Bank Math

Question: If 4^a = 5, 5^b = 6, 6^c = 7, 7^d = 8, then the value of abcd is = ?

Solution:
8 = 7^d
= (6^c)^d
= 6^cd
= (5^b)^cd
= 5^bcd
= (4^a)^bcd
= 4^abcd
⇒ 4^abcd = 8
⇒ (2²)^abcd = 2³
⇒ 2abcd = 3
∴ abcd = 3/2

Question: The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25%, then the value of x is:
 
Answer:
Let,
The cost price of each article is Taka 1
∴ Cost Price of x articles = x Taka

Selling Price of x articles = 20 Taka
∴ Profit = (20 - x) Taka

x টাকায় লাভ হয় 20 - x টাকা
∴ 1 টাকায় লাভ হয় (20 - x)/ x টাকা
∴ 100 টাকায় লাভ হয় = [(20 - x) × 100]/x টাকা
    
প্রশ্নমতে,
[(20 - x) × 100]/x = 25
⇒ 100(20-x) = 25x
⇒ 125x = 2000
∴ x = 2000/125
= 16

আরেকভাবে,
মনে করি, প্রতিটি আর্টিকেলের ক্রয়মূল্য (cost price) = ১০০ টাকা।
তাহলে ২০টি আর্টিকেলের ক্রয়মূল্য = ২০ × ১০০ = ২০০০ টাকা।

প্রশ্ন অনুসারে, এই ২০০০ টাকা x টি আর্টিকেলের বিক্রয়মূল্য (selling price) এর সমান।

লাভ ২৫% হলে,
ক্রয়মূল্য (cost price) = ১০০ টাকা। হলে,
অর্থাৎ, প্রতি আর্টিকেলের বিক্রয়মূল্য (SP) = ১২৫ টাকা।

তাহলে x টি আর্টিকেলের SP = x × ১২৫
ATQ,
 x × ১২৫  = ২০০০
⇒ x = ২০০০/১২৫
∴ x = ১৬ টাকা

Question: What should be the value of "P" so that, the expression (9 - 12x + Px²) becomes a perfect square?

Solution:
Given Expression,
(9 - 12x + Px²)
= (3)² - 2 × 3 × 2x + (2x)² + Px² - (2x)²
= (3 - 2x)² + Px² - 4x²

∴ the expression becomes a perfect square if,
Px² - 4x² = 0
⇒ Px² = 4x²
∴ P = 4

মনে করি, মোট ফ্রিসবি এর পরিমাণ x টি
প্রশ্নমতে, 7x = 10000 + 2x
⇒ 7x - 2x = 10000
⇒ 5x = 10000
∴ x = 10000/5 = 2000

Let money invested by Rubel = Tk. x
Money invested by Tushar = 9/10 x = 0.9x
Money invested by Himu = 9/10x × 110/100 = 0.99x
Also, x + 0.9x + 0.99x = 5780
= x= 5780/2.89
= 2000

Therefore, amount invested by Rubel is Tk. 2000.

Question: Two trains of equal lengths takes 10 seconds and 15 seconds respectively to cross a telegraph post. If the length of each train be 120 miters, in what time ( in seconds) will they cross each other traveling in opposite direction?

Solution:
Speed of the first train = (120/10) m/sec = 12 m/sec

And,
Speed of the second train = (120/15) m/sec = 8 m/sec

∴ Relative speed = (12 + 8)m/sec = 20 m/sec

∴ Required time = (120 + 120)/20 sec
= 240/20
= 12 seconds

Question: A grocer mixes two types of pulses, one costing 20 taka per kg and the other 25 taka per kg.If the mixture is sold at 22 taka per kg, what is the ratio of two types of pulses in the mixture?

Solution:
মনে করি,
প্রথম প্রকার ডালের পরিমাণ = x কেজি
দ্বিতীয় প্রকার ডালের পরিমাণ = y কেজি

প্রথম প্রকার ডালের x কেজির মূল্য = 20x টাকা
দ্বিতীয় প্রকার ডালের y কেজির মূল্য = 25y টাকা

প্রশ্নমতে,
20x + 25y = 22(x + y)
⇒ 20x + 25y = 22x + 22y
⇒ 22x - 20x= 25y - 22y
⇒ 2x = 3y
∴ x/y = 3 : 2

(10)²⁰⁰÷(10)¹⁹⁶
=(10)^200−196
= 10⁴
=10000

Question: A train 110 metres long is running at a speed of 60 km/h. In what time will it pass a man who is running at 6 km/h in the direction opposite to that in which the train is going?

Solution:
Given that,
Length of train = 110 meters
Speed of train = 60 km/h
Speed of man = 6 km/h (in opposite direction)

Now, Since the train and the man are moving in opposite directions, their relative speed is the sum of their individual speeds.
 S_relative = S_train + S_man = (60 + 6)km/h = 66 × (5/18) = 55/3 m/s

Than, 
The train passes the man when it covers its own length (110 m) at relative speed.

∴ Time = Distance/Relative speed
= 110/(55/3)
= 110 × (3/55)
= 2 × 3
= 6 seconds

∴ The train will pass the man in 6 seconds.

Dog : Hare
= (3 × 3) leaps of hare : (5×1) leaps of hare
= 9 : 5

Question: 4 log 2 + log 7 =?

Solution: 
4 log 2 + log 7
= log 2⁴ + log 7
= log 16 + log 7
= log (16 × 7)
= log 112

Let his initial income is 100 taka
He spends = 75 taka
So, his savings is = 100 - 75 = 25 taka
Income increases by 20%
So new income is 120 taka
At 10% increase, the expense is = 75 + 10% of 75 = 83.5
So, now the savings is = 120 - 83.5 = 37.5
His savings increases = 37.5 - 25 = 12.5 taka
In percent = (12.5×100) / 25 = 50%

Question: The radius of a wheel is 14 cm. How many revolutions will it make in travelling 88 kilometers?

Solution:
আমরা জানি,
চাকার পরিধি = 2πr = 2 × (22/7)​ × 14 = 88 সে. মি.

∴ মোট দূরত্ব = 88 কি. মি.
= 88 × 1000 × 100
= 8800000 সে. মি.

∴ ঘূর্ণন সংখ্যা = 8800000/88​ = 100000 টি

Question: A man deposits certain amount in his bank account. After a few days. he withdraws half of the money deposited and deposits Tk. 500 more. If he has a balance of Tk. 2000 in his bank account, find the amount deposited initially.

Solution:
ধরি,
প্রথমে সে বিনিয়োগ করে ক টাকা 

শর্তমতে,
ক - (ক/২) + ৫০০ = ২০০০
⇒ ক - (ক/২) = ১৫০০
⇒ ক/২ = ১৫০০
∴ ক = ৩০০০

Let,
The length of the rectangle be l and breadth be b.
Then the original area of the rectangle = A = lb....(i)
Now,
the length of the new rectangle is l
And, the breadth of the new rectangle = 120/100 × b
= 6b/5
Then,
The area of the new one = (6/5)lb
Therefore,
The difference between two rectangles = B
= (6/5)lb - lb
= (1/5)lb...(ii)
From (i) and (ii),
B = (1/5)A [A = lb from equation no 1]
⇒ 5B = A.
Answer: 5B = A.

Let the two numbers be are a and b
∴ a + b = 14 .......(i)
a - b = 10 .......(ii)
by adding equation (i) and (ii) we get
2a = 24
∴ a = 12 and b = 2
∴ product of these two numbers = 12 × 2 = 24.

Question: A wall measures 10 meters in length, 5 meters in height, and 30 cm in thickness. Each brick used for construction measures 20 cm × 10 cm × 5 cm. How many bricks are needed to build the wall?

Solution:
Length = 10 m = 1000 cm
Height = 5 m = 500 cm
Thickness = 30 cm

∴ Volume of wall = Length × Height × Thickness
= 1000 × 500 × 30
= 15,000,000 cm³

Volume of one brick = 20 × 10 × 5 = 1000 cm³

∴ Number of bricks = Volume of wall ÷ Volume of one brick
= 15,000,000 ÷ 1000
= 15,000

0.125125... = 0.125 = 125/999

SR = PQ = 2 m
PS = QR = 10√3
tan 30° = TS/PS
1/√3 = TS/10√3
TS = 10√3/√3
= 10
TR = TS + SR
= 10 + 2
= 12 m.

Question: A room measures 7.5 m in length and 3.2 m in width. If the cost of paving is Tk. 750 per square metre, what is the total cost?

Solution:
Given that,
Length = 7.5 m
Width = 3.2 m
And rate = Tk. 750 per square metre

Now,
Area of the room is = Length × Width
= 7.5 × 3.2
= 24 m²

Now, Total cost = Area × Rate
= 24 × 750
= Tk. 18000

Question: How many times are the hands of a clock at right angle in a day?

Solution:
In 12 hours, they are at right angles 22 times.
∴ In 24 hours, they are at right angles 44 times.

Question: Which trigonometric ratio is undefined in value?

Solution:
sin90° = 1 
cos90° = 0
sec0° = 1
cosec0° = ∞(Undefined)

Question: A is twice as old as B. 12 years ago, A was five times as old as B. Find the present age of A.

Solution:
Let the present age of B is = x years.
Then the present age of A is = 2x years (since A is twice as old as B).

12 years ago,
Age of A = 2x - 12
and age of B = x - 12

According to the problem,
2x - 12 = 5(x - 12)
⇒ 2x - 12 = 5x - 60
⇒ 5x - 2x = - 12 + 60 
⇒ 3x = 48
⇒ x = 48/3
∴ x = 16

∴ Present age of B = 16 years
∴ Present age of A = 2 × 16 = 32 years

4b² + 1/b² = 2
(2b)² + (1/b)² + 4 = 2 + 4
(2b)² + 2×2b×1/b + (1/b)² = 6
(2b + 1/b)² = 6
So (2b + 1/b) = (6)^1/2.
Now 8b³ + 1/b³
= (2b)³ + (1/b)³
= (2b + 1/b)³ - 3×2(2b + 1/b)
= (6^1/2)³ - 6(6)^1/2
= 6(6)^1/2 - 6(6)^1/2
= 0

Number of liters of water in 125 liters of the mixture = 20% of 150
= 1/5 of 150
= 30 liters

Let us Assume that another 'P' litres of water is added to the mixture to make water 25% of the new mixture.
So, the total amount of water becomes (30 + P) and the total volume of the mixture becomes (150 + P).

Thus, (30 + P) = 25% of (150 + P)
⇒ 30 + P = 25/100 of (150 + P)
⇒ 30 + P = (1/4) × (150 + P)
⇒ 120 + 4P = 150 + P
⇒ 4P - P = 150 - 120
⇒ 3P = 30
⇒ P = 10

10 Litres of water is added to the mixture to make water 25% of the new mixture.

আমরা জানি, পঞ্চভূজের সমষ্টি = {(5 - 2) × 180}° = 540°
পঞ্চভুজের কোণের অনুপাতের সমষ্টি = 9 + 10 + 12 + 14 + 15 = 60
∴ পঞ্চভুজের বৃহত্তম এবং ক্ষুদ্রতম কোণের সমষ্টি = [{(9+15)/60} × 540]° = 216°

Question: Find the rate of discount being given on a shirt whose selling price is Tk. 420 after deducting a discount of Tk. 80 on its marked price.

Solution:
দেওয়া আছে,
বিক্রয়মূল্য = 420 টাকা
কমিশন = 80 টাকা

∴ গায়ে লিখা দাম = (420 + 80) টাকা 
= 500 টাকা

500 টাকায় কমিশন দেয় 80 টাকা
∴ 1 টাকায় কমিশন দেয় 80/500 টাকা
∴ 100 টাকায় কমিশন দেয় (80 × 100)/500 টাকা
= 16 টাকা

Question: Two trains 240 metres and 270 metres in length are running towards each other on parallel lines, one at the rate of 60 kmph and another at 48 kmph. How much time will they take to cross each other? 

Solution:
Given that,
Length of first train = 240 m
Length of second train = 270 m
Speed of first train = 60 km/h
Speed of second train = 48 km/h

∴ Relative speed = 60 + 48 = 108 km/h
= 108 × (5/18) m/s
= 30 m/s

And total distance to be covered to completely cross each other,
= Sum of lengths of both trains
= 240 m + 270 m
= 510 m

We know, 
Time taken = Distance/Relative speed
= 510m/30 m/s
= 17 seconds

So the two trains will take 17 seconds to cross each other.

Question: Tamim can do a work in 12 days and Sakib in 15 days. If they work on it together for 5 days, then the fraction of the work that is left is:

Solution:
Tamim's 1 day's work = 1/12
Sakib's 1 day's work = 1/15

∴ (Tamim + Sakib)'s 1 day's work = (1/12 + 1/15) part
= (5 + 4)/60 part
= 9/60 part
= 3/20 part

∴ (Tamim + Sakib)'s 5 day's work = [5 × (3/20)] part
= 15/20 part
= 3/4 part

Therefore, Remaining work = (1 - 3/4) part
= 1/4 part