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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 ৬ / ১১ · ৫০১–৬০০ / ১,০৭৬

৫০১.
Alim was hired for a job for 7 days. Each day, he was paid Tk. 10 more than what he was paid for the previous day of work. The total amount he was paid in the first 4 days of work equaled the total amount he was paid in the last 3 days. What was his starting pay?
  1. ক
    ক) TK. 90
  2. খ
    খ) Tk. 138
  3. গ
    গ) Tk. 153
  4. ঘ
    ঘ) Tk. 160
  5. ঙ
    ঙ) None
সঠিক উত্তর: ক
ক) TK. 90
উত্তর
সঠিক উত্তর: ক
ক) TK. 90
ক
ব্যাখ্যা
তার প্রথমদিনের আয় x টাকা।
প্রশ্নমতে, x+x+10+x+20+x+30 = x+40+x+50+x+60
বা,4x + 60 = 3x + 150
বা, x = 90
৫০২.
A fort had arrangements for 150 boys for 45 days. After 10 days, 25 boys left the fort. Then after how much time the food will be consumed completely if the consumption of food remains the same for the remaining boys?
  1. ক
    45 days
  2. খ
    47 days
  3. গ
    42 days
  4. ঘ
    None of these
সঠিক উত্তর: গ
42 days
উত্তর
সঠিক উত্তর: গ
42 days
গ
ব্যাখ্যা
Question: A fort had arrangements for 150 boys for 45 days. After 10 days, 25 boys left the fort. Then after how much time the food will be consumed completely if the consumption of food remains the same for the remaining boys?

Solution:
ATQ,
25 people left the fort after 10 days, but still, the remaining food will be consumed at the same rate.
That means if the 150 boys continue till the end, then the remaining food would last for 150 boys for (45 - 10) = 35 days.
Now
After 10 days the remaining food will be (boys × days) 150 × 35 = 5250 unit
But the 25 boys left the for after 10 days
i.e., 125 boys will consume the 5250 unit food in x days
Now,
x = 5250/125 = 42 days.
That means the remaining food will last for 42 days.
৫০৩.
Three machines, individually, can do a certain job in 4, 5, and 6 hours, respectively. What is the greatest part of the job that can be done in one hour by two of the machines working together at their respective rates?
  1. ক
    11/30
  2. খ
    9/20
  3. গ
    3/5
  4. ঘ
    11/15
  5. ঙ
    5/6
সঠিক উত্তর: খ
9/20
উত্তর
সঠিক উত্তর: খ
9/20
খ
ব্যাখ্যা
Question: Three machines, individually, can do a certain job in 4, 5, and 6 hours, respectively. What is the greatest part of the job that can be done in one hour by two of the machines working together at their respective rates?

Solution:
Machine A in one hour can perform 1/4 of job
Machine B in one hour can perform 1/5 of job
Machine C in one hour can perform 1/6 of job

Machine A and B Together, their rate is 1/4 + 1/5 = 9/20.

Machine A and C Together, their rate is 1/4 + 1/6 = 5/12.

Machine B and C Together, their rate is 1/5 + 1/6 = 11/30.

The question asks for the greatest part of the job that can be done in one hour by two of the machines working together at their respective rates. So we want to find the maximum of the rates we just calculated.

The maximum is 9/20, which is achieved when machines A and B work together.
৫০৪.
A and B together can complete a work in 3 days. They start together but after 2 days, B left the work. If the work is completed after two more days, B alone could do the work in
  1. ক
    ক) 6 days
  2. খ
    খ) 7 days
  3. গ
    গ) 9 days
  4. ঘ
    ঘ) 5 days
সঠিক উত্তর: ক
ক) 6 days
উত্তর
সঠিক উত্তর: ক
ক) 6 days
ক
ব্যাখ্যা
(A+B)'s one day's work = 1/3 part
(A+B) works 2 days together = 2/3 part
Remaining work = 1−2/3 = 1/3 part
1/3 part of work is completed by A in two days
Hence, one day's work of A = 1/6
Then, one day's work of B = 1/3−1/6 = 1/6
So, B alone can complete the whole work in 6 days.
৫০৫.
A tank has two leakages. The first leakage alone can empty the tank in 9 minutes and the second alone would have done it in 6 minutes. If water leaks out at a constant rate, how long does it take both the leakage together to empty the tank?
  1. ক
    ক) 3.6 min
  2. খ
    খ) 3.1 min
  3. গ
    গ) 3.5 min
  4. ঘ
    ঘ) 4.0 min
সঠিক উত্তর: ক
ক) 3.6 min
উত্তর
সঠিক উত্তর: ক
ক) 3.6 min
ক
ব্যাখ্যা

In the 1st case,
1/9 part of the tank is emptied in 1 min
In the 2nd case,
1/6 part of the tank is emptied in 1 min
So, together in 1 min, they empties = (1/9 + 1/6) = 5/18 part

Now, 5/18 part is emptied in 1 minute
∴ 1 or, total part is emptied in = 18/5 = 3.6 minutes

৫০৬.
4 men can do one work in 8 days.  How many men can do the half work in 2 days?
  1. ক
    ক) 6 men
  2. খ
    খ) 8 men
  3. গ
    গ) 10 men
  4. ঘ
    ঘ) 5 men
সঠিক উত্তর: খ
খ) 8 men
উত্তর
সঠিক উত্তর: খ
খ) 8 men
খ
ব্যাখ্যা
Question: 4 men can do one work in 8 days.  How many men can do the half work in 2 days?

Solution: 
From the first part of the question, we get - 4 men can do one work in 8 days.
here, 
men, M1 = 4,
work, W1 = 1,
day, D1 = 8.

From the second part of the question, we get - How many men can do the half work in 2 days?
here, 
men, M2 =?
work, W2 = 1/2,
day, D = 2.

we know,
M1 × W2 × D1 = M2 × W1 × D2
M2 = (M1 × W2 × D1) / (W1 × D2)
= {4 × (1/2) × 8}/ (1 × 2)
= 16/2
= 8 men
৫০৭.
Machine A produces bolts at a uniform rate of 120 every 40 seconds and Machine B produce bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts?
  1. ক
    ক) 22
  2. খ
    খ) 25
  3. গ
    গ) 28
  4. ঘ
    ঘ) 32
  5. ঙ
    ঙ) 56
সঠিক উত্তর: খ
খ) 25
উত্তর
সঠিক উত্তর: খ
খ) 25
খ
ব্যাখ্যা
Question: Machine A produces bolts at a uniform rate of 120 every 40 seconds and Machine B produce bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts?

Solution:
Given
Machine a produces bolts at a uniform rate of 120 every 40 second
∴ machine a produce 120/40 = 3 bolts per second

Machine b produces bolts at a uniform rate of 100 every 20 second
∴ machine b produces 100/20 = 5 bolts per second

Both machine produce 3 + 5 = 8 bolts running simultaneously per second.

Machine produces 8 bolts for 1 second
∴ For 200 bolts the time taken is 200/8 seconds
= 25 seconds

The machine takes 25 seconds to produce 200 bolts.
৫০৮.
A wheel that has 6 cogs is meshed with a larger wheel of 12 cogs. If the smaller wheel has made 22 revolutions, then find the number of revolutions made by the larger wheel.
  1. ক
    11
  2. খ
    13
  3. গ
    15
  4. ঘ
    17
  5. ঙ
    None of these
সঠিক উত্তর: ক
11
উত্তর
সঠিক উত্তর: ক
11
ক
ব্যাখ্যা
Question: A wheel that has 6 cogs is meshed with a larger wheel of 12 cogs. If the smaller wheel has made 22 revolutions, then find the number of revolutions made by the larger wheel.

Solution:
As number of cogs increase, the revolutions made decrease. Hence, this is a problem related to indirect proportion.
Let the number of wheels be x.
More cogs (↑),Less revolutions (↓)

12 : 6 : : 22 : x
⇒ 12 × x = 6 × 22
⇒ x = (6 × 22)/12
∴ x = 11
৫০৯.
Mr. Power attends to, on the average, 12 TV service calls per day. Mr. Fixit attends to, on the average, 16 service calls per day. Mr. Power's average charge is 3/2 as much as that Mr. Fixit, who earns Tk. 7200 per month. The monthly earnings of Mr. Power, in Taka, is-
  1. ক
    3600
  2. খ
    7200
  3. গ
    8100
  4. ঘ
    None
সঠিক উত্তর: গ
8100
উত্তর
সঠিক উত্তর: গ
8100
গ
ব্যাখ্যা
Question: Mr. Power attends to, on the average, 12 TV service calls per day. Mr. Fixit attends to, on the average, 16 service calls per day. Mr. Power's average charge is 3/2 as much as that Mr. Fixit, who earns Tk. 7200 per month. The monthly earnings of Mr. Power, in Taka, is-

Solution:
Mr. Power's average charge is 3/2 as much as that Mr. Fixit

Let,
Mr. Power got 3x Taka per call
Mr. Fixit got 2x Taka per call

ATQ,
For Mr. Fixit
16 × 30 × 2x = 7200
⇒ x = 7200/960
∴ x = 7.5

∴ Mr. Power erned = 12 × 30 × 3x = 360 × 3 × 7.5 = 8100 taka
৫১০.
No. of books bound by a child is half the number of books a man can bind. 20 men and 16 children take 10 days to bind all the books. If there are 8 men and 12 children working on the assignment, how many days will they take to bind all the books?
  1. ক
    ক) 20 days
  2. খ
    খ) 24 days
  3. গ
    গ) 25 days
  4. ঘ
    ঘ) 40 days
সঠিক উত্তর: ক
ক) 20 days
উত্তর
সঠিক উত্তর: ক
ক) 20 days
ক
ব্যাখ্যা

We know,
M1D1T1W2 = M2D2T2W1 [Men = M; Days = D; Time/Hours = T; Work = W]

1 child binds half the number of books bound by 1 adult
∴ 1 adult = 2 children
Take work done = 1
∴ (20 adults + 16 children) x 10 days x 1 = (8 adults + 12 children) x ? days x 1
∴ 56 children x 10 days x 1 = 28 children x ? days x 1 [Convert either all adults to children or all children to adults.]
∴ ? = 20 days = they will need these many days.

৫১১.
A man's regular pay is Taka 30 per hour up to 40 hours. Overtime is twice the payment for regular time. If he was paid Taka 1680, how many hours of overtime did he work?
  1. ক
    ক) 8
  2. খ
    খ) 16
  3. গ
    গ) 28
  4. ঘ
    ঘ) 48
সঠিক উত্তর: ক
ক) 8
উত্তর
সঠিক উত্তর: ক
ক) 8
ক
ব্যাখ্যা

40 ঘন্টার জন্য regular pay = (30 × 40) = 1200 টাকা।
Overtime এর টাকার পরিমান = (1680 - 1200) টাকা = 480 টাকা
যেহেতু, Overtime এর প্রতিদিনের টাকার পরিমান Regular Payment এর দ্বিগুন,
সেহেতু মোট overtime কাজ করার সময় = (480 ÷ (30×2) ঘন্টা = 8 ঘন্টা

৫১২.
A group of men can do a job in 25 days. But 10 men did not turn up for the job and the remaining men did the job in 30 days. The original number of men in group was
  1. ক
    ক) 40
  2. খ
    খ) 60
  3. গ
    গ) 50
  4. ঘ
    ঘ) 70
সঠিক উত্তর: খ
খ) 60
উত্তর
সঠিক উত্তর: খ
খ) 60
খ
ব্যাখ্যা
The group of men can do a job = 25 Days

Formula used:

W = E × T     Where, W = Work, E = Efficiency, and T = Time

Calculation:
Let the initially the number of men be X

According to the question
⇒ 25 × X = 30 × (X - 10)
⇒ 25X = 30X - 300
⇒ 30X - 25X = 300
⇒ 5X = 300
⇒ X = 300/5 
⇒ X= 60

The initially the number of men = 60

∴ The required result will be 60.
৫১৩.
P and Q together can do a job in 10 days. Q and R together can do it in 12 days. P and R together can do it in 15 days. If all of them work together, in how many days can they finish the work?
  1. ক
    3 days
  2. খ
    5 days
  3. গ
    9 days
  4. ঘ
    6 days
  5. ঙ
    None of these
সঠিক উত্তর: ঙ
None of these
উত্তর
সঠিক উত্তর: ঙ
None of these
ঙ
ব্যাখ্যা
Question: P and Q together can do a job in 10 days. Q and R together can do it in 12 days. P and R together can do it in 15 days. If all of them work together, in how many days can they finish the work?

Solution:
(P + Q) 1 দিনে করে কাজটির 1/10 অংশ
(Q + R ) 1 দিনে করে কাজটির 1/12 অংশ
(P + R) 1 দিনে করে কাজটির 1/15 অংশ

∴ (P + Q + Q + R + P + R) 1 দিনে করে কাজটির = (1/10) + (1/12) + (1/15) অংশ
(2P + 2Q + 2R) 1 দিনে করে কাজটির = (6 + 5 + 4)/60 অংশ
2(P + Q + R) 1 দিনে করে কাজটির = 1/4 অংশ
(P + Q + R) 1 দিনে করে কাজটির = 1/8 অংশ

(P + Q + R) 1/8 অংশ কাজ করে 1 দিনে
∴ (A + B + C) 1 বা সম্পূর্ণ অংশ কাজ করে = (8 × 1)/1 = 8 দিনে
৫১৪.
৩০ × ১৬ ফুটের একটি মেঝে মেরামত করতে ২৪৯৬ টাকা ব্যয় হল। প্রতি বর্গফুটে ব্যয় কত টাকা?
  1. ক
    ক) ৪.২০ টাকা
  2. খ
    খ) ৬.২০ টাকা
  3. গ
    গ) ৫.২০ টাকা
  4. ঘ
    ঘ) ৫.৫০ টাকা
সঠিক উত্তর: গ
গ) ৫.২০ টাকা
উত্তর
সঠিক উত্তর: গ
গ) ৫.২০ টাকা
গ
ব্যাখ্যা
প্রশ্ন: ৩০ × ১৬ ফুটের একটি মেঝে মেরামত করতে ২৪৯৬ টাকা ব্যয় হল। প্রতি বর্গফুটে ব্যয় কত টাকা?

সমাধান: 
মেঝের ক্ষেত্রফল = ৩০ × ১৬ বর্গফুট 
= ৪৮০ বর্গফুট 

প্রতি বর্গফুটে ব্যয় (২৪৯৬/৪৮০) টাকা 
= ৫.২০ টাকা 
৫১৫.
A can do a piece of work in 30 days. B is 20% more efficient than A. The number of days taken by B to do the same piece of work is:
  1. ক
    ক) 22 days
  2. খ
    খ) 25 days
  3. গ
    গ) 20 days
  4. ঘ
    ঘ) 26 days
সঠিক উত্তর: খ
খ) 25 days
উত্তর
সঠিক উত্তর: খ
খ) 25 days
খ
ব্যাখ্যা
Question: A can do a piece of work in 30 days. B is 20% more efficient than A. The number of days taken by B to do the same piece of work is:

Solution:
A ও B কাজটি করার সময়ের অনুপাত 
= 120 : 100
= 6 : 5

ধরি,
A কাজটি করতে x দিন সময় নেয়
6 : 5 :: 30 : x
⇒ 6/5 = 30/x
⇒ x = (5 × 30)/6
⇒ x = 25 দিন

B কাজটি করতে পারে 25 দিনে।
৫১৬.
3 pumps, working 4 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work, to empty the tank in one day?
  1. ক
    ক) 4 hours
  2. খ
    খ) 5 hours
  3. গ
    গ) 6 hours
  4. ঘ
    ঘ) 6.5 hours
সঠিক উত্তর: গ
গ) 6 hours
উত্তর
সঠিক উত্তর: গ
গ) 6 hours
গ
ব্যাখ্যা

Given that,
3 pumps, working 4 hours a day, can empty a tank in 2 days.
Therefore, it means that:
3 pumps take a total of 8 hours to empty the tank.
Hence, 1 pump will take 8 × 3 = 24 hours

As the number of pumps decreases, the time required increases.
So, if 4 pumps work, the time required decreases.
∴ 24/4 = 6 hours needed to empty the tank in 1 day.

৫১৭.
If Machine A can produce 500 units in 3 hours and Machine B can produce 500 units in 6 hours, in how many hours can Machines A and B, working together at these constant rates, produce 500 units?
  1. ক
    2 hours
  2. খ
    3 hours
  3. গ
    4.5 hours
  4. ঘ
    5 hours
সঠিক উত্তর: ক
2 hours
উত্তর
সঠিক উত্তর: ক
2 hours
ক
ব্যাখ্যা

Question: If Machine A can produce 500 units in 3 hours and Machine B can produce 500 units in 6 hours, in how many hours can Machines A and B, working together at these constant rates, produce 500 units?

Solution:
মেশিন A, 3 ঘণ্টায় তৈরি করে ৫০০টি ইউনিট।
∴ 1 ঘণ্টায় মেশিন A তৈরি করে = 500/3 ইউনিট

মেশিন B, 6 ঘণ্টায় তৈরি করে ৫০০টি ইউনিট।
∴ 1 ঘণ্টায় মেশিন B তৈরি করে = 500/6 ইউনিট

তারা একত্রে ১ ঘণ্টায় তৈরি করে = (500/3) + (500/6) ইউনিট
= (1000 + 500)/6
= 1500/6 = 250 ইউনিট

তারা একত্রে 250টি ইউনিট তৈরি করে 1 ঘণ্টায়।
∴ তারা একত্রে 500টি ইউনিট তৈরি করতে সময় নেবে = 500/250 = 2 ঘণ্টা

৫১৮.
P, Q and R can do a job in 20, 30 and 60 days respectively. In how many days can P do the job if he is assisted by Q and R every third day?
  1. ক
    11 days
  2. খ
    15 days
  3. গ
    17 days
  4. ঘ
    16 days
সঠিক উত্তর: খ
15 days
উত্তর
সঠিক উত্তর: খ
15 days
খ
ব্যাখ্যা
Question: P, Q and R can do a job in 20, 30 and 60 days respectively. In how many days can P do the job if he is assisted by Q and R every third day?

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

Now, 1/5 jobs is done in 3 days
Whole job  will be done in (3 × 5) = 15 days.
৫১৯.
If two jeans and three shirts cost Tk. 4000 and three jeans and two shirts costs Tk. 3500, than how much does a shirt?
  1. ক
    Tk. 500
  2. খ
    Tk. 1000
  3. গ
    Tk. 1200
  4. ঘ
    Tk. 1500
সঠিক উত্তর: খ
Tk. 1000
উত্তর
সঠিক উত্তর: খ
Tk. 1000
খ
ব্যাখ্যা
Question: If two jeans and three shirts cost Tk. 4000 and three jeans and two shirts costs Tk. 3500, than how much does a shirt?

Solution: 
মনেকরি 
১টি  jeans এর দাম = x টাকা 
১টি shirt এর দাম = y টাকা 

এখানে 
2x + 3y = 4000.......................(1)
3x + 2y = 3500.........................(2)

(1) × 3 - (2) × 2 ⇒
6x + 9y - 6x - 4y = 12000 - 7000
⇒ 5y = 5000
∴ y = 1000

১টি shirt এর দাম = 1000 টাকা
৫২০.
Working alone, Rasel can complete a certain kind of job in 8 hours. Rasel and Shima working together at their respective rates, can complete one of these jobs in 6 hours. In how many hours can Shima working alone, complete one of these jobs?
  1. ক
    24 hours
  2. খ
    20 hours
  3. গ
    22 hours
  4. ঘ
    18 hours
সঠিক উত্তর: ক
24 hours
উত্তর
সঠিক উত্তর: ক
24 hours
ক
ব্যাখ্যা

Question: Working alone, Rasel can complete a certain kind of job in 8 hours. Rasel and Shima working together at their respective rates, can complete one of these jobs in 6 hours. In how many hours can Shima working alone, complete one of these jobs?

Solution:
Working together Rasel and Shima can complete the job in 6 hours
which means
(1/Rasel) + (1/Shima) = (1/6)
⇒ (1/8) + (1/Shima) = (1/6)      [Rasel can complete the job in 8 hours]
⇒ (1/Shima) = (1/6) - (1/8)
⇒ (1/Shima) = (4 - 3)/24
∴ (1/Shima) = 1/24

so, Shima working alone can complete the job in 24 hours

৫২১.
A can type 10 pages in 5 minutes. B can type 5 pages in 10 minutes. Working together, how many pages can they type in 30 minutes?
  1. ক
    ক) 78
  2. খ
    খ) 73
  3. গ
    গ) 75
  4. ঘ
    ঘ) 76
সঠিক উত্তর: গ
গ) 75
উত্তর
সঠিক উত্তর: গ
গ) 75
গ
ব্যাখ্যা
A can type 10 pages in 5 minutes.
B can type 5 pages in 10 minutes.

Concept Used:
Total Work = Time × Efficiency

Calculation:
A can type 10 pages in 5 minutes.
A can type 1 minute = 10/5 = 2

B can type 5 pages in 10 minutes.
B can type 1 minute = 5/10 = 1/2

Both (A + B)'s type in 1 minutes = 2 + 1/2 = 5/2

The number of pages in 30 minutes = 30 × 5/2 = 75
∴ The number of pages in 30 minutes by both (A + B) is 75.
৫২২.
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
    14 days
  2. খ
    12 days
  3. গ
    10 days
  4. ঘ
    16 days
সঠিক উত্তর: গ
10 days
উত্তর
সঠিক উত্তর: গ
10 days
গ
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.
৫২৩.
Karim does a job 60 days quicker than Asad. If Karim works three times faster than Asad, how long will it take him to finish the job by himself?
  1. ক
    15 days
  2. খ
    20 days
  3. গ
    30 days
  4. ঘ
    45 days
সঠিক উত্তর: গ
30 days
উত্তর
সঠিক উত্তর: গ
30 days
গ
ব্যাখ্যা

Question: Karim does a job 60 days quicker than Asad. If Karim works three times faster than Asad, how long will it take him to finish the job by himself?

Solution:
ধরি,
কাজটি আসাদ করে = x দিনে 
করিম করে = (x - 60) দিনে 

প্রশ্নমতে,
x - 60 = 3x
⇒ 3x - x = 60
⇒ 2x = 60
⇒ x = 60/2
⇒ x = 30

৫২৪.
If 6 men can complete a task in 10 days, how long will it take for 15 men to complete the same task?
  1. ক
    4 days
  2. খ
    5 days
  3. গ
    6 days
  4. ঘ
    8 days
সঠিক উত্তর: ক
4 days
উত্তর
সঠিক উত্তর: ক
4 days
ক
ব্যাখ্যা
Question: If 6 men can complete a task in 10 days, how long will it take for 15 men to complete the same task?

Solution:
Formula used: M1 × T1 = M2 × T2
Where M1 and M2 is men and T1 and T2 is time

∴ 6 × 10 = 15 × t
⇒ t = 60/15
∴ t = 4 days
৫২৫.
A wall of 100 meters can be built by 7 men or 10 women in 10 days. How many days will 14 men and 20 women take to build a wall of 600 metres ?
  1. ক
    ক) 12
  2. খ
    খ) 15
  3. গ
    গ) 18
  4. ঘ
    ঘ) 21
সঠিক উত্তর: খ
খ) 15
উত্তর
সঠিক উত্তর: খ
খ) 15
খ
ব্যাখ্যা
Let the required number of days be x
7 men = 10 women
(14 men and 20 women) = (20 + 20) women = 40 women
More length, More days (Direct proportion)
More women, Less days (Indirect proportion)
Length 100 : 600
                          ⟩ :: 10 : x
 Women 40 : 10
100 × 40 × x = 600 × 10 × 10
x = 15
৫২৬.
If 5 identical machines, operating at a constant speed, can manufacture 200 pencils in one minute, how many pencils will 12 such machines produce in 10 minutes at the same rate?
  1. ক
    2800 pencils
  2. খ
    4000 pencils
  3. গ
    4800 pencils
  4. ঘ
    1800 pencils
  5. ঙ
    3800 pencils
সঠিক উত্তর: গ
4800 pencils
উত্তর
সঠিক উত্তর: গ
4800 pencils
গ
ব্যাখ্যা

Question: If 5 identical machines, operating at a constant speed, can manufacture 200 pencils in one minute, how many pencils will 12 such machines produce in 10 minutes at the same rate?

Solution:
Given,
In 1 minute, 5 machines can produce 200 pencils
In 1 minute, 1 machine can produce 200/5 = 40 pencils

So, in 10 minutes, 12 machines can produce = (40 × 12 × 10) pencils
= 4800 pencils

৫২৭.
3 pumps, working 4 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work, to empty the tank in one day?
  1. ক
    ক) 5 hours
  2. খ
    খ) 8 hours
  3. গ
    গ) 7 hours
  4. ঘ
    ঘ) 6 hours
সঠিক উত্তর: ঘ
ঘ) 6 hours
উত্তর
সঠিক উত্তর: ঘ
ঘ) 6 hours
ঘ
ব্যাখ্যা
Question: 3 pumps, working 4 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work, to empty the tank in one day?

Solution:
Given that,
3 pumps, working 4 hours a day, can empty a tank in 2 days.
Therefore, it means that:
3 pumps take a total of 8 hours to empty the tank.
Hence, 1 pump will take 8 × 3 = 24 hours

As the number of pumps decreases, the time required increases.
So, if 4 pumps work, the time required decreases.
∴ 24/4 = 6 hours needed to empty the tank in 1 day.

বিকল্প সমাধান:
প্রতিটি পাম্প ১ দিনে চলে ৪ ঘণ্টা 
∴ প্রতিটি পাম্প ২ দিনে চলে ৮ ঘণ্টা 
∴ ৩টি পাম্প ২ দিনে চলে (৮ × ৩) ঘণ্টা 
= ২৪ ঘণ্টা 

তাহলে,
৪টি পাম্পকেও মোট ২৪ ঘণ্টা কাজ করতে হবে।
∴ ১টি পাম্পকে চলতে হবে (২৪/৪) ঘণ্টা 
= ৬ ঘণ্টা 
৫২৮.
If two man or four women or six boys can finish a job in 72 days, then find out the number of days in which that work can be done by one man, two women, and three boys?
  1. ক
    72 days
  2. খ
    24 days
  3. গ
    36 days
  4. ঘ
    48 days
সঠিক উত্তর: ঘ
48 days
উত্তর
সঠিক উত্তর: ঘ
48 days
ঘ
ব্যাখ্যা
Question: If two man or four women or six boys can finish a job in 72 days, then find out the number of days in which that work can be done by one man, two women, and three boys?

Solution:
Given,
2 man = 4 women = 6 boys
1 man = 2 women = 3 boys
∴ 1 man + 2 women + 3 boys = (1 + 1 + 1) men
= 3 men

Here,
2 men can do in = 72 days
∴ 1 men can do in = (72 × 2) days
= 144 days
∴ 3 men can do in = (144 ÷ 3) days
= 48 days
৫২৯.
A can do a work in 10 days and B can do the same work in 15 days. If they start working together but stop working after four days, find the fraction of the work that is left.
  1. ক
    1/3
  2. খ
    4/3
  3. গ
    2/7
  4. ঘ
    1/2
সঠিক উত্তর: ক
1/3
উত্তর
সঠিক উত্তর: ক
1/3
ক
ব্যাখ্যা
Question: A can do a work in 10 days and B can do the same work in 15 days. If they start working together but stop working after four days, find the fraction of the work that is left.

Solution:
A's one day work = 1/10
B's one day work = 1/15

(A + B)'s one day work = 1/10 + 1/15 = (3 + 2)/30 = 5/30 = 1/6
A and B's four day work = (1/6) × 4 = 2/3

Therefore,the remaining work = 1 - 2/3 = (3 - 2)/3 = 1/3
৫৩০.
In a factory, 20 workers can make 20 toys in 10 days working 10 hours per day. Then, in how many days can 25 workers make 30 toys working 20 hours per day? 
  1. ক
    8 days
  2. খ
    12 days
  3. গ
    16 days
  4. ঘ
    6 days
  5. ঙ
    None
সঠিক উত্তর: ঘ
6 days
উত্তর
সঠিক উত্তর: ঘ
6 days
ঘ
ব্যাখ্যা

Question: In a factory, 20 workers can make 20 toys in 10 days working 10 hours per day. Then, in how many days can 25 workers make 30 toys working 20 hours per day?

Solution:
Given:

Number of workers initially = 20
Number of days initially = 10
Hours per day initially = 10
Number of toys initially = 20
Number of workers later = 25
Hours per day later = 20
Number of toys later = 30
Number of days later = ?

∴ (Number of workers initially × Number of days initially × Hours per day initially × Number of toys later)
= (Number of workers later × Number of days later × Hours per day later × Number of toys initially)
⇒ 20 × 10 × 10 × 30 = 25 × (Number of days later) × 20 × 20
⇒ 20 × 10 = 200 → 200 × 10 = 2,000 → 2,000 × 30 = 60,000
⇒ 25 × 20 × 20 = 10,000 × (Number of days later)
⇒ 60,000 = 10,000 × (Number of days later)

∴ Number of days later = 60,000 ÷ 10,000 = 6 days

৫৩১.
If 10 ships require 10 tanks of oil in 10 days. How long is 1 tank of oil enough for a ship?
  1. ক
    7 days 
  2. খ
    8 days 
  3. গ
    9 days 
  4. ঘ
    10 days 
সঠিক উত্তর: ঘ
10 days 
উত্তর
সঠিক উত্তর: ঘ
10 days 
ঘ
ব্যাখ্যা
Question: If 10 ships require 10 tanks of oil in 10 days. How long is 1 tank of oil enough for a ship?

Solution: 
10 ships require 10 tanks of oil in 10 days
10 ships require 1 tanks of oil in 10/10 days = 1 days
1 ships require 1 tanks of oil in 10 × 1 days
= 10 days 
৫৩২.
Mr. Rahim can complete a task in 12 days, while his daughter can complete the same task in 18 days. How many days will it take for them to finish the work if they work together?
  1. ক
    8.2 days
  2. খ
    7.2 days
  3. গ
    10.2 days
  4. ঘ
    21.2 days
  5. ঙ
    None
সঠিক উত্তর: খ
7.2 days
উত্তর
সঠিক উত্তর: খ
7.2 days
খ
ব্যাখ্যা

Question: Mr. Rahim can complete a task in 12 days, while his daughter can complete the same task in 18 days. How many days will it take for them to finish the work if they work together?

Solution:
Given,
Mr. Rahim can do the work in 12 days
∴ in 1 day he can do = 1/12 part

His daughter can do the work in 18 days
∴ in 1 day she can do = 1/18 part

∴ in 1 day, working together they can do = (1/12 + 1/18) part
= (3 + 2)/36 part
= 5/36 part

Hence, together they complete 5/36 of the work in 1 day
∴ They can complete the whole work in = 36/5 days
= 7.2 days

∴ If they work together, the work will be completed in 7.2 days.

৫৩৩.
If 3 men or 9 boys can finish a task in 36 days, how long will it take 9 men and 9 boys to finish the same task?
  1. ক
    6 days
  2. খ
    7 days
  3. গ
    9 days
  4. ঘ
    12 days
  5. ঙ
    18 days
সঠিক উত্তর: গ
9 days
উত্তর
সঠিক উত্তর: গ
9 days
গ
ব্যাখ্যা

Question: If 3 men or 9 boys can finish a task in 36 days, how long will it take 9 men and 9 boys to finish the same task?

Solution:
এখানে, 3 জন পুরুষ = 9 জন বালক
∴ 1 জন পুরুষ = 9/3 = 3 জন বালক

এখন, 9 জন পুরুষ + 9 জন বালক
= (9 × 3) জন বালক + 9 জন বালক
= 27 জন বালক + 9 জন বালক
= 36 জন বালক

প্রশ্নমতে,
9 জন বালক কাজটি করে 36 দিনে
∴ 1 জন বালক কাজটি করে (36 × 9) দিনে
∴ 36 জন বালক কাজটি করে (36 × 9)/36 দিনে
= 9 দিনে

∴ কাজটি 9 দিনে শেষ হবে।

৫৩৪.
A tyre has two punctures. The first puncture alone would have made the tyre 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
    ক) 3(1/5) min
  2. খ
    খ) 3(2/5) min
  3. গ
    গ) 3(3/5) min
  4. ঘ
    ঘ) 3(4/5) min
সঠিক উত্তর: গ
গ) 3(3/5) min
উত্তর
সঠিক উত্তর: গ
গ) 3(3/5) min
গ
ব্যাখ্যা

1-minute work done by both the punctures = (1/9) + (1/6)
= (4 + 6)/36
= 10/36
= 5/18

So both punctures will make the type flat in = (18/5) min
= 3(3/5) min

৫৩৫.
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
    20
  2. খ
    22
  3. গ
    24
  4. ঘ
    30
সঠিক উত্তর: গ
24
উত্তর
সঠিক উত্তর: গ
24
গ
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
or, 6(y - 8) + 4(y - 12) + 3y = 216
or, 6y - 48 + 4y -  48 + 3y = 216
or, 13y = 216 + 96 = 312
or, y = 312/13
or, y = 24
৫৩৬.
Adults can do a job twice as fast as children. If x adults can complete a job in d days, how many children can do the same job in d + 2 days?
  1. ক
    ক) 2dx/(d + 2)
  2. খ
    খ) (d + 2)x/d
  3. গ
    গ) dx/2(d + 2)
  4. ঘ
    ঘ) (2x + d)/(d - x)
সঠিক উত্তর: ক
ক) 2dx/(d + 2)
উত্তর
সঠিক উত্তর: ক
ক) 2dx/(d + 2)
ক
ব্যাখ্যা

x adults = 2x children
to complete in d days 2x children is required.
∴ to complete in (d + 2) days (2x × d)/(d + 2) children is required.

৫৩৭.
76 men complete a job in 33 days. Due to some reason, some men did not join the work and therefore it was completed in 44 days. The number of men who did not report for the work is- 
  1. ক
    19
  2. খ
    20
  3. গ
    21
  4. ঘ
    23
সঠিক উত্তর: ক
19
উত্তর
সঠিক উত্তর: ক
19
ক
ব্যাখ্যা
Question: 76 men complete a job in 33 days. Due to some reason, some men did not join the work and therefore it was completed in 44 days. The number of men who did not report for the work is- 

Solution: 
Let L be the rate of work per men. 

the total work = 76L × 33 

Let,  with x men absent, the total work is = (76 - x)L × 44

 76L × 33 = (76 - x)L × 44
⇒ 76 × 33 = 76 × 44 - 44L 
⇒ 44L = 76 × 44 - 76 × 33 = 76 × 11 
⇒ L = 19
৫৩৮.
Working alone, Protik takes complete April to build a pavement. His friend Mahmud is 25% faster than him at the same work. Working alone, how many days will Mahmud take to build the same pavement?
  1. ক
    17 days
  2. খ
    20 days
  3. গ
    15 days
  4. ঘ
    24 days
  5. ঙ
    21 days
সঠিক উত্তর: ঘ
24 days
উত্তর
সঠিক উত্তর: ঘ
24 days
ঘ
ব্যাখ্যা

April has 30 days. So Protik takes 30 days to build the pavement.
Mahmud is 25% faster than Protik

25% = 25/100 = .25
This means, if Protik is 1, then Mahmud is (1 + 0.25) = 1.25
Protik takes 30 days to do the work.
Mahmud will take = 30/1.25 = 24 days to get the work done.

৫৩৯.
40% of the employees in a factory are workers. All the remaining employees are executives. The annual income of each worker is 390 taka. The annual income of each executive is 420 taka. What is the average annual income of all the employees in the factory together?
  1. ক
    408
  2. খ
    360
  3. গ
    270
  4. ঘ
    410
  5. ঙ
    412
সঠিক উত্তর: ক
408
উত্তর
সঠিক উত্তর: ক
408
ক
ব্যাখ্যা
Let X be the number of employees.
Now, 40% of X = 0.4X
Hence, the number of workers is 2X/5

All the remaining employees are executives,
so the number of executives = X - 2X/5 = 3X/5
The total annual income of all workers together = 2X/5 × 390 = 156X

The total income of all the executives together = 3X/5 × 420 = 252X

Hence, the total income of the employees = 156X + 252X = 408X
The average income of all the employees together = 408
৫৪০.
3 men or 5 women can do a work in 12 days. How long will 6 men and 5 women take to finish the work?
  1. ক
    3 days
  2. খ
    4 days
  3. গ
    6 days
  4. ঘ
    8 days
সঠিক উত্তর: খ
4 days
উত্তর
সঠিক উত্তর: খ
4 days
খ
ব্যাখ্যা
Question: 3 men or 5 women can do a work in 12 days. How long will 6 men and 5 women take to finish the work?

Solution:
According to the question,
3 men = 5 women
As they complete the same work in the same time
6 men + 5 women
= 6 men + 3 men
= 9 men

If, 3 men do work in 12 days
1 man does a work in 12 × 3
∴ 9 men do work in (12 × 3)/9
  = 4 days
৫৪১.
10 men can complete a work in 7 days. But 10 women need 14 days to complete the same work. How many days will 5 men and 10 women need to complete the work?
  1. ক
    5
  2. খ
    7
  3. গ
    4
  4. ঘ
    6
  5. ঙ
    9
সঠিক উত্তর: খ
7
উত্তর
সঠিক উত্তর: খ
7
খ
ব্যাখ্যা

Work is done by 10 men in 1 day = 1/7
Work is done by 1 man in 1 day = (1/7)/10 = 1/70

Work is done by 10 women in 1 day = 1/14
Work is done by 1 woman in 1 day = 1/140

Work done by 5 men and 10 women in 1 day = 5 × (1/70) + 10 × (1/140)
= 5/70 + 10/140 = 1/7
∴ 5 men and 10 women can complete the work in 7 days.

৫৪২.
76 ladies complete a job in 33 days. Due to some reason, some ladies did not join the work and therefore it was completed in 44 days. The number of ladies who did not report for the work is- 
  1. ক
    18
  2. খ
    19
  3. গ
    20
  4. ঘ
    21
সঠিক উত্তর: খ
19
উত্তর
সঠিক উত্তর: খ
19
খ
ব্যাখ্যা
Question: 76 ladies complete a job in 33 days. Due to some reason, some ladies did not join the work and therefore it was completed in 44 days. The number of ladies who did not report for the work is- 

Solution: 
Let L be the rate of work per lady. 

the total work = 76L × 33 

Let,  with x ladies absent, the total work is = (76 - x)L × 44

 76L × 33 = (76 - x)L × 44
⇒ 76 × 33 = 76 × 44 - 44L 
⇒ 44L = 76 × 44 - 76 × 33 = 76 × 11 
⇒ L = 19
৫৪৩.
M takes 10 days more than N to complete a task. If they work together, they finish it in 12 days. How long does N take alone?
  1. ক
    16 days
  2. খ
    20 days
  3. গ
    28 days
  4. ঘ
    22 days
সঠিক উত্তর: খ
20 days
উত্তর
সঠিক উত্তর: খ
20 days
খ
ব্যাখ্যা
Question: M takes 10 days more than N to complete a task. If they work together, they finish it in 12 days. How long does N take alone?

Solution:
Let N's time = x days.
Then, M's time = x + 10 days.

ATQ,
⇒ (1/x) + {1/(x + 10)} = 1/12
⇒ (x + 10 + x)/x(x + 10) = 1/12
⇒ 12(2x + 10) = x(x + 10)
⇒ 24x + 120 = x2 + 10x
⇒ x2 - 14x - 120 = 0
⇒ x2 - 20x + 6x - 120 = 0
⇒ x(x - 20) + 6(x - 20) = 0
⇒ (x - 20)(x + 6) = 0
Now,
x - 20 = 0
∴ x = 20
And
x + 6 = 0
∴ x = - 6  ;[since time cannot be negative]

∴ N takes 20 days to complete the task alone.
৫৪৪.
If 15 men can built a wall 50 meters long in 5 days, what length of a similar wall can be built by 30 men in 2 days? 
  1. ক
    ক) 20 meters   
  2. খ
    খ) 25 meters   
  3. গ
    গ) 30 meters   
  4. ঘ
    ঘ) 40 meters   
সঠিক উত্তর: ঘ
ঘ) 40 meters   
উত্তর
সঠিক উত্তর: ঘ
ঘ) 40 meters   
ঘ
ব্যাখ্যা
Question: If 15 men can built a wall 50 meters long in 5 days, what length of a similar wall can be built by 30 men in 2 days? 

Solution:
In 5 days 15 men can build = 50 meters 
∴ In 1 day 1 man can build = 50/(5 × 15) meters
∴ In 2 days 30 men can build = (50 × 30 × 2)/(5 × 15) meters = 40 meters
৫৪৫.
A can do a piece of work in 8 days and B can do the same piece of work in 12 days. A and B together complete the same piece of work and get Tk. 200 as the combined wages. B’s share of the wages will be -
  1. ক
    ক) Tk. 75
  2. খ
    খ) Tk. 80
  3. গ
    গ) Tk. 85
  4. ঘ
    ঘ) Tk. 90
সঠিক উত্তর: খ
খ) Tk. 80
উত্তর
সঠিক উত্তর: খ
খ) Tk. 80
খ
ব্যাখ্যা

A's share : B's share = Ratio of their 1 day's work
= 1/8 : 1/12
= 3 : 2.
∴ B's share = Tk. 200 × (2/5)
= Tk. 80.

৫৪৬.
Sakib can paint 10 desks in 20 minutes. Tamim can paint the same number of desks in only 10 minutes. Sakib paints the first 5 desks, then Tamim paints for 3 minutes, and finally Sakib finishes the remaining desks. How long will it take them to paint 15 desks in total?
  1. ক
    15 minutes
  2. খ
    21 minutes
  3. গ
    27 minutes
  4. ঘ
    34 minutes
সঠিক উত্তর: গ
27 minutes
উত্তর
সঠিক উত্তর: গ
27 minutes
গ
ব্যাখ্যা

Question: Sakib can paint 10 desks in 20 minutes. Tamim can paint the same number of desks in only 10 minutes. Sakib paints the first 5 desks, then Tamim paints for 3 minutes, and finally Sakib finishes the remaining desks. How long will it take them to paint 15 desks in total?

Solution:
তামিম 10 মিনিটে রঙ করে 10টি ডেস্ক।
∴ 3 মিনিটে সে রঙ করতে পারে = (10 × 3)/10
= 3টি ডেস্ক

সাকিব প্রথমে রঙ করে 5টি ডেস্ক।
∴ এপর্যন্ত মোট সম্পন্ন কাজ = 5 (সাকিব) + 3 (তামিম) = 8টি ডেস্ক
বাকি থাকে = 15 - 8 = 7টি ডেস্ক

সাকিব 10টি ডেস্ক রঙ করতে সময় নেয় 20 মিনিট।
∴ সাকিবের প্রথম 5টি ডেস্ক রঙ করতে সময় লেগেছে = (20 × 5)/10 = 10 মিনিট
∴ সাকিবের অবশিষ্ট 7টি ডেস্ক রঙ করতে সময় লাগবে = (20 × 7)/10 = 14 মিনিট

∴ মোট সময় = 10 (সাকিব) + 3 (তামিম) + 14 (সাকিব) = 27 মিনিট।

৫৪৭.
If a photocopier makes 2 copies in 1/3 second, at the same rate how many copies does it make in 4 minutes?
  1. ক
    1460 copies
  2. খ
    1440 copies
  3. গ
    1240 copies
  4. ঘ
    1640 copies
সঠিক উত্তর: খ
1440 copies
উত্তর
সঠিক উত্তর: খ
1440 copies
খ
ব্যাখ্যা
Question: If a photocopier makes 2 copies in 1/3 second, at the same rate how many copies does it make in 4 minutes?

Solution:
Here, 4 minutes = (4 × 60) seconds = 240 seconds

In 1/3 second he can make 2 copies 
In 1 second he can make (2 × 3) copies = 6 copies
∴ In 240 seconds he can make (6 × 240) copies
= 1440 copies
৫৪৮.
10 cats caught 10 rats in 10 seconds. How many cats are required to catch 100 rats in 100 seconds?
  1. ক
    10
  2. খ
    20
  3. গ
    50
  4. ঘ
    100
সঠিক উত্তর: ক
10
উত্তর
সঠিক উত্তর: ক
10
ক
ব্যাখ্যা
Question: 10 cats caught 10 rats in 10 seconds. How many cats are required to catch 100 rats in 100 seconds?

Solution:
১০ টি ইঁদুর ১০ সেকেন্ডে ধরতে পারে ১০টি বিড়াল
১০ টি ইঁদুর ১ সেকেন্ডে ধরতে পারে ১০ × ১০টি বিড়াল
১ টি ইঁদুর ১ সেকেন্ডে ধরতে পারে (১০ × ১০)/১০ = ১০টি বিড়াল
১০০ টি ইঁদুর ১ সেকেন্ডে ধরতে পারে ১০ × ১০০টি বিড়াল
১০০ টি ইঁদুর ১০০ সেকেন্ডে ধরতে পারে (১০ × ১০০)/১০০ টি বিড়াল
= ১০টি বিড়াল
৫৪৯.
If 20 men can build a wall 56 meters long in 6 days, what length of a similar wall can be built by 35 men in 3 days?
  1. ক
    39 meters
  2. খ
    42 meters
  3. গ
    46 meters
  4. ঘ
    49 meters
সঠিক উত্তর: ঘ
49 meters
উত্তর
সঠিক উত্তর: ঘ
49 meters
ঘ
ব্যাখ্যা
Question: If 20 men can build a wall 56 meters long in 6 days, what length of a similar wall can be built by 35 men in 3 days?

Solution:
In 6 days 20 men can build 56 meters
∴ In 1 day 1 man can build 56/(6 × 20) meters.
∴ In  3 days 35 men can build (56 × 35 × 3)/(6 × 20) meters.
 = 49 meters.
৫৫০.
A canteen requires 105 kgs of wheat for a week. How many kgs of wheat will it require for 58 days?
  1. ক
    840 kgs
  2. খ
    950 kgs
  3. গ
    620 kgs
  4. ঘ
    870 kgs
সঠিক উত্তর: ঘ
870 kgs
উত্তর
সঠিক উত্তর: ঘ
870 kgs
ঘ
ব্যাখ্যা
Question: A canteen requires 105 kgs of wheat for a week. How many kgs of wheat will it require for 58 days? 

Solution:
For 7 days it is required = 105 kgs
∴ For 1 days it is required = (105/7) kgs
∴ For 58 days it is required = (105 × 58)/7 = 870 kgs

∴ 870 kilograms of wheat will be required for 58 days.
৫৫১.
Two friends P & Q take a project and start working on it together. P alone can finish the project in 20 days while Q takes 5 days more than P to finish it. After working together on the project for some days, P has to leave. Q now takes 10 days to finish the remaining project alone. After how many days of working together did P leave?
  1. ক
    ক) 5 days
  2. খ
    খ) 6(2/3) days
  3. গ
    গ) 8(1/5) days
  4. ঘ
    ঘ) 10(2/5) days
সঠিক উত্তর: খ
খ) 6(2/3) days
উত্তর
সঠিক উত্তর: খ
খ) 6(2/3) days
খ
ব্যাখ্যা

In a days P does 1/20 work; And in 1 day Q does 1/25 work
As seen above, Q works alone for 10 days.
In 1 days Q completes (1/25) × 10 = 2/5 work
Remaining work = 1 - (2/5) = 3/5 = Done by P and Q together
Total Work done = Total days x Work done by all in 1 day

Let P and Q work together for total K days.

∴ 3/5 = K × (1/20 + 1/25)
K = 20/3 = 6(2/3) days = Days when P and Q worked together
Thus P leaves after 6(2/3) days.

৫৫২.
Sumon and Noyon can complete a task together in 6 days. Noyon alone can do it in 9 days. How many days will Sumon take to complete the task alone?
  1. ক
    12 days
  2. খ
    15 days
  3. গ
    18 days
  4. ঘ
    21 days
সঠিক উত্তর: গ
18 days
উত্তর
সঠিক উত্তর: গ
18 days
গ
ব্যাখ্যা

Question: Sumon and Noyon can complete a task together in 6 days. Noyon alone can do it in 9 days. How many days will Sumon take to complete the task alone?

Solution:
ধরা যাক, সুমন একা কাজটি করতে x দিন সময় নেয়।
∴ সুমনের 1 দিনের কাজ = 1/x অংশ
∴ নয়নের 1 দিনের কাজ = 1/9 অংশ (যেহেতু সে একা ৯ দিনে করে)

তারা দুজনে মিলে 6 দিনে কাজটি শেষ করে।
∴ তাদের একত্রে 1 দিনের কাজ = 1/6 অংশ

শর্তমতে,
(1/x) + (1/9) = 1/6
⇒ 1/x = (1/6) - (1/9)
⇒ 1/x = (3 - 2)/18
⇒ 1/x = 1/18
⇒ x = 18

∴ সুমন একা কাজটি করতে 18 দিন সময় নেবে।

৫৫৩.
45 men can do a piece of work in 18 days. How many men would be required to do the same work in 30 days?
  1. ক
    ক) 27
  2. খ
    খ) 26
  3. গ
    গ) 25
  4. ঘ
    ঘ) 28
সঠিক উত্তর: ক
ক) 27
উত্তর
সঠিক উত্তর: ক
ক) 27
ক
ব্যাখ্যা
Question: 45 men can do a piece of work in 18 days. How many men would be required to do the same work in 30 days?

Solution:
18 দিনে কাজটি সম্পন্ন করে 45 জন লোক
∴ 1  দিনে কাজটি সম্পন্ন করে (45 × 18) জন লোক
∴ 30  দিনে কাজটি সম্পন্ন করে (45 × 18)/30 জন লোক
= 27 জন লোক
৫৫৪.
A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?
  1. ক
    35 hours
  2. খ
    30 hours
  3. গ
    25 hours
  4. ঘ
    15 hours
সঠিক উত্তর: ক
35 hours
উত্তর
সঠিক উত্তর: ক
35 hours
ক
ব্যাখ্যা
Question: A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

Solution:
Suppose pipe A alone takes x hours to fill the tank.
Then, pipes B and C will take x/2 and x/4 hours respectively to fill the tank.

ATQ,
∴1/x + 2/x + 4/x = 1/5
⇒ 7/x = 1/5
∴ x = 35 hours
৫৫৫.
The speed of three cars is in the ratio of 2 : 3 : 4. The ratio of the times taken by these cars to travel the same distance is-
  1. ক
    5 : 4 : 2
  2. খ
    7 : 5 : 2
  3. গ
    6 : 4 : 3
  4. ঘ
    5 : 7 : 2
সঠিক উত্তর: গ
6 : 4 : 3
উত্তর
সঠিক উত্তর: গ
6 : 4 : 3
গ
ব্যাখ্যা
Question: The speed of three cars is in the ratio of 2 : 3 : 4. The ratio of the times taken by these cars to travel the same distance is-

Solution:
Speed is always inversely proportional to time,
⇒ s ∝ (1/t)

∴ Ratio of times takes = 1/2 : 1/3 : 1/4
= (1/2) × 12 : (1/3) × 12 : (1/4) × 12
= 6 : 4 : 3
৫৫৬.
In a camp of soldiers there was a stock of food for 190 days for 4000 soldiers. After 30 days 800 soldiers left the barrack. For how many days shall the leftover food last for the remaining soldiers?
  1. ক
    100 days
  2. খ
    210 days
  3. গ
    220 days
  4. ঘ
    200 days
সঠিক উত্তর: ঘ
200 days
উত্তর
সঠিক উত্তর: ঘ
200 days
ঘ
ব্যাখ্যা
Question: In a camp of soldiers there was a stock of food for 190 days for 4000 soldiers. After 30 days 800 soldiers left the barrack. For how many days shall the leftover food last for the remaining soldiers?

Solution:
Let, the remaining food last for x days
4000 soldiers had provision for 160 days
3200 soldiers had provision for x days

ATQ,
3200 : 4000 = 160 : x
⇒ 3200x = 4000 × 160
⇒ x = (4000 × 160)/3200
∴ x = 200

The remaining food last for 200 days.
৫৫৭.
A barrack has enough food for 200 soldiers or 400 sailors. If 120 sailors have taken the food, how many soldiers will be able to eat with the remaining food?
  1. ক
    120 soldiers
  2. খ
    140 soldiers
  3. গ
    160 soldiers
  4. ঘ
    180 soldiers
সঠিক উত্তর: খ
140 soldiers
উত্তর
সঠিক উত্তর: খ
140 soldiers
খ
ব্যাখ্যা

Question: A barrack has enough food for 200 soldiers or 400 sailors. If 120 sailors have taken the food, how many soldiers will be able to eat with the remaining food?

Solution:
এখানে,
400 জন নাবিকের খাবার = 200 জন সৈন্যের খাবার।

মোট নাবিক যাদের জন্য খাবার ছিল = 400 জন।
খাবার গ্রহণ করেছে = 120 জন।
অবশিষ্ট খাবার = (400 - 120) জন নাবিকের খাবার
= 280 জন নাবিকের খাবার।

এখন,
400 জন নাবিকের খাবার = 200 জন সৈন্যের খাবার।
∴ 1 জন নাবিকের খাবার = (200/400) জন সৈন্যের খাবার।
∴ 280 জন নাবিকের খাবার = (200 × 280)/400
= (1/2) × 280
= 140 জন সৈন্যের খাবার।

সুতরাং, অবশিষ্ট খাবার দিয়ে 140 জন সৈন্যকে দেওয়া যাবে।

৫৫৮.
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
    ক) 11 days
  2. খ
    খ) 15 days
  3. গ
    গ) 12 days
  4. ঘ
    ঘ) 13 days
সঠিক উত্তর: ঘ
ঘ) 13 days
উত্তর
সঠিক উত্তর: ঘ
ঘ) 13 days
ঘ
ব্যাখ্যা
প্রশ্ন: 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?

সমাধান: 
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 
৫৫৯.
A can complete a work in 20 days, while B can complete the same work in 30 days. If both A and B work together, in how many days will they complete the entire work?
  1. ক
    8 days
  2. খ
    10 days
  3. গ
    12 days
  4. ঘ
    15 days
সঠিক উত্তর: গ
12 days
উত্তর
সঠিক উত্তর: গ
12 days
গ
ব্যাখ্যা

Question: A can complete a work in 20 days, while B can complete the same work in 30 days. If both A and B work together, in how many days will they complete the entire work?

Solution:
A's 1 day work = 1/20
B's 1 day work = 1/30
Together 1 day work = 1/20 + 1/30
= (3 + 2)/60 = 5/60 = 1/12

∴ Total time = 1/Combined work rate
= 1/(1/12) days
= 12 days

৫৬০.
If A can do 1/4 of a work in 3 days and B can do 1/9 of the same work in 4 days, how much will A get if both work together and paid Tk 800 in all?
  1. ক
    650 Tk
  2. খ
    600 Tk
  3. গ
    550 Tk
  4. ঘ
    500 Tk
সঠিক উত্তর: খ
600 Tk
উত্তর
সঠিক উত্তর: খ
600 Tk
খ
ব্যাখ্যা

Question: If A can do 1/4 of a work in 3 days and B can do 1/9 of the same work in 4 days, how much will A get if both work together and paid Tk 800 in all?

Solution:
Whole work is done by A in (3 × 4) = 12 days
∴ A's 1 day's work = 1/12 part
Whole work is done by B in (4 × 9) = 36 days
∴ B's 1 day's work = 1/36 part

A's 1 day's work : B's 1 day's work
= A's wages : B's wages
= 1/12 : 1/36
= 3 : 1

∴ A's share = (800 × 3/4) Tk
= 600 Tk

৫৬১.
By working 5 hours a day, A can complete work in 8 days, and working 6 hours a day, B can complete the same work in 10 days. Working 8 hours a day, they can jointly complete the work in -
  1. ক
    2 days
  2. খ
    3 days
  3. গ
    4 days
  4. ঘ
    5 days
সঠিক উত্তর: খ
3 days
উত্তর
সঠিক উত্তর: খ
3 days
খ
ব্যাখ্যা
Question: By working 5 hours a day, A can complete work in 8 days, and working 6 hours a day, B can complete the same work in 10 days. Working 8 hours a day, they can jointly complete the work in -

Solution:
Working 5 hours a day,
A can complete the work in 8 days.
A can complete the work in = 5 × 8 = 40 hours.

Working 6 hours a day, B can complete the work in 10 days.
B can complete the work in = 6 × 10 = 60 hours.
(A + B)'s 1 hour's work,
= 1/40 + 1/60
= (3 + 2)/120
= 5/120
= 1/24

Hence, A and B can complete the work in 24 hours.
Working 8 hours a day
They require = 24/8 = 3 days to complete the work.
৫৬২.
If 6 workers can build a wall in 10 days, how many workers would be needed to build the same wall in 5 days?
  1. ক
    12 workers
  2. খ
    15 workers
  3. গ
    18 workers
  4. ঘ
    20 workers
সঠিক উত্তর: ক
12 workers
উত্তর
সঠিক উত্তর: ক
12 workers
ক
ব্যাখ্যা
Question: If 6 workers can build a wall in 10 days, how many workers would be needed to build the same wall in 5 days?

Solution: 
6 workers can build a wall in 10 days,
workers would be needed to build the same wall in 5 days = (6 × 10)/5 
= 12 workers
৫৬৩.
A worker earns Tk. 500 on the first day and spends Tk. 200 on the second day, earns Tk. 500 on the third day and again spends Tk. 200 on the fourth day, and so on. On which day would he have had Tk. 2000?
  1. ক
    9th day
  2. খ
    11th day
  3. গ
    13th day
  4. ঘ
    15th day
সঠিক উত্তর: খ
11th day
উত্তর
সঠিক উত্তর: খ
11th day
খ
ব্যাখ্যা

Question: A worker earns Tk. 500 on the first day and spends Tk. 200 on the second day, earns Tk. 500 on the third day and again spends Tk. 200 on the fourth day, and so on. On which day would he have had Tk. 2000?

Solution:
1ম দিনে আয় = 500 টাকা
2য় দিনে ব্যয় = 200 টাকা
∴ প্রতি 2 দিনে প্রকৃত জমা হয় = (500 - 200) = 300 টাকা

শেষ দিনে আয় করার পর কাঙ্ক্ষিত লক্ষ্যে পৌঁছাবে, তাই শেষ দিনের আয় বাদ দিলে থাকে = (2000 - 500) = 1500 টাকা

এখন, 300 টাকা জমা হয় = 2 দিনে
∴ 1500 টাকা জমা হয় = (2 × 1500) / 300 = 10 দিনে

অর্থাৎ, 10 দিনের শেষে তার হাতে জমা থাকবে 1500 টাকা।
পরের দিন অর্থাৎ 11-তম দিনে তিনি আবার 500 টাকা আয় করবেন।
∴ মোট সঞ্চয় হবে = 1500 + 500 = 2000 টাকা

∴ 11-তম দিনে তার কাছে মোট 2000 টাকা থাকবে।

৫৬৪.
Four men and three women can do a job in 6 days. When five men and six women work on the same job, the work gets completed in 4 days. How long will a woman take to do the job, if she works alone on it?
  1. ক
    30 days
  2. খ
    40 days
  3. গ
    54 days
  4. ঘ
    60 days
সঠিক উত্তর: গ
54 days
উত্তর
সঠিক উত্তর: গ
54 days
গ
ব্যাখ্যা
Question: Four men and three women can do a job in 6 days. When five men and six women work on the same job, the work gets completed in 4 days. How long will a woman take to do the job, if she works alone on it?

Solution:
ধরি, ১ জন পুরুষ ১ দিনে করে x অংশ এবং ১ জন মহিলা ১ দিনে করে y অংশ 

4x + 3y = 1/6
⇒ 5 (4x + 3y) = 5/6
⇒ 20x + 15y = 5/6

5x + 6y = 1/4
⇒ 4(5x + 6y) = 1
⇒ 20x + 24y = 1

20x + 24y - 20x - 15y = 1 - (5/6)
⇒ 9y = 1/6
⇒ y = 1/54

অর্থাৎ, ১ জন মহিলা ১/৫৪ অংশ সম্পন্ন করে ১ দিনে 
১ অংশ সম্পন্ন করে ৫৪ দিনে। 
৫৬৫.
If Arif works alone he will take 20 more hours to complete a task than if he works with Babu to complete the task. If Babu work alone, he will take 5 more hours to complete the task than if he works with Arif to complete the task. What is the ratio of the time taken by Arif to than taken by Babu if each of them works alone to complete the task?
  1. ক
    5 : 2
  2. খ
    4 : 3
  3. গ
    3 : 1
  4. ঘ
    2 : 1
  5. ঙ
    None
সঠিক উত্তর: ঘ
2 : 1
উত্তর
সঠিক উত্তর: ঘ
2 : 1
ঘ
ব্যাখ্যা
Question: If Arif works alone he will take 20 more hours to complete a task than if he works with Babu to complete the task. If Babu work alone, he will take 5 more hours to complete the task than if he works with Arif to complete the task. What is the ratio of the time taken by Arif to than taken by Babu if each of them works alone to complete the task?

Solution:
ধরি
আরিফ ও বাবু কাজটি করে x ঘণ্টায়

আরিফ একা কাজটি করে (x + 20) ঘণ্টায়
বাবু একা কাজটি করে (x + 5) ঘণ্টায়

প্রশ্নমতে,
{1/(x + 20)} + {1/(x + 5)} = 1/x
⇒ 1/(x + 20) = (1/x) - {1/(x + 5)}
⇒ 1/(x + 20) = (x + 5 - x)/(x2 + 5x)
⇒ 1/(x + 20) = 5/(x2 + 5x)
⇒ x2 + 5x = 5x + 100
⇒ x2 = 100 
∴ x = 10

আরিফ একা কাজটি করে (10 + 20) ঘণ্টা = 30 ঘণ্টায়
বাবু একা কাজটি করে (10 + 5) ঘণ্টা = 15 ঘণ্টায়

আরিফ ও বাবুর কাজের সময়ের অনুপাত = 30 : 15
= 2 : 1
৫৬৬.
To complete a work, P takes 25% more time than Q. If together they take 20 days to complete the work, how much time will Q take to do it?
  1. ক
    26 days
  2. খ
    30 days
  3. গ
    36 days
  4. ঘ
    25 days
  5. ঙ
    40 days
সঠিক উত্তর: গ
36 days
উত্তর
সঠিক উত্তর: গ
36 days
গ
ব্যাখ্যা

Question: To complete a work, P takes 25% more time than Q. If together they take 20 days to complete the work, how much time will Q take to do it?

Solution:
Let,
Q takes x days to complete the work

Then P will take 25% more time
i.e. 125% of x days
i.e. (5/4)x days

So, the one day’s work of P and Q together will be
(1/x) + {1/(5x/4)} = 1/20

⇒ (1/x) + (4/5x) = 1/20
⇒ (9/5x) = 1/20
⇒ x = 36

∴ Q takes 36 days to complete the work.

৫৬৭.
A can finish a task in 36 days, B in 54 days, and C in 72 days. All three start working together, but A leaves 8 days before the work is completed, and B leaves 12 days before completion. C works alone until the task is finished. How many days does it take to complete the work?
  1. ক
    34 days
  2. খ
    20 days
  3. গ
    14 days
  4. ঘ
    24 days
  5. ঙ
    21 days
সঠিক উত্তর: ঘ
24 days
উত্তর
সঠিক উত্তর: ঘ
24 days
ঘ
ব্যাখ্যা

Question: A can finish a task in 36 days, B in 54 days, and C in 72 days. All three start working together, but A leaves 8 days before the work is completed, and B leaves 12 days before completion. C works alone until the task is finished. How many days does it take to complete the work?

Solution: 
Let the work be completed in y days.
C works for y days

Therefore, A works for (y - 8) days
and, 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 = 1
⇒ 6(y - 8) + 4 (y - 12) + 3y = 216
⇒ 6y - 48 + 4y -  48 + 3y = 216
⇒ 13y = 216 + 96 = 312
⇒ y = 312/13
∴ y = 24

৫৬৮.
James hired some workers to build his house in 20 days. But on the day of starting of the construction, 12 men did not come. Rest of the people then built the entire house in 32 days. How many workers had he initially hired?
  1. ক
    28
  2. খ
    32
  3. গ
    36
  4. ঘ
    40
সঠিক উত্তর: খ
32
উত্তর
সঠিক উত্তর: খ
32
খ
ব্যাখ্যা
Question: James hired some workers to build his house in 20 days. But on the day of starting of the construction, 12 men did not come. Rest of the people then built the entire house in 32 days. How many workers had he initially hired?

Solution:
Initially hired workers = x
Total work = x × 20 (man-days)
After 12 workers didn't come, Remaining workers = x - 12
Total work = (x - 12) × 32 (man-days)

ATQ,
x × 20 = (x - 12) × 32
⇒ 20x = 32x - 384
⇒ 12x = 384
∴ x = 32
৫৬৯.
A certain machine produces 10,000 units of product P per hour. Working continuously at this constant rate, this machine will produce how many units of product P in 11 days?
  1. ক
    2,640,000
  2. খ
    168,000
  3. গ
    264,000
  4. ঘ
    1,680,000
সঠিক উত্তর: ক
2,640,000
উত্তর
সঠিক উত্তর: ক
2,640,000
ক
ব্যাখ্যা
7 days = 11 × 24 hours = 264 hours
In 1 hour, the machine produces 10,000 units
In 168 hours, the machine will produce = 10,000 × 264  units = 2640000 units
৫৭০.
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 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
    22
  2. খ
    14
  3. গ
    25
  4. ঘ
    24
সঠিক উত্তর: ঘ
24
উত্তর
সঠিক উত্তর: ঘ
24
ঘ
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 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. So C works for y days
Therefore, A works for (y - 8) days B works for (y - 12) days.

ATQ,
{(y - 8)/36} + {(y - 12)/54} + (y/72) = 1
⇒ {6(y - 8) + 4(y - 12) + 3y}/216 = 1
⇒ 6y - 48 + 4y - 48 + 3y = 216
⇒ 13y = 216 + 96
⇒ 13y = 312
⇒ y = 312/13
∴ y = 24

৫৭১.
A and B together can complete a work in 3 days. They start together but after 2 days, B left the work. After that the work is completed in next two days, B alone could do the work in-
  1. ক
    4 days
  2. খ
    6 days
  3. গ
    8 days
  4. ঘ
    10 days
সঠিক উত্তর: খ
6 days
উত্তর
সঠিক উত্তর: খ
6 days
খ
ব্যাখ্যা
Question: A and B together can complete a work in 3 days. They start together but after 2 days, B left the work. After that the work is completed in next two days, B alone could do the work in-

Solution:
(A + B)'s one day's work = 1/3 part
(A + B) works 2 days together = 2/3 part.

∴ Remaining work = 1 - (2/3) = 1/3 part

1/3 part of work is completed by A in two days.
Hence, one day's work of A = 1/6

Then, one day's work of B = (1/3) - (1/6)
= (2 - 1)/6
= 1/6

So, B alone can complete the whole work in 6 days.
৫৭২.
If 5 boys write 5 pages in 5 minutes, then 3 boys will write 3 pages in -
  1. ক
    ক) 1 minute
  2. খ
    খ) 3 minutes
  3. গ
    গ) 5 minutes
  4. ঘ
    ঘ) 9 minutes
সঠিক উত্তর: গ
গ) 5 minutes
উত্তর
সঠিক উত্তর: গ
গ) 5 minutes
গ
ব্যাখ্যা
If 5 boys write 5 pages in 5 minutes

Formula used:
Men × Time/Work = Men × Time/Work

Calculation:
According to the question,
⇒ (5 × 5)/5 = (3 × Time)/3
⇒ 5 minutes

∴ 3 boys will write 3 pages in 5 minutes.
৫৭৩.
Machine P prints x units in 20 minutes and machine Q prints 2x units in 10 minutes. In how many minutes will P and Q, working together, print 100x units?
  1. ক
    5 hours and 50 minutes
  2. খ
    6 hours and 40 minutes
  3. গ
    5 hours 
  4. ঘ
    6 hours and 10 minutes
সঠিক উত্তর: খ
6 hours and 40 minutes
উত্তর
সঠিক উত্তর: খ
6 hours and 40 minutes
খ
ব্যাখ্যা

Question: Machine P prints x units in 20 minutes and machine Q prints 2x units in 10 minutes. In how many minutes will P and Q, working together, print 100x units?

Solution:
Machine P প্রতি মিনিটে প্রিন্ট করে = x/20 ইউনিট
Machine Q প্রতি মিনিটে প্রিন্ট করে = 2x/10 = x/5 ইউনিট

∴ P এবং Q একত্রে প্রতি মিনিটে প্রিন্ট করে = (x/20) + (x/5)
= (x + 4x)/20
= 5x/20
= x/4 ইউনিট

এখন,
x/4 ইউনিট প্রিন্ট করতে সময় লাগে 1 মিনিট।
∴ 100x ইউনিট প্রিন্ট করতে সময় লাগবে = (100x × 4)/x = 400 মিনিট
= 6 ঘণ্টা 40 মিনিট

৫৭৪.
P and S are working on an assignment. P takes 6 hours to type 32 pages on a computer, while S takes 5 hours to type 40 pages. How much time will they take, working together on two different computers to type an assignment of 160 pages?
  1. ক
    ক) 6 hours
  2. খ
    খ) 12 hours
  3. গ
    গ) 10 hours
  4. ঘ
    ঘ) 14 hours
সঠিক উত্তর: খ
খ) 12 hours
উত্তর
সঠিক উত্তর: খ
খ) 12 hours
খ
ব্যাখ্যা
Number of pages typed by P in 1 hour = 32/6=16/3
Number of pages typed by S in 1 hour = 40/5 = 8
Number of pages typed by both in 1 hour= (16/3) + 8
                                                                    = (16 + 24)/3
                                                                    =40/3

∴Time taken by both to type 160 pages
 = (160×3/40) hours
 = 12 hours
৫৭৫.
A watch which gains 5 seconds in 3 minutes was set right at 7 a.m. In the afternoon of the same day, when the watch indicated quarter past 4 o'clock, the true time is?
  1. ক
    4 : 00
  2. খ
    4 : 04
  3. গ
    4 : 09
  4. ঘ
    4 : 10
  5. ঙ
    4 : 41`
সঠিক উত্তর: ক
4 : 00
উত্তর
সঠিক উত্তর: ক
4 : 00
ক
ব্যাখ্যা
The watch shows 3 minutes and 05 seconds when actual watch shows just 3 minutes.
3 minutes = 05 seconds
60 minutes = 05×60/3 seconds
                   = 100 seconds
Therefore, 9 × 60 minutes or 9 hours  = (100× 9)  seconds
                                                             = 900/60 minutes
                                                             = 15 minutes
Its shows the time to be 4:15 PM by the faulty watch
7 am to 4 pm is a time gap of 9 hours.
The watch would have gained 15 minutes in total
Thus the true time by a normal running watch would be 4:00PM.
৫৭৬.
If three boys can paint a wall in 2 days, what part of the job can be completed by two boys in 1 day?
  1. ক
    ক) 1/3
  2. খ
    খ) 2/3
  3. গ
    গ) 3/4
  4. ঘ
    ঘ) 4/2
সঠিক উত্তর: ক
ক) 1/3
উত্তর
সঠিক উত্তর: ক
ক) 1/3
ক
ব্যাখ্যা
question: If three boys can paint a wall in 2 days, what part of the job can be completed by two boys in 1 day?

solution: 
given,
M1 = 3 boys (member)
D1 = 2 days ( days to complete the work)
W1 = 1 ( ratio of work )
M2 = 2 boys
D2 = 1 day
W2 = ?

we know,
M1 × D1 × W2 = M2 × D2 × W1
W2 = (M2 × D2 × W1)/ (M1 × D1)
= (2 × 1 × 1)/(3 × 2)
= 1/3
৫৭৭.
If 8 persons working 10 hours a day earn Tk. 12000 per week, then 6 persons working 14 hours a day will earn per week ?
  1. ক
    12600 Tk.
  2. খ
    11500 Tk.
  3. গ
    13600 Tk.
  4. ঘ
    12900 Tk.
সঠিক উত্তর: ক
12600 Tk.
উত্তর
সঠিক উত্তর: ক
12600 Tk.
ক
ব্যাখ্যা
Question: If 8 persons working 10 hours a day earn Tk. 12000 per week, then 6 persons working 14 hours a day will earn per week ?

Solution: 
8 persons working 10 hours = 80 working hours
6 persons working 14 hours = 84 working hours

80 working hours = 12000 Tk
∴ 84 working hours = (84 × 12000)/80 Tk
= 12600 Tk.
৫৭৮.
Tamim can do a work in 12 days and Sakib in 15 days. If they work on it together for 5 days, then the fraction of the work that is left is:
  1. ক
    3/4
  2. খ
    2/4
  3. গ
    1/4
  4. ঘ
    15/20
সঠিক উত্তর: গ
1/4
উত্তর
সঠিক উত্তর: গ
1/4
গ
ব্যাখ্যা

Question: Tamim can do a work in 12 days and Sakib in 15 days. If they work on it together for 5 days, then the fraction of the work that is left is:

Solution:
Tamim's 1 day's work = 1/12
Sakib's 1 day's work = 1/15

∴ (Tamim + Sakib)'s 1 day's work = (1/12 + 1/15) part
= (5 + 4)/60 part
= 9/60 part
= 3/20 part

∴ (Tamim + Sakib)'s 5 day's work = [5 × (3/20)] part
= 15/20 part
= 3/4 part

Therefore, Remaining work = (1 - 3/4) part
= 1/4 part

৫৭৯.
If a typist can type 125 pages, 36 lines each, 11 words to each line in 5 days, how many pages of 30 lines each and 12 words to each line can he type in 6 days?
  1. ক
    ক) 165
  2. খ
    খ) 160
  3. গ
    গ) 145
  4. ঘ
    ঘ) 120
সঠিক উত্তর: ক
ক) 165
উত্তর
সঠিক উত্তর: ক
ক) 165
ক
ব্যাখ্যা

প্রতি লাইনে 11 words, প্রতি page এ 36 লাইন বিশিষ্ট 125 pages এ
word আছে 11× 36 × 125
5 দিনে 11 × 36 × 125 words হয়
1 দিনে (11 × 36 × 125)/5 words হয়
∴ 6 দিনে (11 × 36 × 125 × 6)/5 words হয়
= (11 × 36 × 125 × 6)/(5 × 12) lines
= (11 × 36 × 125 × 6)/(5 × 12 ×30) pages
= 165 pages.

৫৮০.
A certain psychologist charges Tk. 30 more for the first hour of therapy than for each additional hour. If the total charge to a patient who receives 6 hours of therapy is Tk. 300, what is the total charge to a patient who receives only 3 hours of therapy?
  1. ক
    135
  2. খ
    150
  3. গ
    165
  4. ঘ
    192
সঠিক উত্তর: গ
165
উত্তর
সঠিক উত্তর: গ
165
গ
ব্যাখ্যা
Question: A certain psychologist charges Tk. 30 more for the first hour of therapy than for each additional hour. If the total charge to a patient who receives 6 hours of therapy is Tk. 300, what is the total charge to a patient who receives only 3 hours of therapy?

Solution:
Let the charge for first hour = x + 30
then charge for each additional hour = x
x + 30 + 5x = 300
⇒ 6x = 270
⇒ x = 45

Total charge for patient for 3 hours of therapy = x + 30 + 2x = 3x+30
= 3 × 45 + 30
= 165
৫৮১.
If a person has Tk. 2000 and he wants to distribute this to his five children in the manner that each son has Tk. 30 more than the younger one, what will be the share of the youngest child?
  1. ক
    ক) Tk. 175
  2. খ
    খ) Tk. 325
  3. গ
    গ) Tk. 340
  4. ঘ
    ঘ) Tk. 260
  5. ঙ
    ঙ) Tk. 300
সঠিক উত্তর: গ
গ) Tk. 340
উত্তর
সঠিক উত্তর: গ
গ) Tk. 340
গ
ব্যাখ্যা

Assume first child (the youngest) get = Tk. x
According to the question ;
each son having Tk. 30 more than the younger one

Second child will get = Tk. x + 30
Third child will get = Tk. x + 30 + 30 = x + 60
Fourth child will get = Tk. x + 30 + 30 + 30 = x + 90
Fifth child will get = Tk. x + 30 + 30 + 30 + 30 = x + 120

Total amount they got = Tk. 2000

x + (x+30) + (x+60) + (x+90) + (x+120) = 2000
5x + 300 = 2000
5x = 1700
x = Tk. 340
So the youngest child will get Tk. 340.

৫৮২.
The ratio of the amount of work done by (x-1) labours in (x+1) days and (x+1) labours in (x+2) days is 5 : 6. Then the value of x is?
  1. ক
    ক) 16
  2. খ
    খ) 20
  3. গ
    গ) 17
  4. ঘ
    ঘ) 15
সঠিক উত্তর: ক
ক) 16
উত্তর
সঠিক উত্তর: ক
ক) 16
ক
ব্যাখ্যা

From M1D1 = M2D2
⇒ M1D1 / M2D2 = 5/6
⇒ (x−1)(x+1)/(x+1)(x+2) = 5/6
⇒ (x−1)/(x+2) = 5/6
⇒ 6x−6 = 5x+10
⇒ x = 16

৫৮৩.
A man rowed 3 miles upstream in 90 minutes. If the river flowed with a current of 2 miles per hour, how long did the man's return trip take?
  1. ক
    ক) 20 minutes
  2. খ
    খ) 50 minutes
  3. গ
    গ) 30 minutes
  4. ঘ
    ঘ) 60 minutes
সঠিক উত্তর: গ
গ) 30 minutes
উত্তর
সঠিক উত্তর: গ
গ) 30 minutes
গ
ব্যাখ্যা
Question: A man rowed 3 miles upstream in 90 minutes. If the river flowed with a current of 2 miles per hour, how long did the man's return trip take?

Solution:
Let, the velocity of the boat be x mph and  the stream be y mph
In upstream,
In 90 minutes he goes 3 miles
In 60 minutes he goes (3 × 60)/90 = 2 miles

ATQ,
x - y = 2
⇒ x - 2 = 2  [stream's velocity = 2 mph]
⇒ x = 4

So, velocity in downstream = x + y = 2 + 4 = 6 mph

He goes 6 miles in 1 hr
He goes 3 miles in 3/6 = 1/2 hr = 30 minutes
৫৮৪.
A man and a boy received Tk. 820 as wages for 5 days for the work they did together. The man's efficiency in the work was thrice times that of the boy. What are the daily wages of the boy?
  1. ক
    ক) Tk. 40
  2. খ
    খ) Tk. 39
  3. গ
    গ) Tk. 41
  4. ঘ
    ঘ) None of the above
সঠিক উত্তর: গ
গ) Tk. 41
উত্তর
সঠিক উত্তর: গ
গ) Tk. 41
গ
ব্যাখ্যা
Man = 3 boy
Daily wages for them = 820/5
                                   = 164
4 boy (1 man + 1 boy) = 164
4 boy = 164
Or, boy = Tk. 41
৫৮৫.
If a dozen bananas cost 30 Taka, what is the price of the number of bananas that is 2 less than a dozen?
  1. ক
    15 Taka
  2. খ
    22 Taka
  3. গ
    25 Taka
  4. ঘ
    30 Taka
সঠিক উত্তর: গ
25 Taka
উত্তর
সঠিক উত্তর: গ
25 Taka
গ
ব্যাখ্যা

Question: If a dozen bananas cost 30 Taka, what is the price of the number of bananas that is 2 less than a dozen?

Solution:
A dozen bananas costs = 30 Taka.
A dozen = 12 bananas

So, the cost of 12 bananas = 30 Taka
∴ Cost of 1 banana = 30/12 = 2.5 Taka.

Now, 2 less than a dozen of bananas = 12 - 2 = 10 bananas.
Then, cost of 10 bananas,
= 10 × 2.5
= 25 Taka.

৫৮৬.
Fifteen women can do a work in sixteen days. Twelve men can complete the same work in fifteen days. What is the ratio between the capacity of a man and a woman?
  1. ক
    3 : 4
  2. খ
    1 : 3
  3. গ
    1 : 4
  4. ঘ
    4 : 3
সঠিক উত্তর: ঘ
4 : 3
উত্তর
সঠিক উত্তর: ঘ
4 : 3
ঘ
ব্যাখ্যা
Question: Fifteen women can do a work in sixteen days. Twelve men can complete the same work in fifteen days. What is the ratio between the capacity of a man and a woman?

Solution:
Given,
15 women can do a work in 16 days
(15 × 16) = 240 women can complete the work in 1 day
∴ 1 woman's 1 day's work = 1/240

12 men can complete the work in 15 days
(12 × 15) = 180 men can complete the work in 1 day.
∴ 1 man's 1 day's work = 1/180

So, required ratio = 1/180 : 1/240
= 240 : 180
= 4 : 3
৫৮৭.
A man travels 50 km at speed 25 km/hr and next 40 km at 20 km/hr and there after travel 90 km at 15 km/hr. His average speed is:
  1. ক
    ক) 9 km/hr
  2. খ
    খ) 12 km/hr
  3. গ
    গ) 18 km/hr
  4. ঘ
    ঘ) 36 km/hr
সঠিক উত্তর: গ
গ) 18 km/hr
উত্তর
সঠিক উত্তর: গ
গ) 18 km/hr
গ
ব্যাখ্যা
Question: A man travels 50 km at speed 25 km/hr and next 40 km at 20 km/hr and there after travel 90 km at 15 km/hr. His average speed is:

Solution:
We know,
Average speed = Total distance/Total time
= (50 + 40 + 90)/(2 + 2 + 6) km/hr
= 180/10 km/hr
= 18 km/hr.
৫৮৮.
A box contains 90 nuts each of 100 gm and 100 bolts each of 150 gm. If the entire box weighs 34.5 kg, then find the weight of the empty box.
  1. ক
    10.5 kg
  2. খ
    11 kg
  3. গ
    11.5 kg
  4. ঘ
    24 kg
সঠিক উত্তর: ক
10.5 kg
উত্তর
সঠিক উত্তর: ক
10.5 kg
ক
ব্যাখ্যা
Question: A box contains 90 nuts each of 100 gm and 100 bolts each of 150 gm. If the entire box weighs 34.5 kg, then find the weight of the empty box.

Solution:
১টি নাটের ওজন ১০০ গ্রাম 
∴ ৯০টি নাটের ওজন (৯০ × ১০০) গ্রাম 
= ৯০০০ গ্রাম
= ৯ কেজি 

১টি বল্টুর ওজন ১৫০ গ্রাম
∴ ১০০টি বল্টুর ওজন (১৫০ × ১০০) গ্রাম 
= ১৫০০০ গ্রাম 
= ১৫ কেজি

নাট ও বল্টুর মোট ওজন = ৯ + ১৫ = ২৪ কেজি

খালি বাক্সের ওজন = (৩৪.৫ - ২৪) কেজি
= ১০.৫ কেজি
৫৮৯.
A wheel rotates 10 times per minute and moves 20 m during each rotation. How many meters does the wheel move in 1 hour?
  1. ক
    ক) 10,000
  2. খ
    খ) 12,000
  3. গ
    গ) 18,000
  4. ঘ
    ঘ) 20,000
সঠিক উত্তর: খ
খ) 12,000
উত্তর
সঠিক উত্তর: খ
খ) 12,000
খ
ব্যাখ্যা

Number of rotation in one hour = 10 × 60 = 600
So, Distance moved = (600 × 20) = 12000 m

৫৯০.
12 students working for 5 hours a day can solve a certain number of problems in 8 days. How many students are needed to solve five times the original number of problems, if they work at 4 hours a day for 15 days? 
  1. ক
    ক) 90
  2. খ
    খ) 45
  3. গ
    গ) 95
  4. ঘ
    ঘ) 40
সঠিক উত্তর: ঘ
ঘ) 40
উত্তর
সঠিক উত্তর: ঘ
ঘ) 40
ঘ
ব্যাখ্যা
ধরি,
x  জন ছাত্র লাগবে 
প্রশ্নমতে,
x ×  4 × 15 = 12 × 5 × 5 × 8  
x = (12 × 5 × 5 × 8)/(4 × 15)
x = 40
৫৯১.
P men can do a work in t hours, if 2 men leave the work, the additional time required to complete the work is -
  1. ক
    (2t)/(p + 2)
  2. খ
    (2t)/(p - 2)
  3. গ
    (p + 2)/(2t)
  4. ঘ
    (p - 2)/(2t)
সঠিক উত্তর: খ
(2t)/(p - 2)
উত্তর
সঠিক উত্তর: খ
(2t)/(p - 2)
খ
ব্যাখ্যা
Question: P men can do a work in t hours, if 2 men leave the work, the additional time required to complete the work is -

Solution:
P men need = t days
∴ 1 man need = tP days
∴ (P - 2) men need = tP/(P - 2) days

So, additional days = {tP/(P - 2)} - t 
= {tP - t(P - 2)}/(P - 2)
= (tP - tP - 2t)/(P - 2)
= (2t)/(P - 2)
৫৯২.
3 pumps, working 12 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
    12 hours
  2. খ
    16 hours
  3. গ
    18 hours
  4. ঘ
    20 hours
সঠিক উত্তর: গ
18 hours
উত্তর
সঠিক উত্তর: গ
18 hours
গ
ব্যাখ্যা

Question: 3 pumps, working 12 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?

Solution:
3 pumps need 2 days by working 12 hours
1 pump needs 1 day by working 12 × 3 × 2 hours
4 pumps need 1 day by working (12 × 3 × 2)/4 hours
= 18 hours

 

৫৯৩.
There is a group of 7 boys and 4 girls. The two groups working together can do five times as much work as a boy and a girl. Ratio of working capacities of a boy and a girl is:
  1. ক
    2 : 1
  2. খ
    1 : 2
  3. গ
    1 : 3
  4. ঘ
    3 : 1
সঠিক উত্তর: খ
1 : 2
উত্তর
সঠিক উত্তর: খ
1 : 2
খ
ব্যাখ্যা

Question: There is a group of 7 boys and 4 girls. The two groups working together can do five times as much work as a boy and a girl. Ratio of working capacities of a boy and a girl is:

Solution:
Let 1 boy's 1 day's work = x
And 1 girl's 1 day's work = y
Now,
(7 boys + 4 girls)'s work = 7x + 4y
Given,
7x + 4y is equal to 5 times work done by a boy and a girl
Thus,
7x + 4y = 5(x + y)
⇒ 7x + 4y = 5x + 5y
⇒ 7x - 5x = 5y - 4y
⇒ 2x = y
⇒ x/y = 1/2
⇒ x : y = 1 : 2

Hence, the required ratio is 1 : 2

৫৯৪.
A machine produces 300 pens in 5/2 hours. How many pens can it produce in 50 minutes?
  1. ক
    100 pens
  2. খ
    120 pens
  3. গ
    150 pens
  4. ঘ
    200 pens
সঠিক উত্তর: ক
100 pens
উত্তর
সঠিক উত্তর: ক
100 pens
ক
ব্যাখ্যা

Question: A machine produces 300 pens in 5/2 hours. How many pens can it produce in 50 minutes?

Solution:
দেওয়া আছে,
সময় = 5/2 ঘণ্টা = (5/2) × 60 = 150 মিনিট

150 মিনিটে কলম উৎপাদিত হয় = 300টি
1 মিনিটে কলম উৎপাদিত হয় = 300 / 150 = 2টি
50 মিনিটে কলম উৎপাদিত হয় = 2 × 50 = 100টি

∴ মেশিনটি 50 মিনিটে 100টি কলম উৎপাদন করতে পারবে।

৫৯৫.
Fourteen persons can do a work in 18 days. After 5 days of work, 6 workers left the work, and joined back on the last day of the work. In how many days the work got completed?
  1. ক
    ক) 23
  2. খ
    খ) 27
  3. গ
    গ) 25
  4. ঘ
    ঘ) 26
সঠিক উত্তর: খ
খ) 27
উত্তর
সঠিক উত্তর: খ
খ) 27
খ
ব্যাখ্যা
14 persons can do a work in 18 days
After 5 days 6 workers left.
These 5 workers joined on the last day of work.

Concept used:
If each person can do 1unit of work per day then n persons can do 1 × n unit work per day.

Calculation:
Let each person can do 1 unit of work per day

So,
⇒ 14 persons can do = 1 × 14 = 14 units/day
⇒ In 18 days, 14 persons can do = 18 × 14 = 252 units (Total work to be done)

Now,
In first 5 days, 14 persons completed
⇒ 14 × 1 × 5 = 70 units
In last day, 14 persons competed
⇒ 14 × 1 = 14 units

Then,
the total work completed in 6 days = 84 units

Now,
Work left  =168 units
And persons left = 14 - 6 = 8 persons

So,
Time taken by 8 persons to complete to leftover work
⇒ 168/(8 × 1) = 21 days

So,
time taken to complete the whole work = 5 + 21 + 1 = 27 days

∴ The work got completed in 27 days.
৫৯৬.
A and B together can complete a work in 12 days, B and C together can complete the same work in 8 days and A and C together can complete it in 16 days. In total, how many days do A, B and C together take to complete the same work? 
  1. ক
    ক) 73/7 days
  2. খ
    খ) 97/11 days
  3. গ
    গ) 75/13 days
  4. ঘ
    ঘ) 96/13 days
সঠিক উত্তর: ঘ
ঘ) 96/13 days
উত্তর
সঠিক উত্তর: ঘ
ঘ) 96/13 days
ঘ
ব্যাখ্যা
(A + B)'s 1 day work = 1/12
(B + C)'s 1 day work =1/8
(A + C)'s 1 day work =1/16

2( A+B +C ) one day work =(1/12) + (1/8) + (1/16)
                                           = (4 + 6 + 3)/48
                                           = 13/48


( A+B +C ) one day work =13/96

So (A+B+C) done work = 96/13 days
৫৯৭.
Three mechanics A, B, and C can manufacture 120 units in 12, 20, and 30 hours respectively. What is the ratio of the time taken by A alone to complete the work to the time taken by all three working together to complete the same work?
  1. ক
    5 : 2
  2. খ
    4 : 3
  3. গ
    3 : 1
  4. ঘ
    2 : 1
সঠিক উত্তর: ঘ
2 : 1
উত্তর
সঠিক উত্তর: ঘ
2 : 1
ঘ
ব্যাখ্যা
Question: Three mechanics A, B, and C can manufacture 120 units in 12, 20, and 30 hours respectively. What is the ratio of the time taken by A alone to complete the work to the time taken by all three working together to complete the same work?

Solution:
(A + B + C) together can do in 1 hour = (1/12) + (1/20) + (1/30)
= (5 + 3 + 2)/60
= 10/60
= 1/6
So, working together they complete the work in 6 hours.

And A alone takes 12 hours.

∴ Ratio of the time taken by A and (A + B + C) =12 : 6 = 2 : 1
৫৯৮.
A mother can do a job as fast as her two daughters working together. If one daughter does the job alone in 3 hours and the other does it alone in 6 hours, how many hours does it take the mother to do the job alone?
  1. ক
    2
  2. খ
    3
  3. গ
    4
  4. ঘ
    6
  5. ঙ
    None of these
সঠিক উত্তর: ক
2
উত্তর
সঠিক উত্তর: ক
2
ক
ব্যাখ্যা

First daughter, in 1 hour, did 1/3 part
Second daughter, in 1 hour, did 1/6 part
Mother, in 1 hour, did (1/3 + 1/6) part
= 3/6 part
= 1/2 part
so, mother did it in 2 days.

৫৯৯.
A rope can make 70 rounds of the circumference of a cylinder whose radius of the base is 14cm. How many times can it go round a cylinder having radius 20 cm?
  1. ক
    ক) 54 rounds
  2. খ
    খ) 42 rounds
  3. গ
    গ) 49 rounds
  4. ঘ
    ঘ) 52 rounds
সঠিক উত্তর: গ
গ) 49 rounds
উত্তর
সঠিক উত্তর: গ
গ) 49 rounds
গ
ব্যাখ্যা

Let the required number of rounds be x

More radius, less rounds(Indirect proportion)

Hence we can write as
(radius) 14 : 20 :: x : 70
⇒ 14 × 70 = 20x
⇒ 14 × 7 = 2x
⇒ x = 7 × 7
= 49 days.

৬০০.
A truck can carry 24 motorcycles or 36 scooters at a time. If there are 10 motorcycles on the truck, how many scooters can be loaded onto it?
  1. ক
    10 scooters
  2. খ
    11 scooters
  3. গ
    21 scooters
  4. ঘ
    30 scooters
  5. ঙ
    31 scooters
সঠিক উত্তর: গ
21 scooters
উত্তর
সঠিক উত্তর: গ
21 scooters
গ
ব্যাখ্যা

Question: A truck can carry 24 motorcycles or 36 scooters at a time. If there are 10 motorcycles on the truck, how many scooters can be loaded onto it?

Solution:
Here,
24 motorcycles = 36 scooters
∴ 1 motorcycle = 36/24 scooters = 3/2 scooters
∴ 10 motorcycles = (36 × 10)/24 scooters = 15 scooters

∴ Maximum number of scooters that can still be loaded = 36 - 15 = 21 scooters