Bank Math Master

Question: A cylinder has a radius of 7 cm and a height of 10 cm. What is its volume?

Solution: 
Radius, r = 7 cm 
Height, h = 10 cm

We know, 
Volume = πr²h
= (22/7) × (7)² × 10
= 1540 cm³

Question: A factory produces 480 items in 6 days working 8 hours per day. How many items would it produce in 9 days working 10 hours per day?

Solution: 
480 items in 6 days, working 8 hours per day
Total hours worked = 6 × 8 = 48 hours
So, production rate = (480/48) = 10 items/hour

Total hours for 9 days working 10 hours/day
= 9 × 10 = 90 hours

So, total items = 90 × 10 = 900 items

Question: If the volume of a sphere is 288π cm³, what is the surface area of the sphere?

Solution: 
Given that the volume, V = 288π cm³
or, (4/3)πr³ = 288π
or, r³ = 216 
∴ r = 6 cm

Surface area of a sphere, A = 4πr²
= 4π(6)²
= 144π cm²

Question: A contractor undertook to finish a piece of work in 100 days and employed 75 men. After 50 days, 2/5 of the work was completed. How many additional men should be employed to complete the work on schedule?

Solution: 
We know that 75 men completed 2/5 of the work in 50 days. Therefore, the amount of work done by 1 man in 50 days is:
Work done by 1 man in 50 days = 2/(5×75) = 2/375

Remaining work = 1 - 2/5 = 3/5 

Number of men required = Remaining work / Work done by 1 man in 50 days
= (3/5)/(2/375)
= 112.5

Since the number of workers must be an integer, round up to 113 workers.

Additional workers = 113 − 75 = 38

Question: A solid metal sphere of radius 10 cm is melted and recast into solid cones of radius 5 cm and height 12 cm. How many cones can be made?

Solution: 
The volume of a sphere = (4/3)πr³
= (4/3)π(10)³
= (4000π/3) cm³

The volume of a cone = (1/3)πr²h
= (1/3)π(5)²(12)
= 100π cm³

Number of cones = (4000π/3)/100π
= 40/3
= 13. 33
= 13 (approx)

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: A man, a woman, and a child together receive 84 Taka for 6 days' work. A woman's daily wage is twice a child's wage, and a man earns twice as much as a woman's. How much does a woman earn per day?

Solution: 
Let, 
C = Daily wage of the child (in Taka).
W = Daily wage of the woman (in Taka).
M = Daily wage of the man (in Taka).

A.T.Q, 
W = 2C
M = 2(2C) = 4C

Total payment of 6 days = 6 (C + W + M)
84 = 6(C + 2C + 4C)
42C = 84
∴ C = 2

So, the woman earns = 2 × 2 = 4 taka

Question: The surface area of a sphere is 324π cm². What is its radius? 

Solution: 
Let, r is the square of the sphere. 

A.T.Q,
4πr² = 324π
r² = 81
∴ r = 9

The radius of the sphere is 9 cm.

Question: A water pump fills a tank in 8 hours. If two identical pumps work together, how long will it take to fill the same tank?

Solution: 
One pump fills the tank in 8 hours.
Rate of one pump = 1/8 per hour

If two identical pumps work together, their rates add up.
So, the combined rate is = 1/8 + 1/8
= 1/4 per hour

∴ Time = 1/(1/4) = 4 hours

Question: A cylindrical tank with diameter 14 m and height 5 m is filled with water. If the water is transferred to a rectangular tank with base 10 m × 7 m, what will be the height of water in the rectangular tank?

Solution: 
Volume of the cylinder = π(7)²5
= π × 49 × 5
= 245π m³

Volume of the rectangle = 10 × 7 × h (Assuming, height of the rectangle is 'h')
= 70h m³ 

So, 70h = 245π
⇒ h = (245/70)(22/7)
∴ h = 11m 

Question: Six identical machines can produce 540 articles in 12 hours. How many articles would 8 such machines produce in 15 hours?

Solution: 
Total articles produced by 6 machines in 12 hours = 540.
Articles produced by 1 machine in 12 hours = 540/6
Articles produced by 1 machine in 1 hour = 540/(6×12) = 7.5 articles

So, Articles produced by 8 machines in 15 hours = 7.5 × 8 × 15 
= 900 articles

Question: The ratio of the volumes of two spheres is 27:8. What is the ratio of their surface areas?

Solution: 
Volume of a Sphere: V = (4/3)π(r)³
Surface Area of a Sphere: S = 4πr²

Given, 

Question: If 40 workers working at 80% efficiency can complete a task in 15 days, how many workers working at 100% efficiency would be needed to complete the same task in 10 days?

Solution: 
Case-1: 
W = 40 × 0.8 × 15
= 480

Case-2:
Let the required number of workers be x.
W = x × 1 × 10
480 = 10x
∴ x = 48

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

Question: Five machines working together can process 1200 items in 4 days. If one machine breaks down, how many items can the remaining machines process in 6 days?

Solution: 
Total items processed by 5 machines in 4 days = 1200
Items processed by 1 machine in 4 days = (1200/5) items
Items processed by 1 machine in 1 day = {1200/(5 × 4)} items
= 60 items

Therefore, 
Items processed by 4 machines in 6 days = 60 × 6 × 4
= 1440 items

Question: A hollow cylinder has an internal radius of 8 cm, external radius of 12 cm, and height 15 cm. What is the volume of the material used?

Solution: 
Let, 
Internal radius (r) = 8 cm
External radius (R) = 12 cm
Height (h) = 15 cm

Volume of the the material used, 
V = πh(R² - r²) 
= π × 15 (144 - 64)
= π × 15 × 80
= 1200π

Question: A factory has two machines, X and Y. Machine X can produce 6,000 items in 10 days, working 6 hours per day. Machine Y can produce 8,000 items in 8 days, working 10 hours per day. If both machines work together for 8 hours per day, how many days will they take to produce 24,000 items?

Solution: 
Total hours worked by Machine X = 10 days × 6 hours/day = 60 hours.
Rate of X = (6000/60) = 100 items/hour

Total hours worked by Machine Y = 8 days × 10 hours/day = 80 hours.
Rate of Y = (8000/80) = 100 items/hour

Combined rate = Rate of X + Rate of Y = 100 + 100 = 200 items/hour.

So, time required = {24000/(200 × 8)} = 15 days. 

Question: A cube with side length 6 cm fits perfectly inside a hollow spherical ball. What is the total surface area of the sphere?

Solution: 
If a cube fits perfectly inside a sphere, then the diameter of the sphere = space diagonal of the cube

Diagonal of the cube = √3a = 6√3
So, diameter of sphere = 6√3
Radius = 6√3/2 = 3√3

Surface area of the sphere = 4πr²
= 4π(3√3)²
= 108π cm²

Question: If 5 workers can harvest 60 kg of wheat in 3 days, how many kilograms of wheat will 8 workers harvest in 5 days?

Solution: 
5 workers 3 days harvest = 60 kg
1 worker 1 day harvest = (60/15) kg
8 workers 5 days harvest = ( 60 × 40 ) / 15 kg
= 160 kg