Bank Math Master

Question: The area of an isosceles triangle is 25√3 cm², and the measure of each of the equal sides is 10 cm, what is the angle between the equal sides? 

Solution: 
Area of the traingle :
⇒ 1/2absinθ = 25√3
⇒ 10 × 10 sinθ = 50√3
⇒ sinθ = √3/2
∴ θ = 60°

Question: If the radius of a sphere is doubled, how many times does its volume become?

Solution: 
Let the original radius be r. 
Then, original volume = (4/3)πr³

New radius = 2r
∴ New violume = (4/3)π(2r)³
= 8 × (4/3)πr³
= 8 × original volume 

Question: Mr. Shihab moved 3/4th of his lawn in 5/4 hours. Mr. Anik makes twice as fast and finishes the remaining job. How many minutes did Mr. Anik work?

Solution: 
Mr. Shihab moved 3/4th of his lawn in 5/4 hours.
Mr. Shihab completed work in (5/4) × (4/3)
= 5/3 hours

Mr. Anik makes twice as fast
Mr. Anik takes time 5/6 hours
= (5/6) × 60 min 
= 50 min to finish the job 

Mr. Anik takes time to finish the remaining job = 50/4 min 
= 12.5 min

Question: In the diagram, ABCD is a rectangle with AP = PQ = QB. What is the ratio of the areas of ΔCPQ and that of the rectangle? 



Solution: 
let, length of rectangle is AB = x m and breadth BC = y m
area = xy m² 

AP = PQ = BQ = x/3

area of ΔCPQ = (1/2) × (PQ) × BC
=  (1/2) × (x/3) × y
= xy/6

ratio = (xy/6) : xy
= 1 : 6

Question: 5 mat-weavers can wave 5 mats in 5 days. At the same rate, how many mats would be woven by 10 mat-weavers in 10 days? 

Solution: 
5 mat weavers in 5 days wave = 5 mats
∴ 1 mat weavers in 1 day wave = 5/(5 × 5) mats
∴ 10 mat weavers in 10 days wave = (5 × 10 × 10)/(5 × 5)  = 20 mats

Question: Enayet has a triangle in mind. Its longest side has a length of 20 cm and another of its sides has a length of 10 cm. Its area is 80 cm². What is the exact length of its third side?

Solution: 

(1/2) × 20 × h = 80 
h = 8

BD = √(10² - 8²)
= √36 = 6
CD = 20 - 6 = 14

x = √(14² + 8²)
= √(196 + 64)
=  √260
= 2√65 cm

Question: A factory produces 500 bottles of soda in 2 hours. How many bottles will it produce in 6 hours, working at the same rate?

Solution:
Production rate = Number of items produced / Time
Production rate = 500 bottles / 2 hours = 250 bottles per hour

Bottles produced in 6 hours = Production rate × Time
Bottles produced in 6 hours = 250 bottles/hour × 6 hours = 1500 bottles

Question: A contractor employs 40 persons for doing a job in 60 days. After 20 days it was found that only one-fourth of work was finished. How many more persons are to be employed to finish the job as per schedule?

Solution: 
Work remains = 1 - (1/4) part = 3/4 part
Day remaining = 60 - 20 = 40 days

1/4 th work is done in 20 days by 40 person
∴ 1 part is done in 20 days by 40 × 4 person
∴ 1 part is done in 1 day by (40 × 4 × 20) person
∴  3/4 part is done in 40 days by (40 × 4 × 20 × 3)/(40 × 4) = 60 person

∴ He needs = (60 - 40) = 20 more persons.

Question: A rectangular prism has dimensions 8 cm, 4 cm, and 2 cm. Calculate the volume of the prism.

Solution: 
The volume of a rectangular prism can be found using the formula:
Volume = length × width × height
= 8 × 4 × 2 cm³
= 64 cm³

Question: The ratio of total surface area to lateral surface area of a cylinder whose radius is 25 cm and height 100 cm, is - 

Solution: 
Total surface area : Lateral surface area
= (2πrh + 2πr²⁾ : 2πrh
= (h +r) : h
= (100 + 25) : 100
= 125 : 100
= 5 : 4

Question: Nirob cuts a 4 cm cube into 1 cm cubes. What is the percentage increase in the surface area after cutting?

Solution: 
শুরুতে পৃষ্ঠতলের ক্ষেত্রফল = 6 × 4²
= 6 × 16
= 96 

ধরি, n সংখ্যক ঘনক তৈরি করা হলো। 
n 1³ = 4³
n = 64

64 টি ঘনকের প্রত্যেকটির পৃষ্ঠতলের ক্ষেত্রফল = 6 × 1²
= 6 cm²

মোট পৃষ্ঠতলের ক্ষেত্রফল = 64 × 6
= 384 cm²

বৃদ্ধি = 384 - 96 
= 288 cm²

∴ শতকরা বৃদ্ধি = (288/96) × 100%
= 300%

Question: The diameters of two cones are equal, If their slant heights be in the ratio of 5 : 7 then find the ratio of their Curved surface areas.

SolutioN: 
Given,
l₁/l₂ = 5/7

Now,
curved surface area of first cone
= πrl₁

curved surface area of second cone
= πrl₂

Therefore, Ratio
= πrl₁ / πrl₂
= l₁ / l₂
= 5 : 7

Question: If the areas of three adjacent faces of a cuboid are x, y, z respectively, then the volume of the cuboid is :

Solution: 
Let, breadth, width and height are respectively a, b, c

ab = x 
bc = y 
ca = z 

Now
ab × bc × ca = xyz 
⇒ a²b²c² = xyz 
⇒ abc = √xyz

Question: The altitude of an equilateral triangle of side 2√3 m is- 

Solution: 


Altitude AD = 2√3 × sin60 
= 2√3 ×√3/2
= 3 m²

Question: If a photocopier makes 3 copies in 1/4 seconds, at the same rate how many copies does it make in 5 minutes?

Solution: 
In 1/4 seconds he can make 3 copies 
In 1 second he can make 3/(1/4) copies = 12 copies
∴ In 300 seconds he can make (3 × 4 × 300) copies = 3600 copies

Question: A rectangular water reservoir contains 75000 liters of water. If the length of the reservoir is 5 m and the breadth is 3 m, the depth of the reservoir will be - 

Solution: 
1 m³ = 1000 litre
⇒ 75000 litre = 75000/1000
= 75 m³ 

75 =3 × 5 × depth 
∴ depth = 75/15
= 5 m

Question: A square and an equilateral triangle have equal perimeters. If the one side of the square is 9 cm, then what is the area of the triangle? 

Solution:
Let the side of the square be a cm
⇒ a = 9

The perimeter of the square
= 4a
= 4 × 9
= 36 cm

and also perimeter of the equilateral triangle = 36 cm

Each side of the triangle
= 36/3
= 12

Area of the triangle 
= (√3/4) × (12)²
= 36√3 cm²