Bank Math

Question: Two numbers P and Q are such that the sum of 10% of P and 15% of Q is three-fourths of the sum of 20% of P and 18% of Q. Find the ratio of P : Q.

Solution:
10% of P + 15% of Q = 3/4 × (20% of P + 18% of Q)
⇒ 10P/100 + 15Q/100 = 3/4 × (20P/100 + 18Q/100)
⇒ P/10 + 3Q/20 = 3/4 × (P/5 + 9Q/50)
⇒ 2P/20 + 3Q/20 = 3/4 × (10P + 9Q)/50
⇒ (2P + 3Q)/20 = (30P + 27Q)/200
⇒ 10(2P + 3Q) = 30P + 27Q
⇒ 20P + 30Q = 30P + 27Q
⇒ 30Q - 27Q = 30P - 20P
⇒ 3Q = 10P
⇒ P/Q = 3/10
∴ P : Q = 3 : 10

Question: If the average of 'a' numbers is b² and the average of 'b' numbers is a², what is the average of the combined (a + b) numbers?

Solution:
দেওয়া আছে:
'a' সংখ্যার গড় = b² 
∴ a সংখ্যার সমষ্টি = a × b²

'b' সংখ্যার গড় = a² 
∴ 'b' সংখ্যার সমষ্টি = b × a²

∴ মোট সমষ্টি = (a × b²) + (b × a²)
= ab(a + b)

∴ তাদের গড় = মোট সমষ্টি/(a + b)
= ab(a + b)/(a + b)
= ab

Speed = Distance/Time

Relative speed = Speed of train + Speed of boy
50 = Speed of train + 20

Speed of train = 50 – 20 = 30 km/hr

Convert km/hr into m/s
30 km/hr = 30 × (5/18) = 8.33 m/s

Distance = Speed x Time
= 8.33 x 20 = 166.6 m.

(x + y > 5 
(x – y  > 3 
^___________
      2x > 8
     ∴ x > 4

We have an 11-liter solution containing 42% of alcohol in the water.
=> quantity of alcohol in the solution = (11 × 42)/100
Now 3 liter of water is added to the solution.
=> Total quantity of the new solution = 11 + 3 = 14
Percentage of alcohol in the new solution = {(11 × 42)/100}/14 × 100
= (11 × 3)/100
= 33%

Question: If a person invests Tk. 900 at 12% simple interest and Tk. 700 at 8% simple interest, how much interest will he receive after four years?  

Solution:
এখানে,
I₁ = p₁n₁r₁
= (900 × 4 × 12)/100 
= 432 টাকা 

আবার,
I₂ = p₂n₂r₂
= (700 × 4 × 8)/100 
= 224 টাকা

∴ মোট সুদ = (432 + 224) টাকা
= 656 টাকা

Question: A and B started a business investing Tk. 18,000 and Tk. 27,000 respectively. Out of a total profit of Tk. 15,000, what is B’s share?

Solution:
Investment of A = Tk. 18,000
Investment of B = Tk. 27,000

Ratio of investments = 18,000 : 27,000
= 2 : 3

Total profit = Tk. 15,000

B’s share of profit
= 3/(2 + 3) × 15,000
= 3/5 × 15,000
= 9,000

Therefore, B’s share of the profit is Tk. 9,000.

Question: Today is Karim's 18th birthday and his father's 48th birthday. How many years from today will Karim's father be twice as old as Karim at that time?

Solution:
Given that, Today Karim is 18 years old
And his Father is 48 years old
Age difference = 48 - 18 = 30 years (constant)

Let After x years, father's age = 2 × Karim's age

Now, Karim's age after x years = 18 + x
Father's age after x years = 48 + x

ATQ,
48 + x = 2(18 + x)
⇒ 48 + x = 36 + 2x
⇒ 2x - x = 48 - 36
⇒ x = 12

So, in 12 years, Karim's father will be twice as old as Karim.

Question: Find the mother age after 8 years, if the ratio of age of mother and son is 5 ∶ 2 and the product of their ages in years is 1000.

Solution:
Given that,
Mother : Son = 5 ∶ 2
Product of ages = 1000

Let ages of mother and son be 5x, 2x

According to problem,
5x × 2x = 1000
⇒ 10x² = 1000
⇒ x² = 100 = 10²
∴ x = 10

So, age of mother after 8 years = (5x + 8) 
= (5 × 10 + 8)
= 50 + 8
= 58 years

∴ The mother age is 58 years.

If t is odd, then 3t will always be odd
Thus, odd + odd = even (3t + 1 = even number)

Man's/Boat's Speed = X
Stream/Current/River Speed = Y

∴ Downstream speed = X + Y
Upstream speed = X - Y

X+Y = (45+Y) km/hr
1 hour 20 munites = 1 hour + 20/60 = (1 + 1/3) = (4/3) hours
Downstream speed = Distance covered/Time taken

∴ 45 + y = 80/(4/3)
∴ Y = 15 km/hr
X - Y = 45 - 15 = 30 km/hr

Time is taken to go against the stream = 80/30 hours = 2 Hours 40 minutes.

Question: The perimeter of the base of a cube is 48 cm. What is its volume?

Solution: 
Let the side length of the cube be x. 
So, 4x = 48
∴ x = 12 cm

Volume = (12)³
= 1728 cm³

Suppose pipe A can fill the tank in x hours. Then,
pipe B can fill the tank in x/2 hours,
pipe C can fill the tank in x/4 hours.
Part filled by pipe A in 1 hour = 1/x
Part filled by pipe B in 1 hour = 2/x
Part filled by pipe C in 1 hour = 4/x
Therefore, partly filled by pipe A, pipe B, and pipe C together in 1 hour
= 1/x + 2/x + 4/x
= 7/x
i.e., pipe A, pipe B, and pipe C together can fill the tank in x/7 hours.
Given that pipe A, pipe B, and pipe C together can fill the tank in 10 hours
x/7 = 10
⇒ x = 70.

Question: In a class, the number of girls is 20% more than that of the boys. The strength of the class is 66. If 4 more girls are admitted to the class, the ratio of the number of boys to that of the girls is-

Solution:
Let the number of boys be 5k.
Then the number of girls = 20% more than the boys = 5k × 1.2 = 6k

Total students = 66
5k + 6k = 66
⇒ 11k = 66
⇒ k = 66/11 
∴ k = 6

∴ Number of boys = 5 × 6 = 30
∴ Number of girls = 6 × 6 = 36

Now, when 4 more girls are admitted than New number of girls = 36 + 4 = 40

∴ Ratio of boys to girls = 30 : 40 = 3 : 4 (dividing both by 10)

So the ratio of the number of boys to girls is 3 : 4

Question: The members of a club participate in at least one game. Twenty of them play football, 10 play cricket, 12 play hokey. Three of them play cricket only, 4 of them play both the cricket and football but not hockey, 2 of them participate all games. How many people play both cricket and hockey but not football?

Solution:
ফুটবল খেলে = 20 জন
ক্রিকেট খেলে = 10 জন
হকি খেলে = 12 জন
শুধু ক্রিকেট খেলে = 3 জন
ক্রিকেট ও ফুটবল খেলে = 4 জন
ক্রিকেট, ফুটবল ও হকি খেলে = 2 জন
 
হকি ও ক্রিকেট খেলে কিন্তু ফুটবল খেলে না এদের সংখ্যা = { 10 - ( 3 + 4 + 2)} 
= 1 জন।

Let speed of C = X
Then Speed of B = 3X
Then Speed of A = 6X
Ratio of the speed of A and C = 1:6
So, Greater the speed less time taken in journey.
C’s speed is 6 times less than A So A will take
1/6 of the total time taken C to covered same distance.
So, Time taken by A
= 3/(2×6)
= 1/4 hours
= 15minutes

Question: 


Solution: 

Quantity of salt in 6L of sea water,
= (6×4)/100 = 0.24
Percentage of salt in 5L of sea water,
= (0.24×100)/5
= 4(4/5)%

Cost price of motor-car = Tk. 17000
Mark price of motor-car
= Tk 17000 × 100/85 = Tk. 20000
After successive discount,Cost price
= Tk. 20000 × 95/100×90/100
= Tk. 17100 

Question: A shopkeeper marks his goods 40% above the cost price and offers a discount of 20% on the marked price. What is his gain percent?

Solution:
Let,
the cost price (CP) be Tk. 100

Marked Price = 40% more than cost price
= 100 + 40
= 140 Tk.

Discount = 20% of 140
= (20/100) ​× 140
= 28 Tk.

Selling Price (SP)= 140 - 28 = 112 Tk.

∴ Profit = SP - CP = 112 - 100 = 12 Tk.

∴ Gain percent = 12%

Question: Trisha’s grandfather was 8 times older to her 16 years ago. He would be 3 times her age 8 years from now. Eight years ago, what was the ratio of Trisha’s age to that of her grandfather? 

Solution: 
Let, trisha was x years old 16 years ago.
 
grandfather was 8x years 

ATQ, 
8x + 16 + 8 = 3 (x + 16 + 8)
⇒ 8x + 24 = 3x + 72 
⇒ 8x - 3x = 72 - 24 = 48 
⇒ x = 48/5 = 9.6 

 Eight years ago, trisha was 9.6 + 8 = 17.6 
grandfather was = 8 × 9.6 + 8 = 84.8 

ratio = 17.6 : 84.8 
= 176 : 848
= 11 : 53 

Face value of each share = Tk. 20
Dividend per share = 9% of 20 = (9 × 20)/100
= 9/5.
He needs to have an interest of 12% on his money
Money Paid for a share = (9/5) × (12/100)
Money Paid for a share = (9/5) × (100/12)
= 15.
ie, Market Value of the share = Tk. 15.

Question: In a rectangle, the diagonal length is 13 and the width is 5. What is the perimeter of the rectangle? 

Solution: 
Given that, 
Diagonal of rectangle, d = 13
Width, w = 5 

We know, 
Pythagorean theorem, l² + w² = d²
⇒ 13² =  l² + 5²
⇒ 169 = l² + 25
⇒ l²  = 169 - 25
⇒ l² = 144 = 12²
∴ l = 12
∴ length In a rectangle is, l = 12 

Perimeter of a rectangle, P = 2(l + w)
= 2(12 + 5)
= 2 × 17
= 34

So the perimeter of the rectangle is 34.

Let the three numbers be x, y, and z.

Given:Sum of squares of three numbers is 138 and sum of their products taken two at a time is 131
Therefore,
x²+y²+z²=138
xy + yz + zx=131

Formula:
(a + b + c)²= a² + b² + c²+ 2 (ab + bc + ca)
This formula can be used to easily find the sum of three numbers.

Substituting the values, we get
(x + y + z)²= x²+ y²+ z²+ 2 (xy + yz + zx)
(x + y + z )²= 138 + 2(131)
(x + y + z )²= 400
Hence, (x + y + z) = 20.

Question: If the volume of a sphere is 36π cm³, what is the surface area of the sphere?

Solution: 
Given that the volume, V = 36π cm³
or, (4/3)πr³ = 36π
or, r³ = 27
∴ r = 3 cm

Surface area of a sphere, A = 4πr²
= 4π(3)²
= 36π cm²

Question: What is the least square number which is exactly divisible by 2, 3, 10, 18 and 20?
Solution: 

The least or smallest number which is exactly divisible by 2, 3, 10, 18, and 20 is the LCM of 2, 3, 10, 18, and 20. 

LCM (2, 3, 10, 18, 20) = 180

180 = 2² × 3² × 5

So, to become a perfect square 180 needs to be multiplied by 5. 

Now, the least square number which is exactly divisible by 2, 3, 10, 18 and 20

⇒ 180 × 5 = 900

∴ The least-square number which is exactly divisible by 2, 3, 10, 18, and 20 is 900.

Question: Four boys are standing in a line for a photograph. Rifat is to the right of Rafiq. Arif is to the right of Kamal. Kamal is between Rifat and Arif. Rafiq is at the extreme left end. Who is second from the right in the line?

Solution:
রিফাত রফিকের ডানদিকে। রফিক ⇔ রিফাত
কামাল, রিফাত এবং আরিফের মাঝে। রিফাত ⇔ কামাল ⇔ আরিফ
রফিক একদম বামপ্রান্তে। রফিক ⇔ রিফাত ⇔ কামাল ⇔ আরিফ
∴ রফিক ⇔ রিফাত ⇔ কামাল ⇔ আরিফ।

∴ কামাল সারিটির ডানদিক থেকে দ্বিতীয় হবে।

Question: A rectangular tank with a length of 8m and a width of 5m can store 60000 liters. What is the height of the tank?

Solution:
দেওয়া আছে, ট্যাংকের দৈর্ঘ্য (l) = 8m,
প্রস্থ (b) = 5 m, এবং
আয়তন (V) = 60000 লিটার।

ধরি, ট্যাংকটির উচ্চতা হল h মিটার।

আমরা জানি,
আয়তাকার ঘনবস্তুর আয়তন = দৈর্ঘ্য × প্রস্থ × উচ্চতা
= (8 × 5 × h) m³
= 40h m³

এখন, আমরা জানি, 1 m³ = 1000 লিটার।

প্রশ্নমতে,
40h × 1000 = 60000
⇒ 40h = 60000/1000
⇒ h = 60/40
∴ h = 1.5

সুতরাং, ট্যাংকটির উচ্চতা হলো 1.5 মিটার।