Bank Math Master

Question: The surface area of a sphere is same as the curved surface area of a right circular cylinder whose height and diameter are 12 cm each. The radius of the sphere is:

Solution: 
surface area of sphere = 4πr²
Curved Surface area of cylinder =2πr₁h
diameter = 12 cm
radius, r₁ = 6 cm

⇒ 4πr²=2πr₁h
⇒ r²= (6×12)/2
⇒ r² = 36
⇒ r = 6

radius of sphere 6 cm.

Question: Rajib finishes his work in 15 days while Sagor takes 10 days. Find the number of days it will take them to complete the same work together?

Solution: 
রাজিব, ১৫ দিনে করে ১ অংশ 
১ দিনে করে ১/১৫ অংশ  

সাগর, ১০ দিনে করে ১ অংশ 
১ দিনে করে ১/১০ অংশ 

একসাথে ১ দিনে করে = (১/১৫) + (১/১০)
= (২ + ৩)/৩০
= ৫/৩০ 
= ১/৬ অংশ 

সম্পূর্ণ কাজ করবে ৬ দিনে। 

Question: A rectangular water tank is 6 m high, 4m long and 2.5 m high wide. How many liters of water can it hold?

Solution:
Volume = length × width × height 
= 6 × 2.5 × 4 m³
= 60 m³ 

1 m³ = 1000 litre
60 m³ = 60 × 1000 litre
= 60000 litre

Question: 5 workers can complete work in 20 days. In how many days, 10 workers can complete the work?

Solution: 
 5 workers can complete work in 20 days
1 workers can complete work in  5 × 20 days 
= 100 days 
10 workers can complete work in 100/10 days 
= 10 days 

Question: Jihan types 450 words in half an hour. How many words would he type in 7 minutes?

Solution: 
Words per minute= (Number of words) / (Time in minutes)
Words per minute = 450 words / 30 minutes 
= 15 words/minute

 number of words  in 7 minutes:
Number of words = Words per minute × Time in minutes 
= 15 words/minute × 7 minutes
= 105 words

Jihan would type 105 words in 7 minutes.

Question: 2 men or 3 women can earn 192 tk. in a day. Find how much 5 men and 7 women will earn in a day?

Solution: 
২ জন পুরুষের আয় ১৯২ টাকা
১ জন পুরুষে আয় ১৯২/২ টাকা 
= ৯৬ টাকা 
৫ জন পুরুষের আয় = (৫ × ৯৬) টাকা 
= ৪৮০ টাকা 

৩ জন মহিলার আয় ১৯২ টাকা 
১ জন মহিলার আয় ১৯২/৩ টাকা 
= ৬৪ টাকা 
৭ জন মহিলার আয় = (৬৪ × ৭) টাকা 
= ৪৪৮ টাকা 

৫ জন পুরুষ ও ৭ জন মহিলার আয় = ৪৮০ + ৪৪৮ টাকা 
= ৯২৮ টাকা 

Question: A rectangular water reservoir contains 48000 liters of water. If the length of the reservoir is 6m and the breadth is 4m, the depth of the reservoir will be - 

Solution: 
1 m³ = 1000 litre
⇒ 48000 litre = 48000/1000
= 48 m³ 

48 = 6 × 4 × depth 
∴ depth = 48/24
= 2 m

Question: A factory has 6 machines that produce 800 units per day. If three of the machines are out of order, how many units will be produced in a day?

Solution:
Let the total number of units produced by the 6 machines in one day be U.
Each machine produces U/6 units per day.

With three machines out of order, there are 6 - 3 = 3 machines working.
The number of units produced per day by the three working machines is (U/6) * 3 = U/2 units.

So, in a day, 800/2 = 400 units will be produced when three machines are out of order.

Question: 2 men, working 9 hours a day, can build a dam in 2 days. How many hours a day must 3 men work to build the dam in 1 day?

Solution:
2 men need 2 days working 9 hours
∴ 1 man need 2 days working (9 × 2) hours
∴ 1 man need 1 day working (9 × 2 × 2) hours
∴ 3 men need 1 day working (9 × 2 × 2)/3 hours
= 12 hours

Question: A rectangular prism has dimensions 12 cm, 8 cm, and 5 cm. Calculate the volume of the prism.

Solution: 
The volume of a rectangular prism can be found using the formula:
Volume = length × width × height
= 12 × 8 ×5 cm³
= 480 cm³

Problem: A construction team can build a wall in 10 days. How many walls can they build in 100 days if they work at the same rate?

Solution:
The construction team builds 1 wall in 10 days.
In 100 days, they will build 100/10
= 10 walls.

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: 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: 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
and curved surface area of second cone
= πrl₂
Therefore, Ratio
= πrl₁ / πrl₂
= l₁ / l₂
= 5 : 7

Question: A hemispherical bowl of internal radius 9 cm contains a liquid. This liquid is to be filled into cylindrical shaped small bottles of diameter 3 cm and height 8 cm. How many bottles will be needed to empty the bowl?

Solution:
অর্ধগোলকের আয়তন  = (1/2)× 4πr³/3
= (2/3) π9³ ঘনসেমি 

প্রতি সিলিন্ডার আকৃতির বোতলের আয়তন = π (3/2)² × 8
= 18π ঘনসেমি 
ধরি, n সংখ্যক বোতল লাগবে। 

n × 18π = (2/3) π9³
⇒ n = (2/3) π9³/18π
∴ n = 27 

Question: If the side of a cube is increased by 50%, find by what percentage the surface area of the cube is increased?

Solution:
ধরি, ঘনকের এক বাহুর মান ১০০ মিটার 
সমগ্রতলের ক্ষেত্রফল = ৬ × বাহু^২
= ৬ × ১০০^২
= ৬ × ১০০০০ বর্গমিটার 

নতুন বাহুর মান ১০০ + ১০০ এর ৫০% 
= ১০০ + ৫০
= ১৫০ মিটার 

সমগ্রতলের ক্ষেত্রফল = ৬ × বাহু^২ 
= ৬ × ১৫০^২ বর্গমিটার 

ক্ষেত্রফল বৃদ্ধি = ৬ × ১৫০^২ - ৬ × ১০০^২
= ৬ × (১৫০^২ - ১০০^২)
= ৬ × (২২৫০০ - ১০০০০)
= ৬ × ১২৫০০ বর্গমিটার 

∴ শতকরা ক্ষেত্রফল বৃদ্ধি = (৬ × ১২৫০০) ×১০০%/(৬ × ১০০০০)
= ১২৫%