Bank Math Master

Question: A man can do 1 work in 8 hours. How many work he can do at 40 hours?

Solution: 
In 8 hours 1 work is done
In 1 hour 1/8 work is done
∴ In 40 hours 40/8 work = 5 work can be done 

Question: A and B can do a work in 9 days, B and C can do it in 12 days and A and C can do it in 18 days. If all of them work together, in how many days they can finish the work?

Solution:
A + B can do in 1 day = 1/9 part
B + C can do in 1 day = 1/12 part
A + C can do in 1 day = 1/18 part 
2(A + B + C ) can do in 1 day = ((1/9) + (1/12) + (1/18)) = 1/4 part
∴ A + B + C can do in 1 day = 1/(4 × 2) part = 1/8 part 

∴ Total days needed = 1/(1/8) = 8 days

Question: A man works twice as much as his assistant. The man can finish the job in 9 days. Find the number of days in which the man and his assistant can finish the job if they work together? 

Solution: 
Since, Man works twice as much as his assistant 
Assistant can do the work in (2 × 9) days = 18 days 
A man can do in 1 day = 1/9 part
Assistant can do in 1 day = 1/18 part 
They together can do in 1 day = (1/9) + (1/18) = 3/18 = 1/6 part 
They can do the job in = 1/(1/6) days = 6 days 

Question: Mr. Rafiq moved 2/3 of his lawn in 4/3 hours. Mr. Shafiq makes twice a fast and finishes the remaining job. How many minutes did Mr. Shafiq work? 

Solution: 
2/3 of work is done in 4/3 hours 
full work is done in (4/3) × (3/2) hours = 2 hours 
∴ Work left = 1 - (2/3) = 1/3 part

Shafiq can complete the work in = 2/2 hours = 1 hour
Shafiq can do 1/3 part of the work in = 1/3 hour
= (1/3) × 60 minutes 
= 20 minutes

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 man and a boy can finish a job working together for 6 days. The man can work as much as two boys can do. Then in how many days can the man do it alone?

Solution: 
1 man and 1 boy can do in 1 day 1/6 part
2 man and 2 boy can do in 1 day 2/6 part = 1/3 part

Now 1 man = 2 boys 
(2 + 1) man can do in 1 day is 1/3 part
3 man can do in 1 day is 1/3 part
1 man can do in 1 day is 1/(3 × 3) part = 1/9 part

∴ Man alone can do the work in 9 days

Question: A can do a piece of work in 10 days and B in 20 days. They work together but 2 days before the completion of the work, A leaves. In how many days was the work completed?

Solution: 
B's 2 day's work = (1/20) × 2 = 1/10
∴ Remaining work = 1 - (1/10) = 9/10

(A + B)'s 1 day's work = (1/10) + (1/20) = 3/20

Now, 3/20 work is done by A and B in 1 day
9/10 work is done by A and B in (20/3) × (9/10) = 6 days

Hence, total time taken = 6 + 2 = 8 days

প্রশ্নে ভুল থাকার কারণে বাতিল করা হয়েছে?

প্রশ্নটি হওয়া উচিৎ ছিল -
Question: If 5 men and 2 boys working together can do four times as much work per hour as a man and a boy together, compare the work of a man with that of a boy?

Solution:
Let 1 man 1 day work = x
1 boy 1day work = y

then 5x + 2y = 4(x + y)
⇒ 5x + 2y = 4x + 4y
⇒ x = 2y
⇒ x/y = 2/1
⇒ x : y = 2 : 1

Question: 35 men can complete a work in 15 days. Five days after they started working, 15 more men joined them. How man days will they now take to complete the remaining work? 

Solution: 
After working 5 days, days left = 15 - 5 = 10 dyas 
and now total man = 35 + 15 = 50 

35 man needed 10 days
1 man needed (10 × 35) days 
50 man needed (10 × 35)/50 days = 7 days

Question: A man and a boy recieved Tk. 600 as wages for 6 days for the work they did together. The man's efficiency in the work was three times that of the boy. What is the daily wages of the boy?

Solution:
Ratio of 1 day's work of man and boy = 3 : 1
Total wages of the boy = 600 × ( 1/4) = 150
Daily wages of the boy = 150/6 = Tk. 25 

Question: If the volume of a cube is 1728 cm³, then the surface area of the cube will be -

Solution: 
দেওয়া আছে,
আয়তন, a³ = 1728
⇒ a³ = 12³
⇒ a = 12

পৃষ্ঠের ক্ষেত্রফল = 6a²
= 6 × 12²
= 6 × 144
= 864 cm²

Question: If the radius of a cylinder is decreased by 40% and the height is increased by 50% to form a new cylinder, the volume will be decreased by -

Solution:
মনে করি,
ব্যাসার্ধ, r = 10 একক
এবং উচ্চতা, h = 10 একক

আয়তন = πr²h
= π × (10)² × 10
= 1000π একক³

40% হ্রাসে ব্যাসার্ধ = 10 - 4 = 6 একক
50% বৃদ্ধিতে উচ্চতা = 10 + 5 = 15 একক

নতুন আয়তন = π × (6)² × 15
= 540 একক³

আয়তন হ্রাস পেয়েছে = 1000 - 540 = 460 একক³
আয়তন শতকরা হ্রাস পেয়েছে = (460 × 100)/(1000 = 46%

Question: If the numbers representing the volume and surface area of a cube are equal, then the length of the edge of the cube in terms of the unit of measurement will be -

Solution:
ঘনকের ধার a হলে,
আয়তন = a³
পৃষ্ঠের ক্ষেত্রফল = 6a²

প্রশ্নমতে,
a³ = 6a²
⇒ a = 6

Question: A right triangle with sides 6 cm, 8 cm, and 10 cm is rotated on the side of 6 cm to form a cone. The volume of the cone so formed is -

Solution:

We have,
r = 6cm
h = 8cm

∴ Volume = (1/3)πr²h  
= (1/3) × π × 6² × 8 cm³
= 96π cm³

Question: Find the number of lead balls, each with a diameter of 1 cm, that can be made from a sphere of diameter 12 cm.

Solution:
ব্যাসার্ধ, r = 6

গোলকটির আয়তন = (4/3) × π × r³
= (4/3) × π × 6³
= 288π

নতুন বলের ব্যাস 1 সেমি হলে ব্যাসার্ধ = 1/2 সেমি
নতুন বলের আয়তন = (4/3) × π × (1/2)³
= π/6 

নতুন বলের সংখ্যা হবে = 288π/(π/6) = 1728

অপশনে সঠিক উত্তর না থাকায় বাতিল করা হয়েছে।

Question: Find the volume of the cube, if its diagonal is 6​√3 cm.

Question:
দেওয়া আছে,
ঘনকটির কর্ণ,
√3a = 6√3 
⇒ a = 6 

ঘনকটির আয়তন = a³ = 6³ = 216

Question: The ratio of radii of two cones is 6 : 7 and height are in the ratio 7 : 3. What is the ratio of their volume?

Solution:
মনে করি,
কোণক দুটির উচ্চতা যথাক্রমে 7h এবং 3h
এবং ব্যাসার্ধ 6r এবং 7r

আয়তনের অনুপাত =[(1/3) × (6r)² × 7h] / [(1/3) × (7r)² × 3h]
= (36 × 7)/(49 × 3)
= 12/7
= 12 : 7

Question: Three solid spheres of radii 3 cm, 4 cm, and 5 cm respectively are melted and converted into a single solid sphere. Find the radius of this sphere.

Solution:
মনে করি,
নতুন গোলকটির ব্যাসার্ধ  = r cm

প্রশ্নমতে,
(4/3) × π × r³ = {(4/3) × π × 3³} + {(4/3) × π × 4³} + {(4/3) × π × 5³}
⇒ (4/3) × π × r³ = (4/3) × π × {27 + 64 +125)
⇒ r³ = 6³
⇒ r³ = 216
⇒ r = 6

Question: A cube of side 4 cm is cut into cubes of side 1 cm. Calculate the total surface area of all the small cubes.

Solution: 
Side of original cube = 4 cm.

Number of cubes of side 1 cm
= (Volume of the bigger cube)/(Volume of the smaller cube)
= (4 × 4 × 4)/(1 × 1 × 1)
= 64

Now, the surface area of one cube = 6(side)²
= 6 × 1²
= 6 cm²

∴ The surface area of 64 cubes = 6 × 64 cm²
= 384 cm²

Question: A copper sphere of diameter 18 cm is drawn into a wire of diameter 4 mm. Find the length of the wire.

Solution:
ব্যাসার্ধ, r = 18/2 = 9 সেমি
গোলকটির আয়তন = (4/3) × π × r³
= (4/3) × π × 9³
= 972π

তারটি ব্যাসার্ধ = 4/2 = 2 মিমি = 0.2 সেমি
তারটির আয়তন = πr²l
= π × (0.2)² × l
= 0.04πl

শর্তমতে,
0.04πl = 972π
⇒ l = 972/0.04
⇒ l = 24300 cm
⇒ l = 243 m

Question: A cistern 6 m long and 4 m wide contains water up to a depth of 2 m. The total area of the wet surface is -

Solution: 
চৌবাচ্চার উপরের অংশ খোলা থাকে। এখানে ভেজা অংশের ক্ষেত্রফল জানতে চাওয়া হয়েছে। 
তাই মোট ক্ষেত্রফল থেকে উপরের খোলা অংশ বাদ দিতে হবে।

Area of open side = 6 × 4 = 24 square meter

So, the area of total wet surface = 2(6 × 4 + 4 × 2 + 6 × 2) - (6 × 4)
= 88 - 24 
= 64