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Time & Work, Chain Rule

মোট প্রশ্ন১,০৭৬এই পাতা১০০প্রতি পাতা১০০
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
উত্তর
উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Time & Work, Chain Rule

PrepBank · পাতা ৯ / ১১ · ৮০১–৯০০ / ১,০৭৬

৮০১.
If 5 workers can harvest 60 kg of wheat in 3 days, how many kilograms of wheat will 8 workers harvest in 5 days?
  1. ক
    48 kg
  2. খ
    160 kg
  3. গ
    150 kg
  4. ঘ
    145 kg
ব্যাখ্যা

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

৮০২.
If P and Q together can complete a work in 18 days, P and R together in 12 days, and Q and R together in 9 days, then Q alone can do the work in- 
  1. ক
    36 days
  2. খ
    30 days
  3. গ
    18 days
  4. ঘ
    24 days
ব্যাখ্যা

Question: If P and Q together can complete a work in 18 days, P and R together in 12 days, and Q and R together in 9 days, then Q alone can do the work in-

Solution:
One day's work of,
(P + Q) = 1/18 .......(1)

One day's work of,
(P + R) = 1/12 .......(2)

One day's work of,
(Q + R) = 9 .......(3)

Adding(1),(2)and(3),
⇒ 2 × (P + Q + R) = (1/18) + (1/12) + (1/9)
⇒ 2 × (P + Q + R) = 1/4
⇒ (P + Q + R) = 1/8

Now,
⇒ Q = (1/8) - (P + R)
⇒ Q = (1/8) - (1/12)
One day's work of Q = (3 - 2)/24 = 1/24
∴ Q need 24 days

৮০৩.
A garrison of 600 men had provisions for 30 days. After 3 days a reinforcement of 300 men arrived. How many days will the remaining food last now?
  1. ক
    18 days
  2. খ
    16 days
  3. গ
    14 days
  4. ঘ
    20 days
  5. ঙ
    22 days
ব্যাখ্যা

Question: A garrison of 600 men had provisions for 30 days. After 3 days a reinforcement of 300 men arrived. How many days will the remaining food last now?

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

৮০৪.
If 30 men can complete a piece of work in 27 days, in what time 18 men can do another piece of work 3 times as greater?
  1. ক
    135 days
  2. খ
    54 days
  3. গ
    81 days
  4. ঘ
    56 days
ব্যাখ্যা
Question: If 30 men can complete a piece of work in 27 days, in what time 18 men can do another piece of work 3 times as greater?

Solution:
30 men can do a piece of work in 27 days, and we know that men × days = total work
So, we can say that the total work = 30 × 27 = 810 units
Now ATQ,
18 men can do work 3 times greater than 810 units.
i.e., 18 men can do a work 3 × 810 = 2430 units in x days
Men × days = total work (2430 units)
⇒ 18 × x = 2430
∴ x = 2430/18 = 135 days
৮০৫.
A can do a piece of work in 42 days and B is thrice as efficient as A. If B joins A on 22nd day then in how many days will the work be completed?
  1. ক
    ক) 5.75
  2. খ
    খ) 5.25
  3. গ
    গ) 5.35
  4. ঘ
    ঘ) 5.45
ব্যাখ্যা
A alone can complete the work = 42 days

Formula used:
W = E × T     
Where, W = The work, E = The efficiency, and T = The time

Let us assume the work will be completed in X days
⇒ The one day work of A = 1/42
⇒ The one day work of B = 3 × (1/42) = 1/14
⇒ The 21 days work of A = 21 × (1/42) = 1/2
⇒ The remaining work = 1 - 1/2 = 1/2
⇒ The efficiency of A and B together = (1/42 + 1/14) = 4/42 = 2/21
⇒ The remaining work will be completed = (1/2) × (21/2) = 21/4 = 5.25 days

∴ The required result will be 5.25 days.
৮০৬.
A man completes 2/3 of a job in 12 days. At this rate, how many more days will it take him to finish the
  1. ক
    ক) 6 days
  2. খ
    খ) 4 days
  3. গ
    গ) 3 days
  4. ঘ
    ঘ) 9 days
ব্যাখ্যা
Question: A man completes 2/3 of a job in 12 days. At this rate, how many more days will it take him to finish the job?

Solution:
Rest part of job = 1 - 2/3 = 1/3

2/3 of the work done in 12 days
1 of the work done in (12 × 3)/2 days
1/3 of the work done in (12 × 3)/(2 × 3) days
= 6 days
৮০৭.
In a hostel, Food was available for 2000 students for 54 days, but after 15 days more students joined hostel and food lasts only for 20 more days. Determine how many students came in hostel?
  1. ক
    ক) 1650
  2. খ
    খ) 1700
  3. গ
    গ) 1900
  4. ঘ
    ঘ) 2100
ব্যাখ্যা

As it is given that food is there for 54 days for 2000 students,
But after 15 days, remaining food was sufficient for 2000 students for 39 days only ( as 54-15 = 39 )
Let x number of students came in hostel after 15 days,
=> Total number of students after 15 days = 2000 + x
=> Remaining food was enough for (2000 + x) students for 20 days
So here is indirect proportion, as more people and less food
=> 2000 : ( 2000 + x ) :: 20 : 39
=> 2000× 39 = (2000 + x )× 20
=> 3900 = 2000 + x
=> x = 3900 - 2000
=> x = 1900

৮০৮.
P can complete a work in 10 days, Q in 15 days, and R in 30 days. P stops working 2 days before the completion of the work, and Q stops 3 days before completion. R continues working alone till the end. What was the total number of days taken to complete the entire work?
  1. ক
    4 days
  2. খ
    7 days
  3. গ
    10 days
  4. ঘ
    15 days
ব্যাখ্যা

Question: P can complete a work in 10 days, Q in 15 days, and R in 30 days. P stops working 2 days before the completion of the work, and Q stops 3 days before completion. R continues working alone till the end. What was the total number of days taken to complete the entire work?

Solution:
ধরি, সম্পূর্ণ কাজটি শেষ হতে মোট সময় লাগে y দিন।

∴ P কাজ করেছে (y - 2) দিন
P-এর কাজের অংশ = (y - 2)/10

Q কাজ করেছে (y - 3) দিন
Q-এর কাজের অংশ = (y - 3)/15

R পুরো y দিন কাজ করেছে, তাই তার কাজের অংশ = y/30

শর্তমতে,
(y - 2)/10 + (y - 3)/15 + y/30 = 1
⇒ {3(y - 2) + 2(y - 3) + y} / 30 = 1
⇒ 3y - 6 + 2y - 6 + y = 30
⇒ 6y - 12 = 30
⇒ 6y = 42
∴ y = 7

∴ সম্পূর্ণ কাজ সম্পন্ন করতে 7 দিন সময় লেগেছে।

৮০৯.
Two workers can complete a job in 24 days while working together. First one of the two workers works alone for 16 days then the other worker works for 24 days alone. If it is known that only 20% of assigned work is left after 40 days, then find out the time (in days) taken by the slower worker to complete the remaining work.
  1. ক
    12 days
  2. খ
    16 days
  3. গ
    20 days
  4. ঘ
    24 days
ব্যাখ্যা
Question: ­Two workers can complete a job in 24 days while working together. First one of the two workers works alone for 16 days then the other worker works for 24 days alone. If it is known that only 20% of assigned work is left after 40 days, then find out the time (in days) taken by the slower worker to complete the remaining work.

Solution:
Let,
the rate of the first worker be 'a' and of the second worker be 'b'.
in 24 days of working together they can complete the total work.
∴ 24a + 24b = 1...........(i)

first worker worked for 16 days alone and second worker worked for 24 days alone to complete 80% of the work.
∴ 16a + 24b = 8/10 = 4/5..............(ii)

subtracting (ii) from (i) we get,
24a + 24b - 16a - 24b = 1 - 4/5
8a = 1/5
a = 1/40

putting a = 1/40 om equation(1) we get,
6/10 + 24b = 1
24b = 1 - 6/10
b = 1/60

so, b is the slower worker.
b can complete the work in 60 days.
as 20% of the work is remaining,
total time to complete the remaining work by b is = 20% of 60 = 12 days
৮১০.
If m men can do a work in r days, then the number of days taken by (m + n) men to do it is:
  1. ক
    ক) (m + n)/mr
  2. খ
    খ) (m + n)/mn
  3. গ
    গ) m/r(m + n)
  4. ঘ
    ঘ) mr/(m + n)
ব্যাখ্যা
Question: If m men can do a work in r days, then the number of days taken by (m + n) men to do it is:

Solution:
m men can do a work in r days.
1 men can do a work in mr days.
∴ (m + n) men can do a work in mr/(m + n) days.
৮১১.
A canteen requires 420 bananas for a week. Totally, how many bananas will it require for the months of January, February, and March in 2024?
  1. ক
    ক) 4860
  2. খ
    খ) 5000
  3. গ
    গ) 5220
  4. ঘ
    ঘ) 5460
ব্যাখ্যা
Question: A canteen requires 420 bananas for a week. Totally, how many bananas will it require for the months of January, February, and March in 2024?

Solution:
January = 31 days
February = 29 days
March = 31 days

Total = 31 + 29 +31 = 91 days

In 7 days the canteen requires 420 bananas
In 1 day the canteen requires 420/7 bananas
In 91 days the canteen requires (420 × 91)/7 bananas
= 5460 bananas
৮১২.
A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 4 days. B can do the work alone in-
  1. ক
    12 days
  2. খ
    8 days
  3. গ
    14 days
  4. ঘ
    10 days
ব্যাখ্যা

Question: A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 4 days. B can do the work alone in-

Solution:
Suppose A, B and C take x, x/2 and x/3 days respectively to finish the work.
Then,
1/x + 2/x + 3/x = 1/4
⇒ 6/x = 1/4
⇒ x/6 = 2
∴ x = 24

So, B takes (24/2) = 12 days to finish the work.

৮১৩.
মুসতাকিম 35 কিলোমিটার দূরুত্বের কিছু অংশ 4 kmph বেগে এবং কিছু অংশ 5 kmph বেগে অতিক্রম করে। সে যদি পুরো দূরুত্ব 5 kmph বেগে যেত তাহলে ঐ একই সময়ে আরো 2 কিলোমিটার বেশি দূরুত্ব অতিক্রম করতে পারতো। মোট ভ্রমণে তাঁর কত সময় লেগেছিল?
  1. ক
    7 ঘণ্টা 40 মিনিট
  2. খ
    7 ঘণ্টা 24 মিনিট
  3. গ
    6 ঘণ্টা 24 মিনিট
  4. ঘ
    8 ঘণ্টা 24 মিনিট
  5. ঙ
    কোনটিই নয়
ব্যাখ্যা
প্রশ্ন: মুসতাকিম 35 কিলোমিটার দূরুত্বের কিছু অংশ 4 kmph বেগে এবং কিছু অংশ 5 kmph বেগে অতিক্রম করে। সে যদি পুরো দূরুত্ব 5 kmph বেগে যেত তাহলে ঐ একই সময়ে আরো 2 কিলোমিটার বেশি দূরুত্ব অতিক্রম করতে পারতো। মোট ভ্রমণে তাঁর কত সময় লেগেছিল?

সমাধান:
ধরি, 
4 kmph বেগে যায় x কি.মি.
5 kmph বেগে যায় (35 - x) কি.মি.

প্রশ্নমতে,
(x/4) + (35 - x)/5 = (35 + 2)/5
⇒ (5x + 140 - 4x)/20 = 37/5
⇒ x + 140 = 148
∴ x = 8

মোট সময় লাগবে = (8/4) + (35 - 8)/5
= 2 + 5.4
= 7.4
= 7 ঘণ্টা 24 মিনিট।
৮১৪.
A man and a boy can finish a job working together for 4 days. The man can work as much as two boys can do. Then in how many days can the man do it alone?
  1. ক
    6 days
  2. খ
    8 days
  3. গ
    10 days
  4. ঘ
    12 days
ব্যাখ্যা
Question: A man and a boy can finish a job working together for 4 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 1/4 part
2 man and 2 boy can do in 1 day 2/4 part = 1/2 part

Now 1 man = 2 boys 
(2 + 1) man can do in 1 day is 1/2 part
3 man can do in 1 day is 1/2 part
1 man can do in 1 day is 1/(2 × 3) part = 1/6 part

∴ Man alone can do the work in 6 days.
৮১৫.
Two pipes, P and Q can fill a cistern in 12 and 15 minutes respectively. Both are opened together, but at the end of 3 minutes, P is turned off. In how many more minutes will Q fill the cistern?
  1. ক
    ক) 5 minutes 15 seconds
  2. খ
    খ) 6 minutes 15 seconds
  3. গ
    গ) 8 minutes 15 seconds
  4. ঘ
    ঘ) 12 minutes
ব্যাখ্যা
P can fill cistern in 12 minutes
P fills cistern in 1 minute = 1/12 part

Q can fill cistern in 15 minutes
Q fills cistern in 1 minute = 1/15 part
P and Q together can fill cistern in 1 minute,
=1/12 + 1/15 = 9/60 part
So, they can together fill cistern in 3 minute,
= 3 × 9/60 = 9/20 part
Rest Cistern
= 1− 9/20
  = 11/20 part
cistern must be filled by Q in (11/20) ÷ (1/15)
= 8 minutes 15 seconds
৮১৬.
A tank can be filled with water by two pipes A and B together in 36 minutes. If the pipe B was stopped after 30 minutes, the tank is filled in 40 minutes. The pipe B can along fill the tank in:
  1. ক
    45 minutes
  2. খ
    60 minutes
  3. গ
    75 minutes
  4. ঘ
    90 minutes
ব্যাখ্যা
Question: A tank can be filled with water by two pipes A and B together in 36 minutes. If the pipe B was stopped after 30 minutes, the tank is filled in 40 minutes. The pipe B can along fill the tank in:

Solution: 
Let the pipes fill the tank in x minutes.
Part of tank filled by pipes A and B in one minute:
1/36 Part of the tank filled by pipe A in 1 minute = 1/36 − 1/x
According to the question,
30×1/x + 40(1/36 − 1/x) = 1
⇒ 30/x + 40/36 − 40/x = 1
⇒ 30/x + 10/9 − 40/x = 1
⇒ 10/x = 1/9
Hence, x = 90 minutes.
৮১৭.
4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 10 women complete it?
  1. ক
    ক) 40 days
  2. খ
    খ) 36 days
  3. গ
    গ) 32 days
  4. ঘ
    ঘ) 34 days
ব্যাখ্যা

Let 1 man's 1 day's work = x and 1 woman's 1 day's work = y
Then, 4x + 6y = 1/8 and 3x + 7y = 1/10
Solving the two equations, we get
x = 11/400, y = 1/400
∴ 1 woman's 1 day's work = 1/400
⇒ 10 women's 1 day's work = ((1/400)×10) = 1/40
Hence, 10 women will complete the work in 40 days

৮১৮.
A does 20% less work than B. If A can complete a piece of work in 15/2 hours, then B can do it in-
  1. ক
    ক) 5 hours
  2. খ
    খ) 7 hours
  3. গ
    গ) 6 hours
  4. ঘ
    ঘ) 8 hours
ব্যাখ্যা
Question: A does 20% less work than B. If A can complete a piece of work in 15/2 hours , then B can do it in-

Solution: 
Ratio of times taken by A and B = 100 : 80 = 5 : 4
Let
B takes  x days to do the work 

Here
5 : 4 = (15/2) : x
5/4 = (15/2)/x 
5x = (15 × 4/2)
5x = 30
x = 6 hours
৮১৯.
3 pumps working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?
  1. ক
    ক) 9
  2. খ
    খ) 10
  3. গ
    গ) 11
  4. ঘ
    ঘ) 12
ব্যাখ্যা

Let the required number of working hours per day be x.
More pumps , Less working hours per day (Indirect Proportion)
Less days, More working hours per day (Indirect Proportion)
Pumps 4 : 3 and Days 1 : 2 } : : 8 : x
Or, (4 × 1 × x) = (3 × 2 × 8)
Or, x = 12

৮২০.
21 binders can bind 1400 books in 15 days. How many binders will be required to bind 800 books in 20 days?
  1. ক
    ক) 7
  2. খ
    খ) 9
  3. গ
    গ) 12
  4. ঘ
    ঘ) 14
ব্যাখ্যা

Let,
The required number of binders is x.
Less books, Less binders [Direct proportion]
More days, Less binders [Indirect proportion]
Books {(1400 : 800) , Days (20 : 15)} :: 21 : x
∴ 1400 × 20 × x = 800 × 15 × 21
⇒ x = (800 × 15 × 21)/(1400 × 20)
= 9.

৮২১.
Time taken by A to finish a piece of work is twice the time taken B and thrice the time taken by C. If all three of them work together, it takes them 2 days to complete the entire work. How much work was done by B alone?
  1. ক
    ক) 2 days
  2. খ
    খ) 6 days
  3. গ
    গ) 3 days
  4. ঘ
    ঘ) 5 days
ব্যাখ্যা

Time taken by A = x days

Time taken by B = x/2 days

Time Taken by C = x/3 days

⇒ {(1/x) + (2/x) + (3/x) = 1/2

⇒ 6/x = 1/2

⇒ x = 12

Time taken by B = x/2 = 12/2 = 6 days.

৮২২.
A laborer is paid Tk. 30 per hour for an 8-hour day. If he works overtime (above 8 hours) in a single day, he is paid 1.5 times the rate he is paid for regular hours. If the laborer receives Tk. 330 for a single day's work. How long did he work on that day?
  1. ক
    9 hours 30 minutes
  2. খ
    10 hours
  3. গ
    9 hours
  4. ঘ
    10 hours 30 minutes
ব্যাখ্যা

Question: A laborer is paid Tk. 30 per hour for an 8-hour day. If he works overtime (above 8 hours) in a single day, he is paid 1.5 times the rate he is paid for regular hours. If the laborer receives Tk. 330 for a single day's work. How long did he work on that day?

Solution:
শ্রমিকের নিয়মিত বেতন (8 ঘণ্টার জন্য) = 8 ঘণ্টা × 30 টাকা/ঘণ্টা = 240 টাকা।

শ্রমিকের অতিরিক্ত কাজের (ওভারটাইম) আয় = 330 টাকা - 240 টাকা
= 90 টাকা।

শ্রমিকের প্রতি ঘণ্টার ওভারটাইম বেতন = 30 টাকা × 1.5 = 45 টাকা।

শ্রমিকের ওভারটাইম কাজের সময় = 90 টাকা ÷ 45 টাকা/ঘণ্টা
= 2 ঘণ্টা।

শ্রমিকের মোট কাজের সময় = 8 ঘণ্টা (নিয়মিত) + 2 ঘণ্টা (ওভারটাইম)
= 10 ঘণ্টা।

৮২৩.
Working 8 hours a day, A can finish an assignment in 12 days while B takes 2 days less to finish it, working 2 hours more than A, every day. The contractor decides to put them to work together on the assignment. Each working day now is 8 hours. How many days will they take to finish the assignment?
  1. ক
    ক) 60/11 days
  2. খ
    খ) 39/12 days
  3. গ
    গ) 300/49 days
  4. ঘ
    ঘ) 8 days
ব্যাখ্যা

A completes the work in 12 days at 8 hour per day = 12 x 8 = 96 hours
In 1 hour, A completes (1/96) amount of work

B completes the work in 10 days (2 days less than A) working 10 hours every day (i.e. 2 hours more than A per day) = 10 x 10 = 100 hours
In 1 hour, B completes (1/100) amount of work

Working together, in 1 hour A and B complete (1/96) + (1/100) = (49/2400) amount of work
So they complete the entire work in (2400/49) hours

Since they work for 8 hours per day, they need = (2400/49) × (1/8) = 300/49 days.

৮২৪.
639 persons can repair a road in 12 days working 5 hours a day. In how many days will 30 persons working 6 hours a day complete the work ?
  1. ক
    ক) 222 days
  2. খ
    খ) 213 days
  3. গ
    গ) 214 days
  4. ঘ
    ঘ) 215 days
ব্যাখ্যা

According to the question,
Let, D is number of days
(639×12×5)/1 road = (30×6×D)/1 road
⇒ D = 213 days

৮২৫.
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?
  1. ক
    600 items
  2. খ
    720 items
  3. গ
    900 items
  4. ঘ
    1080 items
ব্যাখ্যা

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

৮২৬.
X can complete a certain work in the same time in which Y and Z together can do it. If X and Y together can finish it in 10 days and Z alone in 15 days, then Y alone can do it in?
  1. ক
    ক) 60 days
  2. খ
    খ) 25 days
  3. গ
    গ) 18 days
  4. ঘ
    ঘ) 20 days
ব্যাখ্যা

In 1 day Z can do = 1/15
In 1 day X + Y can do = 1/10
Given, X = Y + Z
So, X + Y + Z = 1/15 + 1/10 = 1/6
OR, Y + Z + Y + Z = 2(Y + Z) = 1/6
OR, Y + Z = 1/12
OR, Y = 1/12 -1/15 = 1/60
∴ Y alone can do the work in 60 days.

অপশনে একটু মডিফিকেশন করা হয়েছে। 

৮২৭.
10 women can do a work in 8 days. 8 men can complete the same work in 7 days. What is the ratio between the capacity of a man and a woman?
  1. ক
    7 : 3
  2. খ
    10 : 7
  3. গ
    11 : 5
  4. ঘ
    9 : 7
ব্যাখ্যা

Question: 10 women can do a work in 8 days. 8 men can complete the same work in 7 days. What is the ratio between the capacity of a man and a woman?

Question: 
(10 × 8) = 80 women can complete the work in 1 day.
∴ 1 woman's 1 day's work = 1/80 part

(8 × 7) = 56 men can complete the work in 1 day.
1 man's 1 day's work = 1/56 part

So, required ratio = 1/56 : 1/80
= (1/56)/(1/80)
= 80/56
= 10/7
= 10 : 7

৮২৮.
A tyre has two punctures. The first puncture alone would have made the type flat in 9 minutes and the second alone would have done it in 6 minutes. If air leaks out at a constant rate, how long does it take both the punctures together to make it flat?
  1. ক
    1.5 min
  2. খ
    1.75 min
  3. গ
    3.5 min
  4. ঘ
    3.6 min
ব্যাখ্যা
Question: A tyre has two punctures. The first puncture alone would have made the type flat in 9 minutes and the second alone would have done it in 6 minutes. If air leaks out at a constant rate, how long does it take both the punctures together to make it flat?

Solution:
1st puncture make it flat in 9 min
∴ 1st puncture works in 1 min = 1/9 portion

2nd puncture make it flat in 6 min
∴ 2nd puncture works in 1 min = 1/6 portion 

1 minute’s work of both the punctures =(1/9 + 1/6) portion
= (6 + 9)/54
= 15/54
= 5/18

So, both the punctures will make the tyre flat in 18/5 min = 3.6 min.
৮২৯.
A tank has two leakages. The first leakage alone can empty the tank in 9 min and the second alone would have done it in 6 min. If water leaks out at a constant rate, how long does it take both the leakage together to empty the tank?
  1. ক
    ক) 3.1 min
  2. খ
    খ) 3.5 min
  3. গ
    গ) 4.0 min
  4. ঘ
    ঘ) 3.6 min
ব্যাখ্যা
Formula:
If a leakage empties a tank in 'a' hours and an another empties the same tank in 'b' hours and if water leaks at a constant rate,
the net part filled in 1 hour = 1/a + 1/b
∴ Time taken to empty the tank = ab/(a + b) hours
-----------------------------------------------------------------------------
Solution:
For 1st leakage,
in 9 minutes, 1st leakage can empty a full tank
in 1 minute, 1st leakage can empty 1/9 part of a tank

For 2nd leakage,
in 6 minutes, 2nd leakage can empty a full tank
in 1 minute, 2nd leakage can empty 1/6 part of a tank

In 1 minute, both leakage can empty (1/9 + 1/6) or 5/18 part of a tank

5/18 part of a tank can be empty in 1 minute
Full tank can be empty in 18/5 minutes or 3.6 minutes
---------------------------------------------------------------
Alternative way:
Let the total work done to empty the tank be LCM of 9 and 6 i.e. 18 unit
Tank emptied by 1st leakage in 1 minute = 18/9 = 2 unit
Tank emptied by 2nd leakage in 1 minute = 18/6 = 3 unit
∴ To completely empty the tank, Time taken together = 18/(2 + 3) = 18/5 = 3.6 min
-------------------------------------------------
Alternative way:
1st leakage empties the tank = 9 minutes
2nd leakage empties the tank = 6 minutes
Time taken to empty the tank = (9 × 6)/(9 + 6) = 54/15 = 18/5 = 3.6 min
৮৩০.
If 0.497 mark has the value of one dollar, what is the value to the nearest dollar of 350 marks?
  1. ক
    $176
  2. খ
    $524
  3. গ
    $704
  4. ঘ
    $696
ব্যাখ্যা
Question: If 0.497 mark has the value of one dollar, what is the value to the nearest dollar of 350 marks?

Solution:
0.497 marks = 1 dollar
∴ 1 mark = 1/0.497 dollars
∴ 350 marks = 350/0.497 = 704.22 dollars ≅ 704 dollars
৮৩১.
A tap can completely fill a water tank in 8 hours. The water tank has a hole in it through which the water leaks out. The leakage will cause the full water tank to get empty in 12 hours. How much time will it take for the tap the tank completely with the hole?
  1. ক
    ক) 8 hours 
  2. খ
    খ) 16 hours 
  3. গ
    গ) 24 hours 
  4. ঘ
    ঘ) 12 hours 
ব্যাখ্যা
Net part filled in 1 hour = (1/8) - (1/12) 
                                       = (3 - 2)/24 
                                       = 1/24 

The tank will be filled in = (24 × 1)/1 hours 
                                       = 24 hours 
৮৩২.
If 8 men can reap 80 hectares in 24 days, then how many hectares can 36 men reap in 30 days?
  1. ক
    350 hectares
  2. খ
    400 hectares
  3. গ
    425 hectares
  4. ঘ
    450 hectares
ব্যাখ্যা

Question: If 8 men can reap 80 hectares in 24 days, then how many hectares can 36 men reap in 30 days?
 
Solution:
8 জন লোক 24 দিনে ফসল কাটতে পারে = 80 হেক্টর জমির 
∴ 1 জন লোক 1 দিনে ফসল কাটতে পারে = 80/(8 × 24) হেক্টর জমির 
∴ 36 জন লোক 30 দিনে ফসল কাটতে পারে = (80 × 36 × 30)/(8 × 24)
= 450 হেক্টর জমির 

৮৩৩.
A camp had provisions to support 560 soldiers for 20 days. After 12 days, 112 soldiers were reassigned to another camp. For how many days can the remaining soldiers continue without receiving more rations? 
  1. ক
    10
  2. খ
    12
  3. গ
    11
  4. ঘ
    15
ব্যাখ্যা

Question: A camp had provisions to support 560 soldiers for 20 days. After 12 days, 112 soldiers were reassigned to another camp. For how many days can the remaining soldiers continue without receiving more rations?

Solution:
After 12 days, there was support for 560 soldiers for 8 days.
Remaining persons = (560-112) = 448

Less soldiers, more days (inverse proportion)
Let the x is the required number of days

Then, 448 : 560 = 8 : x
Or, x = (560 × 8)/ 448 = 10

Hence, the required number of days is 10.

৮৩৪.
After being dropped a certain ball always bounces back to 1/5 of the height of its previous bounce. After the first bounce it reaches a height of 125 inches. How high (In inches) will it reach after its fourth bounce?
  1. ক
    ক) 1/5
  2. খ
    খ) 8
  3. গ
    গ) 1
  4. ঘ
    ঘ) 5
ব্যাখ্যা
Question: After being dropped a certain ball always bounces back to 1/5 of the height of its previous bounce. After the first bounce it reaches a height of 125 inches. How high (In inches) will it reach after its fourth bounce?

Solution: 
২য় বাউন্সের পরে উঠবে = (1/5) × 125
                                      = 25
৩য় বাউন্সের পরে উঠবে =(1/5) × 25
                                      = 5
৪র্থ বাউন্সের পরে উঠবে =(1/5) × 5
                                      = 1
৮৩৫.
A man deposits certain amount in his bank account. After a few days. he withdraws half of the money deposited and deposits Tk. 500 more. If he has a balance of Tk. 5000 in his bank account, find the amount deposited initially.
  1. ক
    Tk. 15000
  2. খ
    Tk. 12000
  3. গ
    Tk. 10000
  4. ঘ
    Tk. 9000
ব্যাখ্যা
Question: A man deposits certain amount in his bank account. After a few days. he withdraws half of the money deposited and deposits Tk. 500 more. If he has a balance of Tk. 5000 in his bank account, find the amount deposited initially.

Solution:
ধরি,
প্রথমে সে বিনিয়োগ করে ক টাকা 

শর্তমতে,
ক - (ক/২) + ৫০০ = ৫০০০
⇒ ক - (ক/২) = ৪৫০০
⇒ ক/২ = ৪৫০০
∴ ক = ৯০০০
৮৩৬.
If 9 examiners can examine a certain number of answer books in 12 days by working 5 hours a day; for how many hours a day would 4 examiners have to work in order to examine twice the number of answer books in 27 days?
  1. ক
    ক) 5 hours
  2. খ
    খ) 8 hours
  3. গ
    গ) 10 hours
  4. ঘ
    ঘ) 12 hours
ব্যাখ্যা
Question: If 9 examiners can examine a certain number of answer books in 12 days by working 5 hours a day; for how many hours a day would 4 examiners have to work in order to examine twice the number of answer books in 27 days?

Solution:
9 examiners  need 12 days by working 5 hours
1 examiner needs 1 day by working 5 × 9 × 12 hours
4 examiners  need 27 days by working (5 × 9 × 12)/(4 × 27) hours
= 5 hours

So, twice the work need twice time.
∴ So, the answer = (5 ×  2) = 10 hours
৮৩৭.
55 men can finish a work in 42 days. How many additional men must be engaged to complete the work 9 days earlier?
  1. ক
    ক) 15
  2. খ
    খ) 16
  3. গ
    গ) 17
  4. ঘ
    ঘ) 18
ব্যাখ্যা

42 days এ কাজটি complete করতে লাগে 55 men
1 day এ কাজটি complete করতে লাগে 55 × 42 men
33 day এ কাজটি complete করতে লাগে (55 × 42)/33 men
= 70 men.
Additional men লাগবে 70 - 55 = 15 men.

৮৩৮.
Machine A paints y units in 15 minutes and machine B paints 3y units in 30 minutes. In how many minutes will A and B, working together, paint 45y units?
  1. ক
    270 minutes
  2. খ
    240 minutes
  3. গ
    250 minutes
  4. ঘ
    280 minutes
ব্যাখ্যা

Question: Machine A paints y units in 15 minutes and machine B paints 3y units in 30 minutes. In how many minutes will A and B, working together, paint 45y units?

Solution:
Machine A paints per minute = y/15 units
Machine B paints per minute = 3y/30 = y/10 units

∴ A and B together paint per minute = y/15 + y/10
= (2y + 3y)/30
= 5y/30
= y/6 units

So, to paint y/6 units, it takes 1 minute.

Hence, to paint 45y units, time needed = (45y × 6)/y = 270 minutes.

৮৩৯.
How long will it take for two pipes to fill a tank together when they can fill it alone in 10 hours and 15 hours respectively? 
  1. ক
    4 hours
  2. খ
    3 hours
  3. গ
    6 hours
  4. ঘ
    5 hours
ব্যাখ্যা

Question: How long will it take for two pipes to fill a tank together when they can fill it alone in 10 hours and 15 hours, respectively?

Solution:
together in one hour they can fill = 1/10 + 1/15
= 5/30
= 1/6

so, the total time to fill the tank is = 6 hours

৮৪০.
A alone can do a piece of work in 6 days and B alone in 8 days. A and B undertook to do it for Tk. 3200. With the help of C, they completed the work in 3 days. How much is to be paid to C?
  1. ক
    Tk. 400
  2. খ
    Tk. 500
  3. গ
    Tk. 600
  4. ঘ
    Tk. 800
ব্যাখ্যা
Question: A alone can do a piece of work in 6 days and B alone in 8 days. A and B undertook to do it for Tk. 3200. With the help of C, they completed the work in 3 days. How much is to be paid to C?

Solution: 
C's 1 day's work = 1/3 - (1/6 + 1/8)
= (1/3) - (7/24)
= 1/24

A's wages: B's wages: C's wages = 1/6 : 1/8 : 1/24
= 4 : 3 : 1.

C's share (for 3 days) = Tk. 3 × 1/24 × 3200 = Tk. 400.
৮৪১.
P and Q can complete a work in 15 days and 10 days respectively. They started the work together and then Q left after 2 days. P alone completed the remaining work. The work was finished in --- days.
  1. ক
    ক) 16 days
  2. খ
    খ) 24 days
  3. গ
    গ) 20 days
  4. ঘ
    ঘ) 12 days
ব্যাখ্যা

Work done by P in 1 day = 1/15
Work done by Q in 1 day = 1/10

Work done by P and Q in 1 day = 1/15 + 1/10 = 1/6
Work done by P and Q in 2 days = 2 × (1/6) = 1/3

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

Time taken by P to complete the remaining work 2/3 = (2/3) / (1/15) = 10 days

Total time taken = 2 + 10 = 12 days.

৮৪২.
A and B working together can finish a work in 12 days, B and C working together can finish the work in 16 days. If A works for 5 days, B works for 7 days, and C completes the remaining work in 13 days, C alone can complete the work in how many days?
  1. ক
    22 days
  2. খ
    24 days
  3. গ
    26 days
  4. ঘ
    28 days
ব্যাখ্যা
Question: A and B working together can finish a work in 12 days, B and C working together can finish the work in 16 days. If A works for 5 days, B works for 7 days, and C completes the remaining work in 13 days, C alone can complete the work in how many days?

Solution:
ATQ,
A+B = 12 days
B+C = 16 days

Note: Assume the total work = LCM of the given days
Take the LCM of days = LCM of (12 and 16) = 48
Let the total work = 48

Note: One day work = (total work/ days)
Now,
(A+B)'s one day work = 48/12 = 4 unit
(B+C)'s one day work = 48/16 = 3 unit

As per the question:
A works for 5 days
B works for 7 days or (5 + 2) days, that means B works 5 days with A and remaining 2 days with C.
C works 13 days or (2 + 11) days, that means C works 2 days with B and remaining 11 days alone.

That means total work done by (A + B) in 5 days
A + B = 5 days × 4 unit = 20 units
And, total work done by (B + C) = 2 days × 3 unit = 6 units
So, A + B + C finish the 26 units of work.

Remaining work = 48 - 26 = 22 unit work
And C completes the remaining work in 11 days.
i.e., C's one day's works = 22/11 = 2 units.

C alone can finish total work in [total work/ C's one day work] = [48/2] = 24 days.
৮৪৩.
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?
  1. ক
    ক) Tk. 20 
  2. খ
    খ) Tk. 25 
  3. গ
    গ) Tk. 30
  4. ঘ
    ঘ) Tk. 35 
ব্যাখ্যা
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 
৮৪৪.
A certain hotel has 1,400 single rooms and 420 double rooms. Each room is cleaned by one person. If one person can clean a single room every 15 minutes and a double room every 20 minutes, how many cleaning persons are needed to clean all the rooms if each person works for exactly 7 hours?
  1. ক
    68
  2. খ
    70
  3. গ
    76
  4. ঘ
    88
  5. ঙ
    94
ব্যাখ্যা
Question: A certain hotel has 1,400 single rooms and 420 double rooms. Each room is cleaned by one person. If one person can clean a single room every 15 minutes and a double room every 20 minutes, how many cleaning persons are needed to clean all the rooms if each person works for exactly 7 hours?

Solution:
Number of single rooms = 1400
Number of double rooms = 420
Time taken by a person to clean a single room = 15 mins = 15/60 hour = 1/4 hour
Time taken by a person to clean a double room = 20 mins = 20/60 hour = 1/3 hour

∴ Total time to clean 1400 single and 420 double rooms
= 1400 × (1/4) + 420 × (1/3)
= 350 + 140 hours
= 490 hours

Each person works for = 7 hours

∴ Number of persons required = 490/7 = 70
৮৪৫.
P, Q, and R can do a piece of work in 20, 30, and 60 days respectively. In how many days can P do the work if he is assisted by Q and R on every third day?
  1. ক
    12 days.
  2. খ
    25 days.
  3. গ
    35 days.
  4. ঘ
    15 days
ব্যাখ্যা
Question: P, Q, and R can do a piece of work in 20, 30, and 60 days respectively. In how many days can P do the work if he is assisted by Q and R on every third day?

Solution:
P's 2 day's work = (1/20) × 2
= 1/10
(P + Q + R)'s 1 day's work = (1/20) + (1/30) + (1/60)
= 6/60
= 1/10
(P + Q + R) work in 3 days = (1/10) + (1/10)
= 1/5.

Now, 1/5 work is done in 3 days.
∴ The whole work is done in (3 × 5)
= 15 days.
৮৪৬.
A computer programmer needs to print 144 documents. The documents have an average (arithmetic mean) length of 10 pages and the printer takes 15 seconds to print each page. Approximately how many hours will it take to print all the documents if they are printed without interruptions?
  1. ক
    0.5 hr
  2. খ
    2 hr
  3. গ
    2.5 hr
  4. ঘ
    6 hr
  5. ঙ
    24 hr
ব্যাখ্যা
Question: A computer programmer needs to print 144 documents. The documents have an average (arithmetic mean) length of 10 pages and the printer takes 15 seconds to print each page. Approximately how many hours will it take to print all the documents if they are printed without interruptions?

Solution:
Total number of documents = 144
Average length of a document = 144 × 10 = 1440

In 15 sec Pages printed 1
In 1 min or 60 sec Pages printed 60/15 = 4

Pages printed in an hour = 4 × 60 = 240

Time taken = 1440/ 240= 6 hours
৮৪৭.
A canteen requires 553 liters of water for a week. Totally, how many liters will it require for the months of May, June, July?
  1. ক
    8563 liter
  2. খ
    7268 liter
  3. গ
    7347 liter
  4. ঘ
    8278 liter
ব্যাখ্যা
Question: A canteen requires 553 liters of water for a week. Totally, how many liters will it require for the months of May, June, July?

Solution:
একদিনে পানি লাগে = 553/7 = 79 লিটার
May, June, July মাসে মোট দিন = (31 + 30 + 31) = 92 দিন।

∴ মোট পানি লাগবে = (92 × 79) = 7268 লিটার
৮৪৮.
210 men working 12 hours a day can finish a job in 18 days. How many men are required to finish the job in 20 days working 14 hours a day?
  1. ক
    160 
  2. খ
    162 
  3. গ
    164 
  4. ঘ
    166 
ব্যাখ্যা
Question: 210 men working 12 hours a day can finish a job in 18 days. How many men are required to finish the job in 20 days working 14 hours a day?

Solution: 
210 men need (18 × 12) hours
= 216 hours

 to finish the job in 20 days working 14 hours a day men needed = (210 × 216)/(14 × 20)
= 162 
৮৪৯.
Sharif ran a 2 mile race at an average speed of 8 miles per hour. If Arif ran the same race an average speed of 6 mile per hour, how many minutes longer than Sharif did Arif take to complete the race?
  1. ক
    15
  2. খ
    10
  3. গ
    8
  4. ঘ
    5
  5. ঙ
    None of these
ব্যাখ্যা

D = T × S
Sharif takes, 2 = T × 8
or, T = 1/4 hours
or T = 15 minutes
Arif Takes, 2 = T × 3
or, T = 1/3
or, T = 15 minutes
Difference = (20 - 15) = 5 minutes.

৮৫০.
Rasel started a project where the work ratio of a day is twice the previous day. If the work is fully finished in 20 days, how many days it took to finish 1/4th of the work?
  1. ক
    ক) 5 days
  2. খ
    খ) 10 days
  3. গ
    গ) 18 days
  4. ঘ
    ঘ) 15 days
ব্যাখ্যা
Question: Rasel started a project where the work ratio of a day is twice the previous day. If the work is fully finished in 20 days, how many days it took to finish 1/4th of the work?

Solution: 
প্রতিদিনের কাজ পূর্বের দিনের দ্বিগুণ হয়।

২০ দিনে যদি সম্পূর্ণ শেষ হয় তাহলে,
১৯ দিনে শেষ হবে ১/২ কাজ
১৮ দিনে শেষ হবে (১/২)/২ বা, ১/৪ কাজ 

shortcut: 
১ অংশ কাজ হয় ২০ দিনে,
১/৪ অংশ বা ১/২২ অংশ কাজ হয় = (২০ - ২) দিনে = ১৮ দিনে।
যত অংশ দেয়া থাকবে তার হরকে ২ এর ভিত্তি আকারে প্রকাশ করে ২ এর সূচককে মোট দিন থেকে বাদ দিয়ে দিন বের করতে হবে।

একই ভাবে যদি ১/১৬ অংশ বা ১/২৪ অংশ কাজ হবে = (২০ -৪) = ১৬ দিনে।
৮৫১.
A can do a piece of work in 10 days working, 8 hours per day. If B is two - thirds as efficient as A, then in how many days can B alone do the same piece of work, working 5 hours per day?
  1. ক
    ক) 15
  2. খ
    খ) 18
  3. গ
    গ) 20
  4. ঘ
    ঘ) 24
ব্যাখ্যা

Time taken by A alone to do the work = (10 × 8)
= 80 hrs.
Since B is two -thirds as efficient as A,
So the time taken by B to do the work
= 80 × (3/2)hrs
= 120 hrs.
∴ Required time = (120/5)
= 24 days.

৮৫২.
In the beginning, Ram works at a rate such that he can finish a piece of work in 24 hrs, but he only works at this rate for 16 hrs. After that, he works at a rate such that he can do the whole work in 18 hrs. If Ram is to finish this work at a stretch, how many hours will he take to finish this work?
  1. ক
    20 hrs 30 minute
  2. খ
    21 hrs
  3. গ
    21 hrs 20 minute
  4. ঘ
    22 hrs
  5. ঙ
    None of the above
ব্যাখ্যা
Question: In the beginning, Ram works at a rate such that he can finish a piece of work in 24 hrs, but he only works at this rate for 16 hrs. After that, he works at a rate such that he can do the whole work in 18 hrs. If Ram is to finish this work at a stretch, how many hours will he take to finish this work?

Solution:
Ram’s 16 hr work = 16/24 = 2/3.
Remaining work = 1 - (2/3) = 1/3.

Using work and time formula, this will be completed in (1/3) × 18 i.e. 6 hrs.
So, total time to complete work = 16 + 6 = 22 hrs.
৮৫৩.
The water of a pond is increasing doubled the amount daily. If it becomes full in 24 days, when was it half-filled?
  1. ক
    12th day
  2. খ
    18th day
  3. গ
    22th day
  4. ঘ
    20th day
  5. ঙ
    23rd day
ব্যাখ্যা
Question: The water of a pond is increasing doubled the amount daily. If it becomes full in 24 days, when was it half-filled?

Solution: 
As it is an exponential growth,
we should start from the end.

full on the 24th day
half on the 23rd day
1/4 th on the 22nd day
1/8 th on the 21st day
৮৫৪.
How many days are there in y weeks y days?
  1. ক
    ক) 8y
  2. খ
    খ) 14y
  3. গ
    গ) 7y + 1
  4. ঘ
    ঘ) 7y2
ব্যাখ্যা
প্রশ্ন : How many days are there in y weeks y days?
সমাধান : y weeks y days = (7y + y) days = 8y days

৮৫৫.
39 persons can repair a road in 12 days, working 5 hours a day. In how many days will 30 persons, working 6 hours a day, complete the work?
  1. ক
    10 days
  2. খ
    13 days
  3. গ
    14 days
  4. ঘ
    15 days
ব্যাখ্যা
Question: 39 persons can repair a road in 12 days, working 5 hours a day. In how many days will 30 persons, working 6 hours a day, complete the work?

Solution:
৩৯ জন ৫ ঘণ্টা কাজ করে রাস্তা মেরামত করে ১২ দিনে
৩৯ জন ১ ঘণ্টা কাজ করে রাস্তা মেরামত করে ১২ × ৫ দিনে
১ জন ১ ঘণ্টা কাজ করে রাস্তা মেরামত করে ১২ × ৫ × ৩৯ দিনে
৩০ জন ১ ঘণ্টা কাজ করে রাস্তা মেরামত করে (১২ × ৫ × ৩৯)/৩০ দিনে
৩০ জন ৬ ঘণ্টা কাজ করে রাস্তা মেরামত করে (১২ × ৫ × ৩৯)/(৩০ × ৬) দিনে
= ১৩ দিনে 
৮৫৬.
A shopkeeper has sufficient money to buy 50 books. On reduction in the price of each book by Tk. 8, he could buy 10 books more. How much money does he has?
  1. ক
    Tk. 2200
  2. খ
    Tk. 2800
  3. গ
    Tk. 2400
  4. ঘ
    Tk. 3000
ব্যাখ্যা
Question: A shopkeeper has sufficient money to buy 50 books. On reduction in the price of each book by Tk. 8, he could buy 10 books more. How much money does he has?

Solution:
১টি বইয়ে দাম কমে ৮ টাকা
∴ ৫০টি বইয়ে দাম কমে (৫০ × ৮) টাকা 
= ৪০০ টাকা 

সে মোট বই কিনে (৫০ + ১০) টি 
= ৬০টি

১০টি বইয়ের দাম ৪০০ টাকা 
∴ ৬০টি বইয়ের দাম (৪০০ × ৬০)/১০ টাকা 
= ২৪০০ টাকা 

∴ তার কাছে ২৪০০ টাকা আছে।
৮৫৭.
2 men and 7 boys can do a piece of work in 14 days; 3 men and 8 boys can do the same in 11 days. Then, 8 men and 6 boys can do three times the amount of this work in-
  1. ক
    18 days
  2. খ
    21 days
  3. গ
    24 days
  4. ঘ
    25 days
  5. ঙ
    30 days
ব্যাখ্যা
Question: 2 men and 7 boys can do a piece of work in 14 days; 3 men and 8 boys can do the same in 11 days. Then, 8 men and 6 boys can do three times the amount of this work in-

Solution:
(2 × 14) men + (7 × 14) boys = (3 × 11) men + (8 × 11) boys
⇒ 28 men + 98 boys = 33 men + 88 boys
⇒ 5 men= 10 boys
∴ 1man= 2 boys

Therefore,
(2 men + 7 boys) = (2 × 2 +7) boys = 11 boys
(8 men + 6 boys) = (8 × 2 + 6) boys = 22 boys.

Let the required number of days be x.
More boys , Less days (Indirect proportion)
More work , More days (Direct proportion)

Therefore,
(22 × 1 × x) = (11 × 3 × 14)
⇒ x = 21

Hence, the required number of days = 21
৮৫৮.
Akbar and Vijay undertake to do a piece of work for Tk. 480. Akbar alone can do it in 75 days while Vijay alone can do it in 40 days. With the help of Polash, they finish the work in 25 days. How much should Polash get for his work?
  1. ক
    Tk. 40
  2. খ
    Tk. 20
  3. গ
    Tk. 360
  4. ঘ
    Tk. 100
  5. ঙ
    Tk. 60
ব্যাখ্যা
Question: Akbar and Vijay undertake to do a piece of work for Tk. 480. Akbar alone can do it in 75 days while Vijay alone can do it in 40 days. With the help of Polash, they finish the work in 25 days. How much should Polash get for his work?

Solution:
In 25 days, Akbar and Vijay would have done 1/3 and 5/8 of the work.
The remaining work is 1 - (1/3 + 5/8) = 1/24.
This means Pradeep has done 1/24th of the work, so he should be paid 1/24th of the amount i.e. 480 × (1/24) = Tk. 20
৮৫৯.
A can do a piece of work in certain number of days. While B takes three days more than A to complete the same work. Working together, A and B can complete the work in two days. How many days does B take to complete the work?
  1. ক
    ক) 4
  2. খ
    খ) 6
  3. গ
    গ) 7
  4. ঘ
    ঘ) 8
ব্যাখ্যা
Let, A alone can complete the whole work in x days, then
Hence, B alone can complete the whole work in = (x + 3) days

According to the question
⇒ 1/x + 1/(x + 3) = 1/2
⇒ (x + 3 + x)/x(x + 3) = 1/2
⇒ (2x + 3)/(x2 + 3x) = 1/2
⇒ 4x + 6 = x2 + 3x
⇒ x2 - x - 6 = 0
⇒ x2 - 3x - 2x - 6 = 0
⇒ x (x - 3) + 2 (x - 3) = 0
⇒ (x - 3) (x + 2) = 0

Taking,
x - 3 = 0  (x can’t be -2)
x = 3

Hence, x + 3 = 3 + 3 = 6
∴ B alone can complete the whole work in 6 days.

Short Trick:
Let, B alone can compete the whole work in 6 days
A alone can complete the whole work in = 6 - 3 = 3 days

Total work = 6
A and B together can complete the whole work in = 6/(2 + 1) = 2 days (satisfied)

So, we can say B alone can complete the whole work in 6 days.
৮৬০.
Jahir is twice as good as a workman as Kabir. When they work together they can finish a task in 16 days. If Kabir works alone, in how many days he will complete the task?
  1. ক
    24 days
  2. খ
    36 days
  3. গ
    42 days
  4. ঘ
    48 days
ব্যাখ্যা
Question: Jahir is twice as good as a workman as Kabir. When they work together they can finish a task in 16 days. If Kabir works alone, in how many days he will complete the task?

Solution:
(Jahir + Kabir)'s one-day work = 1/16
As per the question, Jahir can finish twice as much work as finished by Kabir in a given duration of time.

Therefore, 2/3 of their one-day work will be completed by Jahir and 1/3 of their one day work will be completed by Kabir.

So, Kabir's one day work will be = (1/16) × (1/3) = 1/48
So, Kabir's will take 48 days to complete the task.
৮৬১.
There is provision of food in fort for 1200 soldiers for 60 days. After 15 days, 200 soldiers leave the fort. Remaining food will last for how many days?
  1. ক
    ক) 54 days
  2. খ
    খ) 58 days
  3. গ
    গ) 64 days
  4. ঘ
    ঘ) 48 days
ব্যাখ্যা
1200 × 45 = 1000 × x
Hence, x = 54 days
৮৬২.
In a camp, food was sufficient for 2000 people for 54 days. After 15 days, more people came and the food lasted only for 20 more days. How many people came?
  1. ক
    ক) 1900
  2. খ
    খ) 1800
  3. গ
    গ) 2000
  4. ঘ
    ঘ) 1940
ব্যাখ্যা

Given that food was sufficient for 2000 people for 54 days
Hence, after 15 days, the remaining food was sufficient for 2000 people for 39 days (∵ 54 - 15 = 39)

Let x number of people come after 15 days.
Then, total number of people after 15 days = (2000 + x)
Then, the remaining food was sufficient for (2000 + x) people for 20 days

More men, Less days (Indirect Proportion)

(men) 2000 : (2000 + x) :: 20 : 39
⇒ 2000 × 39 = (2000 + x) 20
⇒ 100 × 39 = (2000 + x)
⇒ 3900 = 2000 + x
⇒ x = 3900 − 2000
= 1900

৮৬৩.
A hostel had provisions for 250 men for 40 days. If 50 men left the hostel, how long will the food last at the same rate?
  1. ক
    ক) 45 days
  2. খ
    খ) 48 days
  3. গ
    গ) 50 days
  4. ঘ
    ঘ) 60 days
ব্যাখ্যা

A hostel had provisions for 250 men for 40 days

If 50 men leaves the hostel, remaining men = 250 - 50 = 200
We need to find out how long the food will last for these 200 men.

Let the required number of days = x days.

More men, Less days (Indirect Proportion)
(men) 250 : 200 :: x : 40
⇒ 250 × 40 = 200x
⇒ 5 × 40 = 4x
⇒ x = 5 × 10
= 50

৮৬৪.
A team of workers was employed by a contractor who undertook to finish 360 pieces of an article in a certain number of days. Making four more pieces per day than was planned, they could complete the job a day ahead of schedule. How many days did they take to complete the job ?
  1. ক
    10 days
  2. খ
    12 days
  3. গ
    8 days
  4. ঘ
    14 days
ব্যাখ্যা
Question: A team of workers was employed by a contractor who undertook to finish 360 pieces of an article in a certain number of days. Making four more pieces per day than was planned, they could complete the job a day ahead of schedule. How many days did they take to complete the job?

Solution:
Let the team take x days to finish 360 pieces
Then, number of pieces made each day = 360/x
More number of pieces per day, Less days (Indirect proportion)
৮৬৫.
Three workers can do a job in 12 days. Two of the workers work twice as fast as the third. How long would it take one of the faster workers to do the job himself?
  1. ক
    24
  2. খ
    30
  3. গ
    28
  4. ঘ
    32
ব্যাখ্যা
Question: Three workers can do a job in 12 days. Two of the workers work twice as fast as the third. How long would it take one of the faster workers to do the job himself?

Solution: 
তিনজন শ্রমিক একটি কাজ ১২ দিনে করতে পারে। 
তাদের মধ্যে দুজন তৃতীয়জন থেকে দ্বিগুণ গতিতে কাজ করতে পারে। 

ধরি, তৃতীয়জন 2x দিনে কাজটি সম্পন্ন করে। 
বাকি দুজন প্রত্যেকে কাজটি সম্পন্ন করে x দিনে  

তিনজন একসাথে একদিনে কাজ করে = 1/(2x) + 1/x + 1/x
= (1 + 2 + 2)/(2x)
= 5/(2x) অংশ 

তিনজনের সম্পূর্ণ কাজটি করতে সময় লাগে = (2x)/5 দিন 

ATQ,
(2x)/5 = 12 
⇒ 2x = 60
⇒  x = 30 

দ্বিগুণ গতিতে কাজ করা ব্যক্তিরা প্রত্যেকে ৩০ দিনে কাজটি করতে পারে। 
৮৬৬.
If A and B together can complete a work in 18 days, A and C together in 12 days, and B and C together in 9 days, then B alone can do the work in -
  1. ক
    ক) 24 days
  2. খ
    খ) 22 days
  3. গ
    গ) 20 days
  4. ঘ
    ঘ) 18 days
ব্যাখ্যা
Question: If A and B together can complete a work in 18 days, A and C together in 12 days, and B and C together in 9 days, then B alone can do the work in - 

Solution: 
in one day, 
(A + B) do = 1/18
(A + C) do = 1/12
(B + C) do = 1/9

(A + B) + (A + C) + (B + C) = 1/18 + 1/12 + 1/9
2 (A + B + C) = 9/36 = 1/4
A + B + C = 1/8

so, B alone can do in one day = (A + B + C) - (A + C)
= 1/8 - 1/12
= 1/24

hence, it will take B to do the whole work in 24 days
৮৬৭.
P, Q, R are employed to do a work for Tk. 5,750. P and Q together finished 19/23 of work and Q and R together finished 8/23 of work. Wage of Q, in Taka, is -
  1. ক
    ক) 5,000
  2. খ
    খ) 2,000
  3. গ
    গ) 2,500
  4. ঘ
    ঘ) 1,000
ব্যাখ্যা
P + Q = 19/23
Q + R = 8/23

P + Q + Q + R
= P + Q + R + Q
= 19/23 + 8/23
=27/23
(P + Q + R + Q) - (P + Q + R) = 27/23 - 1
Q  =  4/23

Wages of Q in Taka
= 4/23 × 5750
= 1000
৮৬৮.
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?
  1. ক
    ক) 5 days     
  2. খ
    খ) 8 days     
  3. গ
    গ) 10 days     
  4. ঘ
    ঘ) 13 days     
ব্যাখ্যা
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
৮৬৯.
A, B and C can complete a piece of work in 24, 6 and 12 days respectively. Working together, they will complete the same work in:
  1. ক
    ক) 1/24 days
  2. খ
    খ) 7/24 days
  3. গ
    গ) 24/7 days
  4. ঘ
    ঘ) 4 days
  5. ঙ
    ঙ) None of these
ব্যাখ্যা
(A + B + C)'s 1 day's work = (1/24 + 1/6 + 1/12) = 7/24
so, A, B and C together will complete the work in 24/7 days.
৮৭০.
If 8 men or 12 boys can make 60 tables in 6 days, then how many tables will be made by 4 men and 6 boys in 8 days?
  1. ক
    72
  2. খ
    80
  3. গ
    96
  4. ঘ
    104
ব্যাখ্যা

Question: If 8 men or 12 boys can make 60 tables in 6 days, then how many tables will be made by 4 men and 6 boys in 8 days?

Solution:
Here, 8 men = 12 boys
∴ 4 men = (12/2) boys = 6 boys

∴ 4 men and 6 boys = (6 + 6) boys = 12 boys

Now,
12 boys can make 60 tables in 6 days
In 1 day, 12 boys can make (60/6) = 10 tables
∴ In 8 days, 12 boys can make = (10 × 8) tables
= 80 tables

৮৭১.
A job can done by 3 skilled men in 15 days or by 5 boys in 30 days. How many days will they take if they work together?
  1. ক
    16 days
  2. খ
    15 days
  3. গ
    12 days
  4. ঘ
    10 days
ব্যাখ্যা
Question: A job can done by 3 skilled men in 15 days or by 5 boys in 30 days. How many days will they take if they work together?

Solution:
3 men's 1 day work = 1/15 part
and 5 boys 1 day's work = 1/30 part

∴ (3 men's + 5 boy's) 1 day work = (1/15 + 1/30) = (2 +1)/30 = 3/30 = 1/10 part

∴ 3 men and 5 boys will complete the work in 10 days.
৮৭২.
A and B together can complete a job in 12 days. B alone can complete the job in 20 days. In how many days can A alone complete the job?
  1. ক
    15 days
  2. খ
    20 days
  3. গ
    30 days
  4. ঘ
    25 days
ব্যাখ্যা
Question: A and B together can complete a job in 12 days. B alone can complete the job in 20 days. In how many days can A alone complete the job?

Solution:
Let A's time to complete the job alone be x days.
Work rate: 1/x + 1/20 = 1/12
⇒ (20 + x)/20x = 1/12
⇒ 240 + 12x = 20x
⇒ 8x = 240
∴ x = 30
৮৭৩.
In a dairy farm, 50 cows eat 50 bags of husk in 50 days. In how many days one cow will eat one bag of husk?
  1. ক
    1 days
  2. খ
    1/50 days
  3. গ
    50 days
  4. ঘ
    100 days
ব্যাখ্যা
Question: In a dairy farm, 50 cows eat 50 bags of husk in 50 days. In how many days one cow will eat one bag of husk?

Solution:
50 cows can eat 50 bags of husk in 50 days
1 cow can eat 1 bag of husk in (50 × 50)/50 days
= 50 days
৮৭৪.
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? 
  1. ক
    ক) 15
  2. খ
    খ) 20
  3. গ
    গ) 24
  4. ঘ
    ঘ) 30
ব্যাখ্যা
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
৮৭৫.
Rakib can do a job in 15 minutes and his friend takes twice as long to do the same job. If they work together, how long will it take to complete the job?
  1. ক
    8 minutes
  2. খ
    10 minutes
  3. গ
    12 minutes
  4. ঘ
    6 minutes
ব্যাখ্যা

Question: Rakib can do a job in 15 minutes and his friend takes twice as long to do the same job. If they work together, how long will it take to complete the job?

Solution:
রাকিবের সময় = 15 মিনিট, তার বন্ধুর সময় = 30 মিনিট।

1 মিনিটে রাকিব কাজ করে = 1/15 অংশ।
1 মিনিটে তার বন্ধু কাজ করে = 1/30 অংশ।
তাই, 1 মিনিটে তারা একসাথে করে = 1/15 + 1/30
= (2 + 1)/30
= 3/30 = 1/10 অংশ।

1/10 অংশ করতে সময় = 1 মিনিট
∴ সম্পূর্ণ কাজ করতে সময় = 1 ÷ (1/10) = 10 মিনিট।

∴ একসাথে কাজ করলে তারা 10 মিনিটে কাজ শেষ করবে।

৮৭৬.
A can complete a piece of work in 36 days, B in 54 days and C in 72 days. All the three began the work together but A left 8 days before the completion of the work and B 12 days before the completion of work. Only C worked up to the end. In how many days was the work completed?
  1. ক
    18 days
  2. খ
    24 days
  3. গ
    28 days
  4. ঘ
    36 days
ব্যাখ্যা
Question: A can complete a piece of work in 36 days, B in 54 days and C in 72 days. All the three began the work together but A left 8 days before the completion of the work and B 12 days before the completion of work. Only C worked up to the end. In how many days was the work completed?

Solution: 
Let the work be completed in y days. C works for y days
Therefore, A works for (y - 8) days
B works for (y - 12) days.

According to the question,
{(y-8)/36} + {(y - 12)/54} + (y/72) = 1
⇒ 6(y - 8) + 4 (y - 12) + 3y = 216
⇒ 6y - 48 + 4y -  48 + 3y = 216
⇒ 13y = 216 + 96 = 312
⇒ y = 312/13
∴ y = 24
৮৭৭.
If 8 spiders make 8 webs in 8 days, then 1 spider will make 1 web in how many days?
  1. ক
    2
  2. খ
    4
  3. গ
    8
  4. ঘ
    16
ব্যাখ্যা
Question: If 8 spiders make 8 webs in 8 days, then 1 spider will make 1 web in how many days?

Solution:
8 spiders make 8 webs in 8 days
1 spiders make 1 webs in (8 ×8)/8 days
= 8 days
৮৭৮.
A, B and C can do a piece of work in 12, 18 and 36 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?
  1. ক
    18 days
  2. খ
    15 days
  3. গ
    12 days
  4. ঘ
    9 days
ব্যাখ্যা
Question: A, B and C can do a piece of work in 12, 18 and 36 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?

Solution:
A's 2 day's work = (1/12) × 2 = 1/6 part
(A + B + C)' s 1 day's work = (1/12 + 1/18 + 1/36) = 6/36 = 1/6 part

Work done in 3 days = (1/6 + 1/6) = 2/6 = 1/3 part

Now,
1/3 part work is done in 3 days.
∴ Whole work will be done in (3 × 3) = 9 days
৮৭৯.
If 10 men or 20 boys can make 100 footballs in 10 days, then how many footballs will be made by 8 men and 4 boys in 20 days?
  1. ক
    ক) 400
  2. খ
    খ) 300
  3. গ
    গ) 200
  4. ঘ
    ঘ) 100
ব্যাখ্যা
Question: If 10 men or 20 boys can make 100 footballs in 10 days, then how many footballs will be made by 8 men and 4 boys in 20 days?

Solution:
here, 
10 men = 20 boys
8 men = (2 × 8) boys
= 16 boys
if 20 boys can make 100 footballs in 10 days,

8 men and 4 boys or, (16 + 4) = 20 boys can make in 20 days = (100 × 2) = 200 footballs
৮৮০.
C and D together can complete a task in 6 days. C alone can do it in 10 days. How many days would it take D to do this job alone?
  1. ক
    10
  2. খ
    15
  3. গ
    12
  4. ঘ
    18
ব্যাখ্যা

Question: C and D together can complete a task in 6 days. C alone can do it in 10 days. How many days would it take D to do this job alone?

Solution:
C একা ১০ দিনে কাজটি করতে পারে।
∴ C এর একদিনের কাজ = 1/10

C ও D একসাথে ৬ দিনে কাজটি শেষ করে।
∴ তাদের একদিনের সম্মিলিত কাজ = 1/6

তাহলে, D এর একদিনের কাজ = 1/6 - 1/10
= (10 - 6)/60
= 4/60
= 1/15

অর্থাৎ, D এক দিনে কাজ করে 1/15 অংশ।
∴ পুরো কাজ শেষ করতে D এর সময় = 1 ÷ (1/15) = 15 দিন

৮৮১.
If 3 men or 4 women can construct a wall in 44 days, then the number of days that 8 men and 4 women take to construct it is____?
  1. ক
    ক) 12 days
  2. খ
    খ) 14 days
  3. গ
    গ) 16 days
  4. ঘ
    ঘ) 18 days
  5. ঙ
    ঙ) None of these
ব্যাখ্যা
Question: If 3 men or 4 women can construct a wall in 44 days, then the number of days that 8 men and 4 women take to construct it is____?

Solution:
৩ জন পুরুষ = ৪ জন মহিলা
১ জন পুরুষ = ৪/৩ জন মহিলা
৮ জন পুরুষ = (৪ × ৮)/৩ জন মহিলা
= ৩২/৩ জন মহিলা

এখন, 
৮ জন পুরুষ + ৪ জন মহিলা = (৩২/৩) জন মহিলা + ৪ জন মহিলা = ৪৪/৩ জন মহিলা

৪ জন মহিলা কাজ করে ৪৪ দিনে
১ জন মহিলা কাজ করে ৪৪ × ৪ দিনে
৪৪/৩ জন মহিলা কাজ করে (৪৪ × ৪ × ৩)/৪৪  দিনে
= ১২ দিন


৮৮২.
In a certain country, a person born every 8 seconds and a person dies every 12 seconds. Therefore, the birth and death rates amount for a population growth rate of one person every-
  1. ক
    12 seconds
  2. খ
    18 seconds
  3. গ
    20 seconds
  4. ঘ
    24 seconds
  5. ঙ
    None
ব্যাখ্যা
Question: In a certain country, a person born every 8 seconds and a person dies every 12 seconds. Therefore, the birth and death rates amount for a population growth rate of one person every-

Solution:
Let,
x be the number of seconds for the birth of every new person.

We know,
Birth rate - Death rate = Population growth

ATQ,
⇒ 1 person/8 seconds - 1 person/12 seconds = 1 person/x seconds
⇒ (3 - 2) person/24 seconds = 1 person/x seconds
⇒ 1 person/24 seconds = 1 person/x seconds
∴ x = 24 seconds
৮৮৩.
If A and B can complete a piece of work together in 10 days and B alone in 20 days, in how many days can A alone complete the work?
  1. ক
    ক) 30 days
  2. খ
    খ) 40 days
  3. গ
    গ) 60 days
  4. ঘ
    ঘ) 20 days
ব্যাখ্যা
Question: If A and B can complete a piece of work together in 10 days and B alone in 20 days, in how many days can A alone complete the work?

solution: 
A and B complete a work in = 10 days
One day's work of (A + B) = 1/10

B complete the work in = 20 days;
One day's work of B = 1/20

Then, A's one day's work = 1/10 - 1/20
= 1/20

so, A can do the work alone in 20 days.
৮৮৪.
If 12 carpenters, working 6 hours a day, can make 460 chairs in 24 days, how many chairs will 18 carpenters make in 36days, each working 8 hours a day?
  1. ক
    ক) 1260
  2. খ
    খ) 1320
  3. গ
    গ) 920
  4. ঘ
    ঘ) 1380
ব্যাখ্যা

12×6×24×x = 18×8×36×460
x =(18×8×36×460)/(12×6×24)
x =1380
∴ Required number of chairs = 1380

৮৮৫.
12 pumps working 6 hours a day can empty a completely filled reservoir in 15 days. How many such pumps working 9 hours a day will empty the same reservoir in 12 days?
  1. ক
    ক) 16
  2. খ
    খ) 9
  3. গ
    গ) 10
  4. ঘ
    ঘ) 12
ব্যাখ্যা

Apply formula of
M1D1H1/W1= M2D2H2/W2
Let 'P' pumps are required to empty the reservoir.
(12pumps×6hours×15days)/1 reservoir
= (P×9hours×12days)/1 reservoir
P = 10 pumps

৮৮৬.
If a quarter kg of potato costs 60 paise, how many paise will 200 gm cost?
  1. ক
    48 paise
  2. খ
    54 paise
  3. গ
    56 paise
  4. ঘ
    72 paise
  5. ঙ
    None of these
ব্যাখ্যা
Question: If a quarter kg of potato costs 60 paise, how many paise will 200 gm cost?

Solution:
Let the required weight be x kg.
Less weight, Less cost (Direct Proportion)
250 : 200 : : 60 : x
⇒ 250/200 = 60/x
⇒ 250x = (200 × 60)
⇒ x = (200 × 60)/250
∴ x = 48.
৮৮৭.
Rahim can do a piece of work in 20 days. Karim is 25% more efficient than Rahim. The number of days taken by Karim to do the same piece of work is
  1. ক
    ক) 15
  2. খ
    খ) 19
  3. গ
    গ) 18
  4. ঘ
    ঘ) 16
ব্যাখ্যা
Question: Rahim can do a piece of work in 20 days. Karim is 25% more efficient than Rahim. The number of days taken by Karim to do the same piece of work is:

Solution:
রহিম ও করিম কাজটি করার সময়ের অনুপাত 
= 125 : 100
= 5 : 4

ধরি,
রহিম কাজটি করতে x দিন সময় নেয়
5 : 4 :: 20 : x
⇒ x = (4 × 20)/5
⇒ x = 16 দিন

রহিম কাজটি করতে পারে 16 দিনে।
৮৮৮.
A job can be done by 3 skilled workmen in 20 days or by 5 boys in 30 days. How many days will they take if they work together?
  1. ক
    ক) 8 days
  2. খ
    খ) 10 days
  3. গ
    গ) 11 days
  4. ঘ
    ঘ) 12 days
ব্যাখ্যা

3 men's 1 day's work = 1/20;
5 boy's 1 day's work = 1/30.
(3 men + 5 boys)'s 1 day's work = (1/20) + (1/30)
= 5/60
= 1/12.
∴ 3 men and 5 boys will complete the work in 12 days.

৮৮৯.
An employer pays 3 workers X, Y and Z a total of Tk. 36,600 a week. X is paid 125% of the amount Y is paid and 80% of the amount Z is paid. How much does X make a week? 
  1. ক
    ক) 9,000
  2. খ
    খ) 10,800
  3. গ
    গ) 11,700
  4. ঘ
    ঘ) 12,000
ব্যাখ্যা
X = 125% of Y = 125Y/100 = 5Y/4
⇒ Y = 4X/5
Again,
X = 80% of Z = 80Z/100 = 4Z/5
⇒ Z = 5X/4
X : Y : Z  = X : 4X/5 : 5X/4 = 20 : 16 : 25
Sum of ratio = 20 + 16 + 25 = 61
So, X makes a week = (20/61) × 36600
                               = 20 × 600
                                = 12000 tk.
৮৯০.
A man completes 5/8 of a job in 10 days. At this rate, how many more days will it take him to finish the job?
  1. ক
    ক) 4
  2. খ
    খ) 3
  3. গ
    গ) 6
  4. ঘ
    ঘ) 2
ব্যাখ্যা
Work done = 5/8
Balance work =(1 - 5/8) = 3/8
Let the required number of days be x.
Then,(5/8 : 3/8) :: 10 : x
⇒ 5/8 × x = 3/8 × 10
⇒ x = 3/8 × 10 × 8/5
⇒ x = 6.
৮৯১.
On a certain map of India the actual distance of 1450 km between two cities Teknaf and Tetulia is shown as 5 cm. What scale is used to draw the map?
  1. ক
    ক) 1 : 29 × 106
  2. খ
    খ) 3 : 29 × 106
  3. গ
    গ) 5 : 29 × 106
  4. ঘ
    ঘ) 1.5 : 29 × 106
ব্যাখ্যা
cm on the map represents 1450 km
∴ 1 cm on the map represents (1450/5 km = 290 km
Hence,
the scale is 1 cm : 290 km
= 1cm : 29 × 106cm
= 1 : 29 × 106
৮৯২.
A can do a piece of work in 10 days and B can do the same piece of work in 15 days. A and B together complete the same piece of work and get Tk. 600 as the combined wages. A's share of the wages will be- 
  1. ক
    Tk. 300
  2. খ
    Tk. 260
  3. গ
    Tk. 360
  4. ঘ
    Tk. 460
ব্যাখ্যা

Question: A can do a piece of work in 10 days and B can do the same piece of work in 15 days. A and B together complete the same piece of work and get Tk. 600 as the combined wages. A's share of the wages will be-

Solution:
A এর 1 দিনের কাজ = 1/10
B এর 1 দিনের কাজ = 1/15

(A + B) একত্রে 1 দিনের কাজ = (1/10) + (1/15)
= (3 + 2)/30
= 5/30
= 1/6

∴ (A + B) এর 1 দিনের কাজের অনুপাত = (1/10) : (1/15) = 3 : 2

∴ A এর শেয়ার = {(3/5) × 600} = 360 টাকা

৮৯৩.
10 people can do a work in 4 days working 6 hours a day. If 5 new people join the team to complete the work in 2 days, they have to work per day for -
  1. ক
    ক) 8 hours
  2. খ
    খ) 9 hours
  3. গ
    গ) 10 hours
  4. ঘ
    ঘ) 11 hours
ব্যাখ্যা
Question: 10 people can do a work in 4 days working 6 hours a day. If 5 new people join the team to complete the work in 2 days, they have to work per day for -

Solution: 
প্রশ্নের প্রথম অংশ - ১০ জন লোক একটি কাজ দৈনিক ৬ ঘন্টা কাজ করে ৪ দিনে শেষ করে।
এখানে, 
লোক, M1 = 10,
কাজ, W1 = 1,
দিন, D1 = 4,
সময়, H1 = 6.

প্রশ্নের দ্বিতীয় অংশ থেকে - যদি অতিরিক্ত ৫ জন লোক যুক্ত করা হয় তাহলে একই কাজ ২ দিনে শেষ করতে সবার প্রতিদিন কতক্ষণ কাজ করতে হবে?
এখানে,
লোক, M2 = 10 + 5 = 15,
কাজ, W2 = 1,
দিন, D2 = 2,
সময়, H2 = ?

আমরা জানি,
M1 × W2 × D1 × H1 = M2 × W1 × D2 × H2
H2 = (M1 × W2 × D1 × H1)/ (M2 × W1 × D2)
= (10 × 1 × 4 × 6)/(15 × 1 × 2)
= 240/30
= 8 hours
৮৯৪.
A and B undertake to do a piece of work for Tk. 18,750. A can do it in 20 days and B can do it in 30 days. With the help of C, they finished it in 8 days. How much C should be paid for his work in proportion?
  1. ক
    ক) Tk. 6,050
  2. খ
    খ) Tk. 6,250
  3. গ
    গ) Tk. 6,750
  4. ঘ
    ঘ) Tk. 6,850
ব্যাখ্যা
A and B undertake to do a piece of work for Rs. 18,750

A can do it in 20 days and B can do it in 30 days.

Concept used:
Total work = LCM of the time taken by workers alone
Share of the amount is in the ratio of efficiency

Calculation:
LCM of 20 and 30 is 60 i.e total work
So, efficiency of A and B is 60/20 = 3, 60/30 = 2
Efficiency of A + B + C = 60/8 = 7.5
Efficiency of C = 7.5 - 3 - 2 = 2.5

Share of C = (18750/7.5) × 2.5
⇒ 18750/3
⇒ 6250

∴ C should be paid Tk. 6250 for his work in proportion

৮৯৫.
A works twice as fast as B. If B can complete a work in 12 days immediately, the number of days in which A and B can together finish the work- 
  1. ক
    ক) 3 days
  2. খ
    খ) 4 days
  3. গ
    গ) 5 days
  4. ঘ
    ঘ) 6 days
ব্যাখ্যা
Question: A works twice as fast as B. If B can complete a work in 12 days immediately, the number of days in which A and B can together finish the work- 

Solution:
B can complete a work in 12 days
So, A can complete the work in 6 days  

(A + B)'s 1 days work = (1/6 + 1/12) = 1/4 part
So, (A + B) can complete the work in 4 days
৮৯৬.
A rope makes 70 rounds of the circumference of a cylinder whose radius of the base is 14 cm. How many times can it go round a cylinder with radius 20 cm?
  1. ক
    ক) 40
  2. খ
    খ) 49
  3. গ
    গ) 100
  4. ঘ
    ঘ) 120
ব্যাখ্যা

Let,
the required number of rounds be x.
More radius, Less rounds [Indirect proportion]
∴ 20 : 14 :: 70 : x
⇒ (20 × x) = (14 × 70)
⇒ x = (14 × 70)/20
= 49.

৮৯৭.
A and B can complete a task together in 6 days. A can do it alone in 8 days. How many days would it take B to do this task alone?
  1. ক
    20 days
  2. খ
    12 days
  3. গ
    18 days
  4. ঘ
    24 days
ব্যাখ্যা
Question: A and B can complete a task together in 6 days. A can do it alone in 8 days. How many days would it take B to do this task alone?

Solution:
A's 1 day's work 1/8
A and B's 1 day's work 1/6

∴ B's 1 day's work = (1/6) - (1/8) = (4 - 3)/24 = 1/24

1/24 part of task done by B in 1 day
∴ Full task done by B in (24/1) = 24 days

Therefore, B will complete the full task in 24 days.
৮৯৮.
Neha completes 75% of a work in 15 days. Then she calls in Farin, and together they finish the remaining work in 4 days. How long would Farin alone take to complete the whole work?
  1. ক
    80 days
  2. খ
    75 days
  3. গ
    70 days
  4. ঘ
    68 days
ব্যাখ্যা
Question: Neha completes 75% of a work in 15 days. Then she calls in Farin, and together they finish the remaining work in 4 days. How long would Farin alone take to complete the whole work?

Solution:
Here,
Neha does 75% or 3/4 work in 15 days
in one day, she does = (3/4 ÷ 15) = 1/20 part
in 4 days, she does 4/20 = 1/5 part

work remaining = 1/4 - 1/5
= 1/20 part

Farin does 1/20 parts in 4 days
she will complete the work {4 ÷ (1/20)} days
= 80 days
৮৯৯.
A and B can do a work in 12 days. B and C can do it in 15 days. A and C can do it in 20 days. If all of them work together, in how many days can they finish the work?
  1. ক
    ক) 9
  2. খ
    খ) 10
  3. গ
    গ) 12
  4. ঘ
    ঘ) 13
ব্যাখ্যা
(A + B) 1 দিনে করে কাজটির 1/12 অংশ 
(B  + C ) 1 দিনে করে কাজটির 1/15 অংশ 
(C +  A)  1 দিনে করে কাজটির 1/20 অংশ 

(A + B + B + C + C + A) 1 দিনে করে কাজটির  = (1/12) + (1/15) + (1/20) অংশ 
(2A + 2B + 2C) 1 দিনে করে কাজটির  = (5 + 4 + 3)/60 অংশ 
2(A + B + C) 1 দিনে করে কাজটির  = 1/5 অংশ 
(A + B + C) 1 দিনে করে কাজটির = 1/10 অংশ

(A + B + C) 1/10 অংশ কাজ করে 1 দিনে 
(A + B + C) 1 বা সম্পূর্ণ অংশ কাজ করে  = (10 × 1)/1 দিনে 
                                                                = 10 দিনে
৯০০.
Two pipes can fill a tank in 10 and 14 minutes respectively and a waste pipe can empty 4 gallons per minute. If all the pipes working together can fill the tank in 6 minutes, what is the capacity of the tank?
  1. ক
    ক) 120 gallons
  2. খ
    খ) 240 gallons
  3. গ
    গ) 450 gallons
  4. ঘ
    ঘ) 840 gallons
ব্যাখ্যা
Question: Two pipes can fill a tank in 10 and 14 minutes respectively and a waste pipe can empty 4 gallons per minute. If all the pipes working together can fill the tank in 6 minutes, what is the capacity of the tank?

Solution:
Work done by the waste pipe in 1 minute
= (1/6) - {(1/10 + 1/14 )}
= (1/6) - (24/140)
= -1/210 [-ve sign means emptying]

∴ Volume of 1/210 part = 4 gallons
∴ Volume of whole = (4 × 210 ) gallons = 840 gallons