Bank Math Master

Question: A water tank has two taps (Tap-1 and Tap-2). Tap-1 can fill a tank in 8 hours and Tap-2 can empty the tank in 16 hours. How long will they take to fill the tank if both taps are opened simultaneously but Tap-2 is closed after 8 hours?

 
Solution:
Tap-1, in 1 hour it fills = 1/8 part
Tap-2, in 1 hour it empties = 1/16 part

When both taps are open, in 1 hour it fills
= (1/8 - 1/16) part
= (2 - 1)/16 part
= 1/16 part

When both taps are open, in 8 hours it fills
= (1/16 × 8) part
= 1/2 part

∴ Remaining part = (1 - 1/2)
= 1/2 part

As Tap-2 is closed after 8 hours,

∴ Tap-1 alone can fill 1 part in = 8 hours
So, remaining 1/2 part will be filled in
= 8 × 1/2
= 4 hours

∴ Total time required = 8 + 4 = 12 hours

Question: How many cubes of 8 cm edge can be put in a cubical box of 80 cm edge?

Solution:
Number of cubes = (80 × 80 × 80)/(8 × 8 × 8)
= 512000/512
= 1000 cubes

[Note: 1 m = 100 cm, here all dimensions are in cm]

Question: Two ingoing pipes can fill a tank in 6 hours and 8 hours, respectively. An outgoing pipe is attached to these two pipes and thus the tank was filled in 4 hours. In 48 hours, the outgoing pipe alone can empty how many tanks?

Solution:
Let the outgoing pipe take P hours to empty the tank.

So, in 1 hour, total fill-up = (1/6 + 1/8 - 1/P) part
= (3P + 4P - 24)/24P part
= (7P - 24)/24P part

According to the question,
24P/(7P − 24) = 4

⇒ 24P = 28P − 96
⇒ 4P = 96
∴ P = 24

∴ the outgoing pipe take 24 hours to empty the tank.
In 48 hours, it can empty = 48/24 = 2 tanks

Question: A can do a piece of work in 30 days. When he had worked for 10 days, B joined him. If the complete work was finished in 24 days, B can alone finish that work in -

 
Solution:

A's 1 day's work = 1/30 part
A's 24 day's work = 24/30 part = 4/5 part

∴ Remaining work = 1 - 4/5 = 1/5 part

This 1/5 part of work was done by B in = (24 - 10) = 14 days

∴ 1 part of work done by B in = 14 × 5 = 70 days

Question: The length of a rectangle is 25% more than its breadth. What will be the ratio of the area of the rectangle to that of a square whose side is equal to the breadth of the rectangle?

 
Solution: 
Let,
breadth = X metres
∴ length = 125% of X metres
= 125X/100 metres
= 5X/4 metres

∴ Area of the rectangle = (5x/4 × X) m²
∴ Area of the square = (X × X) m²

∴ The ratio = (5X/4 × X) : (X × X)
= 5 : 4

Question: Two pipes can fill a tank with water in 15 and 12 hours respectively and a third pipe can empty it in 4 hours. If the three pipes be opened, the tank will be emptied in-

Solution:
Part of the tank filled by two pipes in 1 hour = 1/15 + 1/12
= (4 + 5)/60 part
= 9/60 part
= 3/20
Part of the tank emptied by the third pipe in 1 hour = 1/4

∴ Net part of the tank emptied in 1 hour = (1/4 - 3/20) part
= (5 - 3)/20 part
= 2/20 part
= 1/10 part

1/10 Part of tank can be emptied in 1 hour
∴ The whole tank will be emptied in = 10 hours

Question: A and B can do a piece of work in 20 days and 30 days respectively. They began to work together but A leaves after some time, and B completed the remaining work in 10 days. After how many days did A leave?

 
Solution:
Work done by B in 10 days = (1/30 × 10) part
= 10/30
= 1/3 part

Remaining work = 1 - 1/3 = 2/3 part

(A + B)'s 1 day's work = (1/20 + 1/30) part
= (3/60 + 2/60) = 5/60
= 1/12 part

Now, 1/12 part work is done by (A + B) in 1 day
∴ 1 part work is done by (A + B) in 12 days

∴ 2/3 part work is done by (A + B) in = 12 × (2/3) = 8 days

∴

Question: If a trapezium has two parallel sides measuring 4 cm and 6 cm, and its area is 100 square centimeters, what is the perpendicular distance (height) between the parallel sides?

Solution:
Given,
Parallel sides of a trapezium = 4 cm, and 6 cm

We know,
Area of trapezium = (1/2)(sum of the parallel sides) × distance between the parallel sides
100 = (1/2)(4 + 6) × distance
⇒ 100 = 5 × distance
⇒ distance = 100/5
∴ distance = 20 cm

So, the distance between the parallel lines of trapezium = 20 cm.

Question: One pipe can fill a tank four times as fast as another pipe. If together the two pipes can fill the tank in 36 minutes, then the faster pipe alone will be able to fill the tank in:

Solution:
Let,
the slower pipe alone fill the tank in x minutes.
Then, Faster pipe alone will fill it in x/4 minutes.

ATQ,
(1/x) + (4/x) = 1/36
⇒ 5/x = 1/36 
∴ x = 180

The slower pipe alone fill the tank in 180 minutes.
the faster pipe alon will be able to fill the tank in = (180 ÷ 4) minutes.
= 45 minutes

Question: If the volume of a sphere is divided by its surface area, the result is 25 cm. the radius of the sphere is -

Solution:
Let,
the radius of the sphere is r cm
∴ the volume of a sphere = (4/3)πr³
∴ the surface area of a sphere = 4πr²

ATQ,
{(4/3)πr³}/(4πr²) = 25
⇒ r/3 = 25
∴ r = 75

∴ the radius of the sphere is 75 cm

Question: A can finish a job in 18 days, B in 12 days, and C in 6 days. B and C begin the work together but have to stop after working for 2 days. How many days will A alone take to complete the remaining work?

Solution:
Work done by B and C in 1 day:
B's 1-day work = 1/12,
C's 1-day work = 1/6
B + C in 1 day = 1/12 + 1/6
= (1 + 2)/12
= 3/12
= 1/4

Work done by B and C in 2 days = 2 × 1/4 = 1/2

Remaining work = 1 - 1/2 = 1/2

A's 1-day work = 1/18

∴ Time for A to finish remaining work = (1/2) ÷ (1/18) 
= 1/2 × 18
= 9 days

Question: If 60 men can complete a job in 50 days, how many extra men need to be hired to finish the same work 10 days earlier?

Solution:
Let the total work = 60 × 50 = 3000 man-days
New time to finish work = 50 - 10 = 40 days

Number of men required = Total work ÷ New time
= 3000 ÷ 40
= 75 men

Extra men needed = 75 - 60 = 15 men

Question: A tank is filled using 20 buckets, each having a capacity of 10 liters. How many buckets would be required to fill the same tank if each bucket can hold 40 liters of water?

Solution:

Total capacity of the tank = 20 × 10 = 200 liters

∴ Number of buckets needed with 40-liter buckets = 200 ÷ 40
= 5 buckets

Question: A cube has a total surface area of 72 m². Determine the length of its diagonal.

Solution:
মনে করি,
ঘনকটির ধার = a 
ঘনকের সম্পূর্ণ পৃষ্ঠের ক্ষেত্রফল = 6a²

প্রশ্নমতে,
6a²  = 72
⇒ a² = 72/6
⇒ a² = 12
⇒ a² = (4 × 3)
⇒ a = 2√3 

∴ ঘনকটির কর্ণের দৈর্ঘ্য = a√3
= 2√3 × √3 
= 2 × (√3)2
= 2 × 3
= 6

∴ ঘনকটির কর্ণের দৈর্ঘ্য = 6 মিটার।

Question: How many bricks are required to build a wall that is 8 m long, 6 m high, and 11 cm thick, if each brick measures 25 cm × 60 cm × 11 cm?

Solution:
Given,
Wall long = 8 m = 800 cm
Wall thick = 11 cm
Wall high = 6 m = 600 cm

∴ Volume of the wall = (800 × 600 × 11) cm³

∴ Volume of the brick = 25 cm × 11 cm × 60 cm

∴ bricks need to build the wall = Volume of the wall ÷ Volume of the brick
= (800 × 600 × 11) ÷ (25 × 11 × 60)
= 320

Question: A can complete 1/5 of a task in 4 days, while B can complete 1/10 of the same task in 5 days. If they work together and receive a total payment of Tk 900, what amount should A receive?

Solution:

Whole work is done by A in (5 × 4) = 20 days
∴ A's 1-day work = (1/5) ÷ 4 = 1/20

Whole work is done by B in (5 × 10) = 50 days
∴ B's 1-day work = (1/10) ÷ 5 = 1/50

Ratio of A's and B's 1-day work = 1/20 : 1/50
= 50 : 20 = 5 : 2

∴ A's share of payment = (5/(5+2)) × 900 
= (5/7) × 900
= 643 Tk

Question: A pipe can fill a cistern in 20 hours. Once the cistern is half full, three additional identical pipes are opened. How long will it take to fill the cistern completely?

Solution:

Work done by 1 pipe in 1 hour = 1/20
∴ Time to fill half the cistern with 1 pipe = (1/2) ÷ (1/20) = 10 hours

After cistern is half full,
three additional identical pipes are opened, 
total pipes = 4
Work done by 4 pipes in 1 hour = 4 × (1/20) = 1/5
Time to fill remaining half = (1/2) ÷ (1/5)
= 2.5 hours
= 2 hours 30 minutes

∴ Total time to fill cistern = 10 + 2.5
= 12.5 hours = 12 hours 30 minutes

Question: A garrison of 600 men has provisions to last 30 days. After 3 days, 300 additional men arrive. How many days will the remaining food now last?

Solution:
After 3 days, food having for = (30 - 3) = 27 days
After arriving 300 men, total men = (600 + 300) = 900 men

600 men can eat the food for 27 days
∴ 1 man can eat the food for (27 × 600) days
∴ 900 men can eat the food for (27 × 600)/900 days
= 18 days

Question: Five identical cubes, each with a side of 5 cm, are placed next to each other. What is the volume of the resulting solid?

Solution:
The new formed is a cuboid of length = 5 × 5 = 25 cm
breadth = 5 cm and height = 5 cm

∴ Volume = (25 × 5 × 5) cm3
= 625 cm3

Question: A 314 cm long copper strip is bent into a round wheel. What is the wheel’s diameter?

Solution:
Length of the strip = Circumference of the wheel = 314 cm

We know,
Circumference C = π × d

wheel’s diameter, d = C ÷ π = 314 ÷ 3.14 = 100 cm