 0
⇒ n(n - 8) + 7(n - 8) = 0
⇒ (n - 8)(n + 7) = 0

∴ n = 8, - 7

So, the number of people be 8.

Simple interest for 3 years
= 12005 - 9800 = 2205
Simple interest for 5 years
= (2205/3) × 5 = 3675

Some of money = 9800 - 3675 = 6125

∴ Rate of interest = (100 × 2205)/(6125 × 3)
= 12%

Let AB be the tower and C and D be the objects.
Then, AB = 150 m,
∠ACB = 45° and
∠ADB = 60°

AB/AD = tan 60° = √3
AD = AB/√3
= 150/√3 m.

AB/AC = tan 45° = 1
AC = AB = 150 m.

∴CD = (AC - AD)
= {150 - (150/√3)} m
= [{150(√3 - 1)/√3} × {(√3)/(√3)}] m
= 50(3 - √3) m
= (50 × 1.27) m
= 63.5 m.

Question: Complete the following series:
350, 520, 738, ?, 1342

Solution:
350 = 7³ + 7
520 = 8³ + 8,
738 = 9³ + 9,
?
1342 = 11³ + 11,
So the pattern is n³ + n 

So missing number is,
10³ + 10 = 1000 + 10 = 1010

Question: The 2nd term of a geometric sequence is 27/8, and the 5th term is 1. What is the common ratio?

Solution:
আমরা জানি, একটি গুণোত্তর ধারার n-তম পদ, a_n = a.r^{n - 1}

দেয়া আছে,
2য় পদ, a₂ = 27/8
⇒ ar = 27/8 …...(1)

5ম পদ, a₅ = 1
⇒ a.r⁴ = 1 …....(2)

এখন, সমীকরণ (2) ÷ সমীকরণ (1) ⇒
(ar⁴)/(ar) = 1/(27/8)
⇒ r³ = 8/27
⇒ r³ = (2/3)³
⇒ r = 2/3

∴ সাধারণ অনুপাত (common ratio) হলো 2/3

The hands of a clock point in opposite directions (in the same straight line) 11 times in every 12 hours. Because between 5 and 7 they point in opposite directions at 6 o'clock only.
So, in a day, the hands point in the opposite directions 22 times.

Question: A problem is assigned to three students. The chances of solving it for the first, second, and third students are 1/3, 1/4, and 1/6, respectively. What is the probability that the problem will be solved by at least one of them?

Solution:
Probability of 1st student solving the problem = 1/3
Probability of 1st student not solving the problem = 1 - 1/3 = 2/3

Probability of 2nd student solving the problem = 1/4
Probability of 2nd student not solving the problem = 1 - 1/4 = 3/4

Probability of 3rd student solving the problem = 1/6
Probability of 3rd student not solving the problem = 1 - 1/6 = 5/6

Probability that none of the students solve the problem = (2/3) × (3/4) × (5/6)
= 5/12

∴ Probability that the problem will be solved = 1 - 5/12
= 7/12

∴ The probability that the problem will be solved is 7/12.

Given,
y₁ = 2x - 5 and y₂ = - x + 10
As, y₂ > y₁
Or, - x + 10 > 2x - 5
Or, 15 > 3x
Or, 3x < 15
Or, x<5

Question: An aeroplane covers a certain distance at a speed of 400 kmph in 3 hours. To cover the same distance in 3/2 hours, it must travel at a speed of:

Solution:
দেওয়া আছে, প্রথম ক্ষেত্রে গতিবেগ = 400 kmph এবং সময় = 3 hours

আমরা জানি, দূরত্ব = গতিবেগ × সময়
∴ দূরত্ব = 400 × 3 = 1200 km

আবার, দ্বিতীয় ক্ষেত্রে অতিক্রান্ত দূরত্ব একই থাকবে।
দূরত্ব = 1200 km এবং সময় = 3/2 hours

আমরা জানি, গতিবেগ = দূরত্ব/সময়
= 1200/(3/2)
= (1200 × 2)/3
= 400 × 2
= 800 kmph

∴ উড়োজাহাজটিকে 800 kmph গতিবেগে চলতে হবে।

Question: If 2x + 3y = 24 and y = 2x, then find y - x.

Solution:
দেওয়া আছে, 2x + 3y = 24 এবং y = 2x
⇒ 2x + 3(2x) = 24
⇒ 2x + 6x = 24
⇒ 8x = 24
⇒ x = 3

∴ y = 2×3 = 6
তাহলে, y - x = 6 - 3 = 3

let, the smaller number be x and bigger number be y
So, y + x = 55 ..... (i) and y - x = 9 ...... (ii)
by solving equation (i) and (ii) we get y = 32 and x = 23

Question: By selling a book for Tk 525, a seller incurs a loss equal to 2/5 of his cost price. If he sells it for Tk 950, what is his gain or loss percentage?

 Solution:
Let cost price = x Tk
If the book’s selling price is 525 Tk, then loss = (x - 525) Tk

∴ x - 525 = x × (2/5)
⇒ 5(x - 525) = 2x
⇒ 5x - 2625 = 2x
⇒ 3x = 2625
⇒ x = 875

If the book’s selling price is 950 Tk, then gain = (950 - 875) = 75 Tk

∴ Gain percentage = (75 × 100)/875 = 8.57%  

প্রশ্ন: ∠P এবং ∠Q পরস্পর পূরক কোণ। যদি ∠P = ২০° + ৪ক এবং ∠Q = ৬ক হয়, তবে ∠Q এর মান কত?

সমাধান:
এখানে,
∠P = ২০° + ৪ক এবং ∠Q = ৬ক

পূরক কোণের ক্ষেত্রে,
∠P + ∠Q = ৯০°
⇒ (২০° + ৪ক) + ৬ক = ৯০°
⇒ ২০° + ৪ক + ৬ক = ৯০°
⇒ ২০° + ১০ক = ৯০°
⇒ ১০ক = ৯০° − ২০°
⇒ ১০ক = ৭০°
∴ ক = ৭°

∴ ∠Q = ৬ × ৭° = ৪২°

speed of the boat = 6 km/hr
Speed downstream = (6+2) = 8 km/hr
Speed upstream = (6-2) = 4 km/hr
Distance travelled downstream = Distance travelled upstream = 32 km
Total time taken
= Time taken downstream + Time taken upstream
= 32/8 + 32/4
= 4 + 8
= 12 hours.

(1/2)x/21 + (1/2)x/24 = 10
⇒ x/21 + x/24 = 20
⇒ 15x = 168 x 20
∴ x = 224 km

Probability of getting head on all tosses = 1/2 × 1/2 × 1/2 × 1/2 = 1/16

Question: The present ages of Rahim and Sakib are in the ratio 4 : 5. After 6 years, the ratio of their ages will be 5 : 6. What is the difference in their present ages?

Solution:
Let, their present ages be 4x and 5x.

After 6 years,
Rahim's age = 4x + 6
Sakib's age = 5x + 6

According to the question,
(4x + 6)/(5x + 6) = 5/6
⇒ 6(4x + 6) = 5(5x + 6)
⇒ 24x + 36 = 25x + 30
⇒ 25x - 24x = 36 - 30
⇒ x = 6

Rahim's present age = 4 × 6 = 24 years
Sakib's present age = 5 × 6 = 30 years

∴ Difference = 30 - 24 = 6 years

Let the number be x
According to the question,

91 - (30/100) = x
⇒ 9100 - 30x = 100x
⇒ 9100 = 130x
⇒ x = 9100/130
Hence, x = 70.

1) Student attempts x questions.
2) Out of 20 questions he answered 15 correctly and of (x – 20) questions he answered 1/3 correctly.
3) The student gets 50% marks.
Therefore,
15 + 1/3(x – 20)= 50% of x
⇒ 15 + 1/3(x – 20) = (50/100) × x
⇒ 15 + 1/3(x – 20) = x/2
⇒ 90 + 2(x - 20) = 3x
⇒ x = 50
Hence, the number of questions attempted by the students = 50.

Let the speed of slower train = S km/hr
Speed of faster = (S + 14) km/hr
Trains meet after 12 hours.

Distance travelled by slower train in 12 hrs. = 12S
Distance travelled by faster train in 12 hrs. = 12(S + 14)

The total distance to be travelled between the two stations is given.
So, 12S + 12(S + 14) = 240
2S + 14 = 20
S = 3 km/hr.

Hence, The speed of the slower train is 3 km/hr.

Question: If (a/b) + (b/a) = 4, find the value of (a³/b³) + (b³/a³).

Solution:
দেওয়া আছে, (a/b) + (b/a) = 4

প্রদত্ত রাশি = (a³/b³) + (b³/a³)
= (a/b)³ + (b/a)³
= {(a/b) + (b/a)}³ - 3 (a/b)(b/a) . {(a/b) + (b/a)}
= 4³ - (3 × 1 × 4)
= 64 - 12
= 52

∴ প্রদত্ত রাশির মান 52

প্রশ্ন: A rectangular tank with a length of 4m and a width of 2m can store 20000 liters. What is the height of the tank?

Solution:
দেয়া আছে,
ট্যাংকের দৈর্ঘ্য (l) = 4 m, প্রস্থ (b) = 2 m, এবং আয়তন (V) = 20000 লিটার।

ধরি, ট্যাংকটির উচ্চতা হল h মিটার।

আমরা জানি,
আয়তাকার ঘনবস্তুর আয়তন, V = l × b × h ঘন একক
= (4 × 2 × h) m³
= 8h m³

এখন,
1 m³ = 1000 লিটার।

প্রশ্নমতে,
(8h × 1000) = 20000
বা, 8h = 20000/1000
বা, 8h = 20
∴ h = 2.5 

সুতরাং, ট্যাংকটির উচ্চতা হল 2.5 মিটার।

Given,
Perimeter of the square = 24 ft
Length of the side of the square = 24/4 =6 ft
So, its area = 6² = 36 ft²

ATQ,
Area of the rectangle is, length × width = 36 ft²
⇒ length = 36/4 = 9 [As the rectangle's width is 4 ft]

∴ Perimeter of the rectangle = 2(9 + 4) = 26 ft

Question: log 2 + log 4 + log 8 + ............ Find the sum of the first 19th term-

Solution:
given that,
log2 + log4 +log8 + ............
= log 2¹ + log 2² + log 2³ + ............
= log 2 + 2 log 2 + 3 log 2 + ............
= (1 + 2 + 3 + .........) × log 2

The sum of the first 19 natural numbers is given by the formula:
Sum = n(n+1)/2
where n = 19
∴ Sum = 19(19 + 1)/2
= 19 × 10
= 190

So, the sum of the first 19 terms = 190 log 2

Question: A cuboid has dimensions in the ratio 1:2:3 and a total surface area of 88 cm². What is its volume?

Solution: 
Let the dimensions be x, 2x, and 3x.

Total surface area of the cube = 2(x.2x + 2x.3x + 3x.x)
= 22x² 

According to the question, 
22x² = 88
x² = 4 
∴ x = 2

So, volume of the cuboid = 2 × 4 × 6 
= 48 cm³

Question: Solution set of the inequality: 3y + 4 ≥ 2y - 5 is-

Solution:
Given that,
3y + 4 ≥ 2y - 5
⇒ 3y + 4 - 2y ≥ 2y - 5 - 2y
⇒ y + 4 ≥ - 5
⇒ y ≥ - 5 - 4
⇒ y ≥ - 9

∴ Solution set of the inequality is  [- 9,  ∞)

Question: The ratio of A : B is 2 : 3, and the ratio of B : C is 4 : 5. If A 16, what is the value of C?

Solution:
Given that,
A : B = 2 : 3
B : C = 4 : 5
And A = 16

Now, 
A : B = 2 : 3 
⇒ A/B = 2/3
⇒ 2B = 3A
⇒ B = (16 × 3)/2  ; [A = 16]
∴ B = 24

And,
B : C = 4 : 5
B/C = 4/5
⇒ 24/C = 4/5
⇒ C = (24 × 5) / 4
∴ C = 30

So the value of C is 30.

Question: Find the length of the altitude of an equilateral triangle of side 3√3 cm.

Solution: 
Given that, 
Side of equilateral triangle = 3√3 cm

We know, 
Altitude (height) of an equilateral triangle is,
h = (√3/2) × side
h = (√3/2) × 3√3
= (3√3 × √3)/2
= (3 × 3)/2
= 9/2
∴ h = 4.5 cm

So the length of the altitude is 4.5 cm or 9/2 cm.

Question: 0.95 expressed as a percent of 1.9 is-

Solution:
Required Percentage = (0.95 ×100)/1.9
= (0.95 × 100 × 10)/(19 × 100)
= 50%

Difference in discount = 8% - 5% = 3%
Due to this 3% man makes Tk. 60 less in profit
That means 3% of Marked Price = 60
(3/100) × M.P. = 60
∴ M.P. = Tk. 2000

Question: At what angle are the hands of a clock inclined at 10 minutes past 5?

Solution: 
The standard formula for the smaller angle θ between the hour and minute hands is.
θ = {|60H - 11M|}/2 ; [where H = hour, M = minutes]
= {|(60 × 5) - (11 × 10)|}/2
= |300 - 110|/2
= 190/2
= 95°

So the hands of the clock are inclined at 95° at 10 minutes past 5.

Question: The speed of a boat in still water is 10 km/h. The time it takes to travel downstream is one-third the time it takes to travel upstream. What is the speed of the stream?

Solution:
Let the speed of the current be = x km/h

Then,
Downstream speed = (10 + x) km/h
Upstream speed = (10 − x) km/h

We know, time = distance/speed

According to the question:
distance/(10 + x) = distance/{3 × (10 - x)}
⇒ (10 + x) = 3(10 - x)
⇒ 4x = 20
⇒ x = 5
∴ The speed of the current = 5 km/h

Question: If m is an integer such that (- 2)^2m = 2^{9 - m}, then what is the value of m?

Solution: 
দেওয়া আছে,
(- 2)^2m = 2^{9 - m}
⇒ 2^2m = 2^{9 - m}
কোনো ঋণাত্মক সংখ্যার Power যদি জোড় সংখ্যা হয় তবে সংখ্যাটি ধনাত্মক হবে। আর যদি Power বিজোড় হয় তবে সংখ্যাটি ঋণাত্মক হবে। যেহেতু m একটি পূর্ণ সংখ্যা সেহেতু 2m একটি জোড় সংখ্যা। তাই (- 2)^2m সংখ্যাটি ধনাত্মক সংখ্যা হবে।

∴ 2m = 9 - m
⇒ 3m = 9
⇒ m = 9/3
⇒ m = 3

Question: In a right triangle, the length of one of the legs is 8 and the length of the hypotenuse is 17. What is the length of the other leg?

Solution:
এখানে,
সমকোণী ত্রিভুজের (right triangle) অতিভুজ (hypotenuse)= 17 একক
সমকোণ সংলগ্ন এক বাহু = 8 একক
সমকোণ সংলগ্ন অপর বাহু = a একক

প্রশ্নমতে,
a² + 8² = 17²
⇒ a² + 64 = 289
⇒ a² = 289 - 64
⇒ a² = 225
⇒ a = √225
∴ a = 15