Bank Math

When P scores 80, Q scores 64.
When R scores 80, P scores 60
Hence, when R scores 150, Q scores (60×64×150)/(80×80) = 90
Therefore, in a game of 150, R can give 60 points to Q.

When B runs 25m, A runs 45/2m
When B runs 1000m,
A runs = (45/2 × 1/25 × 1000)m
= 900m
∴ B beats A by 100 m

Question: Which one is the set of factors of 24?

Solution: 
1 × 24 
2 × 12
3 × 8
4 × 6

∴ The factors of 24 is 1, 2, 3, 4, 6, 8, 12, 24

So the set of factors of 24 is {1, 2, 3, 4, 6, 8, 12, 24}.

Players who play at least one sport = 35 + 30 - 18 = 47
Players who play neither golf nor soccer = 50 - 47 = 3

Let the number is = 100x

Now after 15% of increase = 100x + 15% of 100x = 115x
Now 25% decrease = 115x - 25% of 115x = 115x - 28.75x = 86.25x

Actual decreased = 100x - 86.25x = 13.75x

According to the question,
13.75x = 22
⇒ x = 22/13.75
⇒ 100x = (22/13.75) × 100
⇒ 100x = 160.

∴ Original number = 160.

Question: What is the distance between the points A(- 1, 3) and B(5, - 5)?

Solution:

Question: If a dozen bananas cost 30 Taka, what is the price of the number of bananas that is 2 less than a dozen?

Solution:
A dozen bananas costs = 30 Taka.
A dozen = 12 bananas

So, the cost of 12 bananas = 30 Taka
∴ Cost of 1 banana = 30/12 = 2.5 Taka.

Now, 2 less than a dozen of bananas = 12 - 2 = 10 bananas.
Then, cost of 10 bananas,
= 10 × 2.5
= 25 Taka.

Question: A certain sum of money doubles in 20 years at simple interest. What is the annual rate of interest?

Solution:
Let,
sum = P,
then simple interest = P
and Time, T = 20 years

∴ Required rate = (100 × SI)/(P × T)
= (100 × P)/(P × 20)
= 5% per annum

12000 × (1 + R/100)⁵ = 24000
⇒ (1 + R/100)⁵ = 2
⇒ {(1 + R/100)⁵}⁴ = 2⁴ = 16
⇒ (1 + R/100)²⁰ = 16
⇒ P(1 + R/100)²⁰ = P16
⇒ 12000(1 + R/100)²⁰ = 16 × 12000 = Tk. 1,92,000

Question: Ratio of two number A and B is 3 : 2. If we decrease 60 from A and Add 60 with B then ratio of A and B is 18 : 17. Find original value of A?  

solution:
Given that,
Ratio of A : B = 3 : 2
If A decreased by 60 and B increased by 60
And new ratio = 18 : 17

Let A = 3x and B = 2x

ATQ,
{3x - 60}/{2x + 60} = {18}/{17}
⇒ 17(3x - 60) = 18(2x + 60)
⇒ 51x - 1020 = 36x + 1080
⇒ 51x - 36x = 1080 + 1020
⇒ 15x = 2100
∴ x = 140

∴ Original value of A = 3x = 3 × 140 = 420

∴ The original value of A = 420.

Number of rotation in one hour = 10 × 60 = 600
So, Distance moved = (600 × 20) = 12000 m

Question: A car moves from Dhaka to Cumilla at the original speed at 60 kmph and reached Cumilla in 20 minutes late. If the car increased speed by one fourth of his original speed reached Cumilla on time, find the distance between Dhaka and Cumilla?

Solution:
Let distance = d km.
Original speed = 60 km/h. So time taken = d/60 hours.

But it is 20 minutes late. That means the scheduled time T (in hours) is such that d/60 = T + (20/60) = T + (1/3).

And
When speed is increased by one fourth. So new speed = 60 + (1/4) × 60 = 60 + 15 = 75 km/h.
Then time taken = d/75. And this equals T (on time).

So we have,
d/60 = T + 1(/3) .........(1)
d/75 = T ..........(2)

Now, subtract the second equation from the first,
(d/60) - (d/75) = 1/3
⇒ (5d - 4d)/300 = 1/3
⇒ d = 300/3
∴ d = 100 km

∴ Distance between Dhaka and Cumilla = 100 km

Let the sum be x
Amount after 4 years = 7x /5
SI for 4 years = 7x/ 5 − x = 2x/5
R = (100 × SI)/ PT
= (100 × 2 x /5)/( x × 4)
= 10

Alternative:
The sum makes 2 /5 of itself in 4 years.
That is , 40 % in 4 years. That is 10% each year.
Therefore, rate of interest per annum is 10%.

According to the question,
At 20% loss, 80 Tk selling price হলে cost = 100 Tk.
∴ 200 Tk. selling price হলে cost = (100 × 200)/80
= 250 Tk.

আমরা জানি,
ত্রিভুজের বৃহত্তম বাহুর বিপরীত কোণ ক্ষুদ্রতম বাহুর বিপরীত কোণ অপেক্ষা বৃহত্তর হবে।
তাই, AB > AC হলে অবশ্যই ∠ACB > ∠ABC

Question: A cuboidal room has its length, breadth, and height increased by 10%, 20%, and 50% respectively. Calculate the percentage change in the volume of the cuboid.

Solution:
Let,
Each side of the cuboid be 10 unit initially.
Initial Volume of the cuboid,
= length × breadth × height
= 10 × 10 × 10
= 1000 cubic unit.

After increment dimensions become,
Length = (10 + 10% of 10) = 11 unit.
Breadth = (10 + 20% of 10) = 12 unit.
Height = (10 + 50% of 10) = 15 unit.

Now, present volume = 11 × 12 × 15 = 1980 cubic unit.
Increase in volume = 1980 - 1000 = 980 cubic unit.
∴ percentage increase in volume = (980/1000) × 100 = 98%

Formula for calculating two overlapping sets:
A + B - both + NOT (A or B) = Total

ATQ,
41 (french) + 22 (german) - 9 (both) + NOT = 78
⇒ 54 + NOT = 78
⇒ NOT = 78 - 54 = 24
⇒ So answer is 24

Question: What is the LCM of 8, 10, and 12?

Solution:
Prime factorization:
8 = 2×2×2 = 2³
10 = 2 × 5
12 = 2 × 2 × 3 = 2² × 3

LCM = take the highest power of each prime:
2³ × 3 × 5 = 8 × 3 × 5 = 120

Let the number of students in rooms A and B be x and y respectively.
Then, x - 10 = y + 10
Or, x - y = 20 .... (i)

And x + 20 = 2(y - 20)
Or, x - 2y = -60 .... (ii)

Solving (i) and (ii) we get: x = 100, y = 80.
The required answer A = 100.

Let, Capacity larger jar = x
Say 20 litters of water poured into each jar, because the same amount of water stored for a while in smaller jar
So x×1/4 = 20
Or, x = 80
Total water of larger jar after pouring the water of smaller jar = 20 + 20 = 40
So, the fraction is = 40/80 = 1/2

Let original length = 10
original breadth = 10
Then, original area
= 10 × 10
= 100
Length is increased by 20%
⇒ New length = 10 + 2 (2 is 20% of 10)
= 12
Breadth is increased by 20%
⇒ New breadth = 10 + 2 (2 is 20% of 10)
= 12
New area = 12 × 12
= 144
Increase in are =
= new area - original area
= 144 - 100
= 44
Percentage increase in area
= (increase in area/original area) × 100
= {(44/100) × 100}%
= 44%

Question: Three taps A, B, C can fill an overhead tank in 4, 6 and 12 hours respectively. How long would the three taps take to fill the tank if all of them are opened together?

Solution:
Work with rates (fraction of tank per hour).
A fills 1/4 per hour.
B fills 1/6 per hour.
C fills 1/12 per hour.
∴ Combined rate = 1/4 + 1/6 + 1/12
= (3 + 2 + 1)/12 
= 6/12
= 1/2 (tank per hour).

∴ Time to fill one tank = 1 ÷ (1/2) = 2 hours.

∴ The three taps would take 2 hours to fill the tank if all of them are opened together.

Question: If C is the midpoint of the points A(2, 3) and B(8, 11), find the length of AC.

Solution:
দেওয়া আছে, A(2, 3) এবং B(8, 11), এবং C হলো AB-এর মধ্যবিন্দু।

দূরত্বের সূত্র ব্যবহার করে AB-এর দৈর্ঘ্য নির্ণয় করি।
AB = √{(x₂ - x₁)² + (y₂ - y₁)²}
AB = √{(8 - 2)² + (11 - 3)²}
AB = √(6² + 8²)
AB = √(36 + 64)
AB = √100
AB = 10

যেহেতু C হলো AB-এর মধ্যবিন্দু, তাই AC হবে AB-এর অর্ধেক।
∴ AC = AB/2
= 10/2
= 5

Question: If x and y are positive real numbers, then (2x⁰ - 5y⁰)³ = ?

Solution:
We know that for any positive real number,
x⁰ = 1 and y⁰ = 1

So, (2x⁰ - 5y⁰)³
= (2 × 1 - 5 × 1)³
= (2 - 5)³
= (- 3)³
= - 27

New annual salary = 90,000
Salary increase = 15,000.
Original salary = 90,000 - 15,000.
= 75,000

% Increase = (15,000/ 75,000 )×100
=20%

Question: A and B started a business with investments in the ratio 2:3. After 4 years, A withdrew his capital. B continued alone for 2 more years. If the total profit after 6 years is Tk. 39,000, what is A's share of the profit?

Solution:
Let A's capital = 2x
Let B's capital = 3x

A's investment for 4 years = 2x × 4 = 8x
B's investment for 6 years = 3x × 6 = 18x

Ratio of profits = 8x : 18x = 4 : 9

∴ Total contribution units = 4 + 9 = 13

Here, total profit = Tk. 39,000
∴ A's share = (4/13) × 39000
= (4 × 39000)/13
= Tk. 12,000

14 থেকে 34 পর্যন্ত = 3 টি

40 থেকে 49 পর্যন্ত = 11 টি

54 থেকে 94 পর্যন্ত = 5 টি
_____________________________

                   মোট = 19 টি

Question: 3 pumps, working 12 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?

Solution:
3 pumps need 2 days by working 12 hours
1 pump needs 1 day by working 12 × 3 × 2 hours
4 pumps need 1 day by working (12 × 3 × 2)/4 hours
= 18 hours

Question: There is a group of 7 boys and 4 girls. The two groups working together can do five times as much work as a boy and a girl. Ratio of working capacities of a boy and a girl is:

Solution:
Let 1 boy's 1 day's work = x
And 1 girl's 1 day's work = y
Now,
(7 boys + 4 girls)'s work = 7x + 4y
Given,
7x + 4y is equal to 5 times work done by a boy and a girl
Thus,
7x + 4y = 5(x + y)
⇒ 7x + 4y = 5x + 5y
⇒ 7x - 5x = 5y - 4y
⇒ 2x = y
⇒ x/y = 1/2
⇒ x : y = 1 : 2

Hence, the required ratio is 1 : 2

Question: A machine produces 300 pens in 5/2 hours. How many pens can it produce in 50 minutes?

Solution:
দেওয়া আছে,
সময় = 5/2 ঘণ্টা = (5/2) × 60 = 150 মিনিট

150 মিনিটে কলম উৎপাদিত হয় = 300টি
1 মিনিটে কলম উৎপাদিত হয় = 300 / 150 = 2টি
50 মিনিটে কলম উৎপাদিত হয় = 2 × 50 = 100টি

∴ মেশিনটি 50 মিনিটে 100টি কলম উৎপাদন করতে পারবে।

Question: The speed of a boat down the stream is 125% of the speed in still water. If the boat takes 30 minutes to cover 20 km in still water, then how much time (in hours) will it take to cover 15 km upstream?

Solution: 
Given that, 
Speed downstream = 125% of speed in still water
Distance in still water = 20 km, time = 30 min = 0.5 hr
Distance to travel upstream = 15 km

Speed of boat in still water = Distance/Time = 20/0.5 = 40 km/h
And,
Downstream speed = 125% of still water speed.
∴ Downstream speed = 1.25 × 40 = 50 km/h 

We know,
Downstream speed = Boat speed in still water + Current speed 
⇒ 50 = 40 + Current speed 
⇒  Current speed = 50 - 40 = 10 km/h 

And upstream speed = Boat speed in still water - Current speed = 40 - 10 = 30 km/h 

∴ Time to cover 15 km upstream = Distance/Speed ​= 15/30 = 1/2 = 0.5 hours

So the time required for the boat to cover 15 km upstream is 0.5 hours. 

Question: If 2 is added to the numerator of a fraction, the fraction becomes 1. If 9 is added to the denominator, the fraction becomes 1/2. Find the fraction.

Solution:
ধরি,
লব x, হর y
শর্তমতে,
(x +2)/y = 1
⇒ x + 2 = y   ................(1) 

আবার,
x/(y + 9) = 1/2 
⇒ 2x = y + 9
⇒ 2x - 9 = y ..................(2) 

(1) ও (2) হতে পাই,
2x - 9 = x + 2
⇒ x = 11

x  এর মান (1) নং এ বসিয়ে পাই,
11 + 2 = y
⇒ y = 13 

∴ ভগ্নাংশটি 11/13

Question: The square root of (6 + 5√2)(6 - 5√2) is:

Solution:
√{(6 + 5√2)(6 - 5√2)}
= √{6² - (5√2)²}   [∵ (a + b)(a - b) = a² - b²]
= √{36 - (25 × 2)}
= √{36 - 50}
= √(-14)
= √{14(- 1)}
= √14 × √(- 1)
= i√14   [যেখানে i² = - 1]

Let the ages of children be x, (x + 3), (x + 6), (x + 9) and (x + 12) years.
Then, x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 50
⇒ 5x = 20
⇒ x = 4.
∴ Age of the youngest child = x = 4 years.

Question: Three partners shared the profit in a business in the ratio 5 : 7 : 8. They had partnered for 14 months, 8 months and 7 months respectively. What was the ratio of their investments?

Solution:
Let their investments be Tk. x for 14 months, Tk. y for 8 months and Tk. z for 7 months respectively.
Then, 14x : 8y : 7z = 5 : 7 : 8.
Now,
14x/8y = 5/7
⇒ 98x = 40y
∴ y = (49/20) x

And,
14x/7z = 5/8
⇒ 112x = 35z
∴ z = (112/35) x = (16/5) x.

x : y : z = x : (49/20) x : (16/5) x = 20 : 49 : 64.