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Time & Work, Chain Rule

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Time & Work, Chain Rule

PrepBank · পাতা ১ / ১১ · ১–১০০ / ১,০৭৬

১.
If 400 gm of sugar costs 60 taka, how many taka will 1.2 kg cost?
  1. ক
    180 taka
  2. খ
    250 taka
  3. গ
    320 taka
  4. ঘ
    290 taka
ব্যাখ্যা

Question: If 400 gm of sugar costs 60 taka, how many taka will 1.2 kg cost?

Solution:
400 gm চিনির দাম = 60 taka
∴ 1 gm চিনির দাম = 60/400 taka
= 3/20 taka

এখন, 1.2 kg = 1200 gm

∴ 1200 gm চিনির দাম = 1200 × (3/20) taka
= (1200 × 3)/20 taka
= 3600/20 taka
= 180 taka

২.
A group of workers promises to complete a piece of work in 10 days. But five of them do not report for work. If it took the remaining workers 12 days to complete the work, then the number of workers originally hired was-
  1. ক
    15
  2. খ
    25
  3. গ
    30
  4. ঘ
    45
ব্যাখ্যা
Question: A group of workers promises to complete a piece of work in 10 days. But five of them do not report for work. If it took the remaining workers 12 days to complete the work, then the number of workers originally hired was-

Solution:
Let, the workers promised were = x
The workers worked were = x - 5

ATQ,
10x = 12(x - 5)
⇒ 10x = 12x - 60
⇒ 12x - 10x = 60
⇒ 2x = 60
∴ x = 30
৩.
A and B can do a work in 9 days, B and C can do it in 12 days and A and C can do it in 18 days. If all of them work together, in how many days they can finish the work?
  1. ক
    ক) 5 days 
  2. খ
    খ) 8 days 
  3. গ
    গ) 10 days 
  4. ঘ
    ঘ) 15 days 
ব্যাখ্যা
Question: A and B can do a work in 9 days, B and C can do it in 12 days and A and C can do it in 18 days. If all of them work together, in how many days they can finish the work?

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

∴ Total days needed = 1/(1/8) = 8 days
৪.
A canteen requires 672 bananas for a week. Totally, how many bananas will it require for the months of March, April and May?
  1. ক
    8632
  2. খ
    8836
  3. গ
    8328
  4. ঘ
    8832
ব্যাখ্যা
Question: A canteen requires 672 bananas for a week. Totally, how many bananas will it require for the months of March, April and May?

Solution:
Total number of days = (31 + 30 + 31) = 92
Let the number of bananas be x

ATQ,
7/92 = 672/x
⇒ 7x = 92 × 672
⇒ x = (92 × 672)/7
∴  x = 8832
৫.
24 workers can complete a construction job in 15 days, working 6 hours a day. How many additional workers are needed to complete the same job in 10 days, working 8 hours a day?
  1. ক
    6
  2. খ
    3
  3. গ
    12
  4. ঘ
    15
ব্যাখ্যা

Question: 24 workers can complete a construction job in 15 days, working 6 hours a day. How many additional workers are needed to complete the same job in 10 days, working 8 hours a day?

Solution:
দৈনিক 6 ঘণ্টা করে কাজ করে 15 দিনে শেষ করতে লোক লাগে 24 জন
∴ দৈনিক 1 ঘণ্টা করে কাজ করে 1 দিনে শেষ করতে লোক লাগে (24 × 15 × 6) জন
∴ দৈনিক 8 ঘণ্টা করে কাজ করে 10 দিনে শেষ করতে লোক লাগে (24 × 15 × 6)/(10 × 8) জন
= 27 জন

∴ অতিরিক্ত লোক লাগবে = (27 - 24) = 3 জন

৬.
A does one-third as much work as B in one-fourth of the time. If together they take 12 days to complete a work, how much time shall B alone take to do it?
  1. ক
    16 days
  2. খ
    28 days
  3. গ
    24 days
  4. ঘ
    20 days
ব্যাখ্যা
Question: A does one-third as much work as B in one-fourth of the time. If together they take 12 days to complete a work, how much time shall B alone take to do it?

Solution:
Let B takes x days to do the work.
A takes 1/4 of x time to do 1/3 of the work.
∴ the work will be done by A in (x/4) × 3 days
= 3x/4 
ATQ,
1/x + 4/3x = 1/12
⇒ 7/3x = 1/12
⇒ x = 28
∴ B alone will take 28 days.
৭.
Mina has 3 Taka more than Babu has, but 5 Taka less than Shelly has. If Mina has 450 taka, how much money Shelly and Babu have altogether?
  1. ক
    902 taka
  2. খ
    900 taka
  3. গ
    850 taka
  4. ঘ
    952 taka
ব্যাখ্যা
Question: Mina has 3 Taka more than Babu has, but 5 Taka less than Shelly has. If Mina has 450 taka, how much money Shelly and Babu have altogether?

Solution:
মিনার কাছে আছে = 450 টাকা
বাবুর কাছে আছে = 450 - 3 টাকা = 447 টাকা
শেলির কাছে আছে = 450 + 5 টাকা = 455 টাকা

বাবু ও শেলির কাছে আছে = 447 + 455 টাকা
= 902 টাকা
৮.
X is 50% more efficient than Y. How much time will they, working together, take to complete a job which Y alone could have done in 25 days?
  1. ক
    7 days
  2. খ
    10 days
  3. গ
    14 days
  4. ঘ
    18 days
ব্যাখ্যা

Question: X is 50% more efficient than Y. How much time will they, working together, take to complete a job which Y alone could have done in 25 days?

Solution:
X, Y এর থেকে 50% বেশি দক্ষ।
⇒ X : Y = 150 : 100 = 3 : 2

Y এক দিনে কাজ করে = 2 ইউনিট
X এক দিনে কাজ করে = 3 ইউনিট

মোট কাজ = Y এর দৈনিক কাজ × Y এর দিন = 2 × 25 = 50 ইউনিট

একসাথে এক দিনে কাজ করে = 3 + 2 = 5 ইউনিট
তাহলে কাজ শেষ করতে সময় লাগবে = 50 ÷ 5 দিন = 10 দিন

∴ সুতরাং, X এবং Y একত্রে কাজটি শেষ করতে 10 দিন সময় নেবে।

৯.
If 5 persons working 7 hours a day earn Tk. 7000 per week, then 8 persons working 5 hours a day will earn per week?
  1. ক
    ক) Tk. 7500
  2. খ
    খ) Tk. 8000
  3. গ
    গ) Tk. 8250
  4. ঘ
    ঘ) Tk. 9100
ব্যাখ্যা
Question: If 5 persons working 7 hours a day earn Tk. 7000 per week, then 8 persons working 5 hours a day will earn per week?

Solution:
Earning of 5 × 7 = 35 hours = 7000 Tk.
Earning of 1 hour is = 7000/35 Tk.
Earning of 8 × 5 = 40 hours is = (7000 × 40)/35 = Tk. 8000
১০.
A farmer grows paddy on one acre and gets 400 kg of it. If each kilogram of paddy yields 700 grams of rice, how much rice does he get in total?
  1. ক
    250 kg
  2. খ
    260 kg
  3. গ
    280 kg
  4. ঘ
    350 kg
ব্যাখ্যা

Question: A farmer grows paddy on one acre and gets 400 kg of it. If each kilogram of paddy yields 700 grams of rice, how much rice does he get in total?

Solution:
1 কেজি ধানে চাল হয় = 700 গ্রাম
= 700/1000 কেজি
= 0.7 কেজি

∴ 400 কেজি ধানে চাল হয় = (400 × 0.7) কেজি = 280 কেজি 

১১.
এক ব্যক্তি ৩০ ঘণ্টায় একটি নির্দিষ্ট দূরত্ব অতিক্রম করেন। যাত্রাপথের অর্ধেক দূরত্ব তিনি ২১ কি.মি./ঘণ্টা এবং বাকী অর্ধেক দূরত্ব ২৪ কি.মি./ঘণ্টা গতিতে চলেন। মোট দূরত্ব কত ছিল?
  1. ক
    ৬৯৪ কি.মি.
  2. খ
    ৬৭৫ কি.মি.
  3. গ
    ৬৮৮ কি.মি.
  4. ঘ
    ৬৭২ কি.মি.
  5. ঙ
    কোনটিই নয়
ব্যাখ্যা
প্রশ্ন: এক ব্যক্তি ৩০ ঘণ্টায় একটি নির্দিষ্ট দূরত্ব অতিক্রম করেন। যাত্রাপথের অর্ধেক দূরত্ব তিনি ২১ কি.মি./ঘণ্টা এবং বাকী অর্ধেক দূরত্ব ২৪ কি.মি./ঘণ্টা গতিতে চলেন। মোট দূরত্ব কত ছিল?

সমাধান:
ধরি,
দূরত্ব = ক কি.মি.
ব্যক্তিটি ৩০ ঘণ্টায় ক দূরত্ব অতিক্রম করেন।
যাত্রাপথের অর্ধেক দূরত্ব অতিক্রম করতে সময় লাগে = অর্ধেক দূরত্ব/যাত্রাপথের অর্ধেক দূরত্ব যাওয়ার গতিবেগ
= (ক/২)/২১
= ক/৪২ ঘণ্টা

যাত্রাপথের বাকি অর্ধেক দূরত্ব অতিক্রম করতে সময় লাগে = বাকি অর্ধেক দূরত্ব/যাত্রাপথের বাকি অর্ধেক দূরত্ব যাওয়ার গতিবেগ
= (ক/২)/২৪
= ক/৪৮ ঘণ্টা

প্রশ্নমতে,
ক/৪২ + ক/৪৮ = ৩০
⇒ (৮ক + ৭ক)/৩৩৬ = ৩০
⇒ ১৫ক = ৩০ × ৩৩৬
∴ ক = ৬৭২ কি.মি.
১২.
If 12 workers can complete a project in 15 days, how many days will it take for 9 workers to complete the same project assuming they all work at the same rate?
  1. ক
    18
  2. খ
    20
  3. গ
    24
  4. ঘ
    25
ব্যাখ্যা

Question: If 12 workers can complete a project in 15 days, how many days will it take for 9 workers to complete the same project assuming they all work at the same rate?

Solution: 
12 workers can complete work in 15 days
∴ 1 workers can complete work in  = 12 × 15 days 
∴ 9 workers can complete work in = (12 × 15)/9 days 
= 20 days 

It will take 9 workers 20 days to complete the same project. 

১৩.
If 6 spiders make 6 webs in 6 days, then one spider will make one web in how many days?
  1. ক
    ক) 9
  2. খ
    খ) 7
  3. গ
    গ) 8
  4. ঘ
    ঘ) 6
ব্যাখ্যা
Number of Spider M1 = 6
Number of days D1 = 6
Number of web W1 = 6

Number of Spider M2 = 1
Number of days D2 = ?
Number of web W2 = 1

Therefore,
(M1 × D1)/W1 = (M2 × D2)/W2
(6 × 6)/6 = (1 × D2)/1
D2 = 6

Number of days D2 = 6
১৪.
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
    6
  2. খ
    4
  3. গ
    5
  4. ঘ
    7
ব্যাখ্যা

Question: A man completes 5/8 of a job in 10 days. At this rate, how many more days will it takes him to finish the job?

Solution:
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

১৫.
P can complete work in 12 days working 8 hours a day. Q can complete the same work in 8 days working 10 hours a day. If both p and Q work together, working 8 hours a day, in how many days can they complete the work?
  1. ক
    61/11
  2. খ
    81/11
  3. গ
    72/11
  4. ঘ
    60/11
ব্যাখ্যা
Question: P can complete work in 12 days working 8 hours a day. Q can complete the same work in 8 days working 10 hours a day. If both p and Q work together, working 8 hours a day, in how many days can they complete the work?

Solution:
P can complete the work in (12 × 8) hrs = 96 hrs 
Q can complete the work in (8 × 10) hrs = 80 hrs 
Therefore, P's 1 hour work = 1/96 and Q's 1 hour work = 1/80

(P + Q)'s 1 hour's work = (1/96) + (1/80)
= 11/480

So both P and Q will finish the work in 480/11 hrs  
Therefore, Number of days of 8 hours each = (480/11) × (1/8) = 60/11
১৬.
M does one-third as much work as N in one-fourth of the time. If together they take 12 days to complete a work, how much time shall N alone take to do it?
  1. ক
    14 days
  2. খ
    24 days
  3. গ
    28 days
  4. ঘ
    32 days
ব্যাখ্যা
Question: M does one-third as much work as N in one-fourth of the time. If together they take 12 days to complete a work, how much time shall N alone take to do it?

Solution:
Let N takes x days to do the work.
M takes 1/4 of x time to do 1/3 of the work.
∴ The work will be done by M in (x/4) × 3 days
= 3x/4 

ATQ,
1/x + 4/3x = 1/12
⇒ 7/3x = 1/12
⇒ 3x = 84
∴ x = 28

∴ N alone will take 28 days.
১৭.
There are 200 questions on a 3-hour examination. Among these questions 50 are mathematics problems. It is suggested that twice as much time be spent on each math problem as for each other question. How many minutes should be spent on mathematics problems?
  1. ক
    32 minutes
  2. খ
    60 minutes
  3. গ
    72 minutes
  4. ঘ
    100 minutes
ব্যাখ্যা
Question: There are 200 questions on a 3-hour examination. Among these questions 50 are mathematics problems. It is suggested that twice as much time be spent on each math problem as for each other question. How many minutes should be spent on mathematics problems?

Solution:
Total exam time 3 hours = 3 × 60 minutes = 180 minutes
Total number of questions 200

Let
x minutes be the time spent on each non-math question.
∴ the time spent on each math problem would be 2x minutes.

∴ the total time spent on non-math questions (200 - 50) × x minutes
= 150x minutes

The total time spent on math problems 50 × 2x minutes
= 100x minutes

ATQ,
150x + 100x = 180
⇒ 250 x = 180
⇒ x = 180 / 250
∴ x = 0.72 minutes 

∴ Time for each math problem = 2 × x = 2 × 0.72 = 1.44 minutes
∴ Time for 50 math problems = 1.44 × 50 = 72 minutes.
১৮.
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
    8
  2. খ
    12
  3. গ
    9
  4. ঘ
    15
  5. ঙ
    14
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
১৯.
If 7 spiders make 7 webs in 7 days, then 1 spider will make 1 web in how many days?
  1. ক
    1
  2. খ
    3
  3. গ
    7
  4. ঘ
    14
  5. ঙ
    None of these
ব্যাখ্যা
Question: If 7 spiders make 7 webs in 7 days, then 1 spider will make 1 web in how many days?

Solution:
7 spiders make 7 webs in 7 days
1 spiders make 1 webs in (7 ×7)/7 days
= 7 days
২০.
A ferry can travel twice as fast when empty as when it is full. It travels 20 miles with full load, spends 1 hour for unloading and returns to its original post empty. It took 11 hours to complete the journey. What is the speed of the ferry when it is empty?
  1. ক
    ক) 5
  2. খ
    খ) 6
  3. গ
    গ) 6.5
  4. ঘ
    ঘ) 8
ব্যাখ্যা
ধরি,
ফেরিটির মাল বোঝাই অবস্থায় গতি বেগ x মাইল /ঘণ্টা 
ফেরিটির খালি অবস্থায় গতি বেগ 2x মাইল/ঘণ্টা

মাল বোঝাই অবস্থায়
ফেরিটির 20 মাইল যেতে সময় লাগে = 20/x ঘণ্টা

খালি অবস্থায়
ফেরিটির 20 মাইল যেতে সময় লাগে =20/2x  = 10/x ঘণ্টা

প্রশ্নমতে,
20/x + 10/x = 11 - 1
(20 + 10)/x = 10 
30/x = 10 
10x = 30 
x = 3 

ফেরিটির খালি অবস্থায় গতি বেগ 2 × 3 = 6  মাইল/ঘণ্টা
২১.
The price of 357 mangoes is Tk. 1517.25. Find the approximate price of 49 dozens of such mangoes?
  1. ক
    Tk. 3099
  2. খ
    Tk. 3599
  3. গ
    Tk. 4099
  4. ঘ
    Tk. 2499
ব্যাখ্যা
Question: The price of 357 mangoes is Tk. 1517.25. Find the approximate price of 49 dozens of such mangoes?

Solution:
We know that 1 dozen = 12 piece
49 dozens = 49 × 12 = 588 mangoes

357 mangoes = Tk. 1517.25
∴ 1 mango = Tk. 1517.25/357
∴ 588 mangoes = Tk. (1517.25 × 588)/357
= Tk. 2499
২২.
If 9 engines consume 24 metric tons of coal, when each is working 8 hours per day, how much coal should be available for 8 engines, each running 13 hours per day, it is given that 3 engines of the former type consume as much as 4 engines of later type.
  1. ক
    20 metric tons
  2. খ
    15 metric tons
  3. গ
    26 metric tons
  4. ঘ
    17 metric tons
ব্যাখ্যা
Question: If 9 engines consume 24 metric tons of coal, when each is working 8 hours per day, how much coal should be available for 8 engines, each running 13 hours per day, it is given that 3 engines of the former type consume as much as 4 engines of later type.

Solution:
We have:
The lesser engines, less coal consumed
More working hours, more coal consumed
Both the cases are directly proportional.
If three engines of former type consume 1 unit, 1 engine will consume 1/3 unit.
If four engines of latter type consume 1 unit, 1 engine will consume ¼ units.
And, less rate of consumption, less coal consumed.
Now, Number of engines = 9: 8
Working hours = 8:13
Therefore, rate of consumption = 1/3 : 1/4

Let the coal consumed by 8 engines is x metric tones
9 × 8 × 1/3 : 8 × 13 × 1/4 = 24 : x
⇒ 9 × 8 × (1/3) × x = 8 × 13 × (1/4) × 24
⇒ 24x = 624
∴ x = 26
২৩.
Reza sells 500 shares in a company via a stock broker who charges a flat Tk. 20 commission rate on all transactions under Tk. 1000. His bank account is credited with Tk. 692 from the sale of the shares. What price were his shares sold at?
  1. ক
    ক) 109
  2. খ
    খ) 131
  3. গ
    গ) 142.4
  4. ঘ
    ঘ) 168.9
ব্যাখ্যা
কমিশনসহ রেজার শেয়ারের বিক্রয়মূল্য ৬৯২+২০ = ৭১২ টাকা।
তাহলে ৫০০ শেয়ারের বিক্রয়মূল্য ৭১২/৫০০ = ১.৪২৪ বা ১৪২.৪ সেন্টস।
২৪.
A tap can fill a tank in 6 hours. After half the tank is filled three more similar taps are opened. What is the total time taken to fill the tank completely? 
  1. ক
    ক) 4 hrs 15 min
  2. খ
    খ) 3 hrs 45 min
  3. গ
    গ) 3 hrs 24 min
  4. ঘ
    ঘ) 4 hrs 51 min
ব্যাখ্যা
Time taken by one tap to fill half the tank = 3 hrs.
Part filled by one tap in 1 hour = 1/6
Part filled by four taps in 1 hour = (4×1/6) = 2/3
Remaining part = (1−1/2) = 1/2
2/3 of the tank is filled by four taps in 1 hour.
So, 1/2 of the tank is filled in = 3/2×1/2=3/4 hours
3/4 hours = 3/4 × 60 = 45 min
So, the total time taken = 3 hrs + 45 min = 3 hrs 45 min or 225 min
-------------------------------------------------
Alternative way:
A tap can fill a tank in 6 hours.
A tap can fill half of a tank in 6/2 or 3 hours.
4 tap can fill half of a tank in 3/4 hours = 45 min
Total time taken = 3 hours 45 min
২৫.
An old man is walking on a foggy road at a speed of x km/h. Due to low visibility, the old man see only up to 600 meters. If a car overtakes the man from behind with the speed of 15 km/hr then the man can see the car for 216 seconds. Find the speed of the man?
  1. ক
    2 km/h
  2. খ
    3 km/h
  3. গ
    5 km/h
  4. ঘ
    6 km/h
ব্যাখ্যা
Question: An old man is walking on a foggy road at a speed of x km/h. Due to low visibility, the old man see only up to 600 meters. If a car overtakes the man from behind with the speed of 15 km/hr then the man can see the car for 216 seconds. Find the speed of the man?

Solution:
Distance up to old man see = 600/1000 = 0.6km
Time for which the man can see the car = 216/(60 × 60) = 0.06 hour

ATQ,
0.6/(15 - x) = 0.06
⇒ 15 - x = 10
∴ x = 5

So the old man is walking at a speed of 5 km/h
২৬.
Rajib can do a work in 5 days while Farez can do the same work in 3 days. Both of them finish the work together and get Tk. 240. What is Rajib's share?
  1. ক
    ক) Tk. 75
  2. খ
    খ) Tk. 80
  3. গ
    গ) Tk. 85
  4. ঘ
    ঘ) Tk. 90
ব্যাখ্যা
Question: Rajib can do a work in 5 days while Farez can do the same work in 3 days. Both of them finish the work together and get Tk. 240. What is Rajib's share?

Solution:
Rajib's wages : Farez's wages = Rajib's 1 day's work : Farez's 1 day's work 
= 1/5 : 1/3
= 3 : 5

Sum of the ratio = 3 + 5 = 8

So, Rajib's share = (3/8) × 240 = Tk. 90
২৭.
Two workers A and B are engaged to do a work. A working alone takes 8 hours more to complete the job than if both worked together. If B worked alone, he would need 9/2 hours more to complete the job than they both working together. What time would they take to do the work together? 
  1. ক
    5 hours
  2. খ
    6 hours
  3. গ
    8 hours
  4. ঘ
    4 hours
ব্যাখ্যা
Question: Two workers A and B are engaged to do a work. A working alone takes 8 hours more to complete the job than if both worked together. If B worked alone, he would need 9/2 hours more to complete the job than they both working together. What time would they take to do the work together? 

Solution: 
Let,
both together can do the work in x hours.
A can do it in = (x + 8) hour
B can do it in = {x + (9/2)} hour
= (2x + 9)/2 hour

so,
{1/(x + 8)} + {2/(2x + 9)} = 1/x
or, (2x + 9 + 2x + 16)/{(x + 8)(2x + 9)} = 1/x
or, x(4x + 25) = (x + 8)(2x + 9)
or, 4x2 + 25x = 2x2 + 25x + 72
or, 2x2 = 72
or, x2 = 36
∴ x = 6
২৮.
3 workers can do a job in 12 days. Two of the workers work twice as fast as the third. How long it will take if one of the faster workers do the job himself?
  1. ক
    ক) 12 minutes
  2. খ
    খ) 15 minutes
  3. গ
    গ) 18 minutes
  4. ঘ
    ঘ) None of these
ব্যাখ্যা
৩ জন শ্রমিক একটি কাজ ১২ দিনে করতে পারে।
এই ৩ জনের মধ্যে ২ জন ৩য় জনের তুলনায় দ্বিগুণ কাজ করতে পারে।
প্রথম দুজনের ১ জন শ্রমিকের কাজ = ২ × ৩য় শ্রমিকের কাজ।
সুতরাং ৩ জন শ্রমিকের কাজ = ৫ × ৩য় শ্রমিকের কাজ।
৫ × ৩য় শ্রমিক কাজটি করতে পারে = ১২ দিনে
৩য় শ্রমিক কাজটি করতে পারে = ৫ × ১২ দিনে = ৬০ দিনে।
২ × ৩য় শ্রমিক কাজটি করতে পারে = ৬০/২ দিনে = ৩০ দিনে
নির্ণেয় সময় = ৩০ দিন
২৯.
40 workers can build 40 engines working 6 hours a day. How many workers need to be appointed extra to boost the production to double if they work 8 hours a days?
  1. ক
    20 workers
  2. খ
    22 workers
  3. গ
    25 workers
  4. ঘ
    30 workers
  5. ঙ
    32 workers
ব্যাখ্যা
Question: 40 workers can build 40 engines working 6 hours a day. How many workers need to be appointed extra to boost the production to double if they work 8 hours a days?

Solution:
6 hours to build 40 engines by 40 workers
1 hour to build 1 engine by = (40 × 6)/40 workers
8 hours to build 80 engine by = (6 × 80)/8 workers
= 60 workers

∴ extra workers = (60 - 40) = 20 workers
৩০.
A team of 10 workers can complete a task in 6 days by working 4 hours per day. If the same task is assigned to 15 workers, how many hours per day should they work in order to finish it in 2 days?
  1. ক
    9 hours per day
  2. খ
    8 hours per day
  3. গ
    6 hours per day
  4. ঘ
    4 hours per day
ব্যাখ্যা

Question: A team of 10 workers can complete a task in 6 days by working 4 hours per day. If the same task is assigned to 15 workers, how many hours per day should they work in order to finish it in 2 days?

Solution:
10 persons can do the work in 6 days by working 4 hours a day
∴ Total man-hours = 10 × 6 × 4 = 240 hours

Let the required hours per day for 15 persons to finish in 2 days = x hours
∴ Total man-hours = 15 × 2 × x = 30x

Equating total work:
30x = 240
⇒ x = 240/30
⇒ x = 8

∴ 15 persons need to work 8 hours per day to complete the work in 2 days.

৩১.
Working alone, Anik can complete a task in ‘a’ days and Bijoy in ‘b’ days. They take turns in doing the task with each working 2 days at a time. If Anik starts they finish the task in exactly 10 days. If Bijjoy starts, they take half a day more. How long does it take to complete the task if they both work together?
  1. ক
    54/11 days
  2. খ
    210/41 days
  3. গ
    36/7 days
  4. ঘ
    None of these
ব্যাখ্যা
Question: Working alone, Anik can complete a task in ‘a’ days and Bijoy in ‘b’ days. They take turns in doing the task with each working 2 days at a time. If Anik starts they finish the task in exactly 10 days. If Bijjoy starts, they take half a day more. How long does it take to complete the task if they both work together?

Solution: 
Let, Anik can complete work in x days and Bijoy can do in y days

If Anik starts, 
(6/x) + (4/y) = 1 
⇒ 4x + 6y = xy 

If Bijoy starts, 
(6/y) + (4.5/x) = 1 
⇒ 4.5y + 6x = xy 

24x + 36y - 18y - 24x = 6xy - 4xy
⇒ 18y = 2xy 
⇒ x = 9

36 + 6y = 9y
⇒ 3y = 36
⇒ y = 12

both do one day = (1/9) + (1/12)
= (4 + 3)/36
= 7/36

if they both work together days needed = 36/7 days
৩২.
A man completes 1/8 of a job in 10 days. At this rate, how many more days will it take him to finish the job? 
  1. ক
    50
  2. খ
    70
  3. গ
    40
  4. ঘ
    30
ব্যাখ্যা

Question: A man completes 1/8 of a job in 10 days. At this rate, how many more days will it take him to finish the job?

Solution:
Work done = 1/8
Balance work = 1 - (1/8) = 7/8

A man completes 1/8 of a job in 10 days
A man completes 1 part of a job in 10 × 8 days
= 80 days

∴ More days required = 80 - 10 
= 70 days.

৩৩.
A case contains c cartons. Each carton contains b boxes, and each box contains 100 paper clips. How many paper clips are contained in 2 cases?
  1. ক
    100bc
  2. খ
    (100b)/c
  3. গ
    200bc
  4. ঘ
    (200b)/c
  5. ঙ
    200/(bc)
ব্যাখ্যা
Question: A case contains c cartons. Each carton contains b boxes, and each box contains 100 paper clips. How many paper clips are contained in 2 cases?

Solution:
We are given that a case contains c cartons. Each carton contains b boxes, and each box contains 100 paper clips. Thus we can say the following:

c × b × 100 = 100bc paper clips per case.
Thus, 2 cases would contain 2 × 100bc = 200bc paper clips.
৩৪.
A group of men decided to do a job in 3 days. But since 30 men dropped out every day, the job completed at the end of the 5th day. How many men were there at the beginning?
  1. ক
    ক) 120
  2. খ
    খ) 150
  3. গ
    গ) 180
  4. ঘ
    ঘ) 210
ব্যাখ্যা
Question: A group of men decided to do a job in 3 days. But since 30 men dropped out every day, the job completed at the end of the 5th day. How many men were there at the beginning?

solution: 
ধরি,
শুরুতে লোক ছিলো = x

প্রশ্নমতে,
3x = x + (x - 30) + (x - 60) + (x - 90) + (x - 120)
3x = 5x - 300
2x = 300
x = 150

∴ শুরুতে ১৫০ জন লোক ছিল।
৩৫.
Mr. X had Tk. 1000 in his savings account. Every month in the first week he needs money, so he withdraws Tk. 500, but by the end of the month, he deposits Tk. 750. After how many months, the original amount will grow three times?
  1. ক
    9 months
  2. খ
    6 months
  3. গ
    7 months
  4. ঘ
    8 months
ব্যাখ্যা
Question: Mr. X had Tk. 1000 in his savings account. Every month in the first week he needs money, so he withdraws Tk. 500, but by the end of the month, he deposits Tk. 750. After how many months, the original amount will grow three times?

Solution: 
Initial money Mr. X had = Tk.1000
Three times the initial money ⇒ 1000 × 3 = Tk. 3000
Money to be deposited ⇒ 3000 - 1000 = Tk. 2000

Net every month ⇒Tk. 750 - Tk. 500 = Tk. 250
Months required to make Tk. 2000 = 2000/250 = 8 months.
৩৬.
A contractor employed 30 men to do a piece of work in 38 days. After 25 days, he employed 5 men more and the work was finished one day earlier. How many days he would have been behind if he had not employed additional men?
  1. ক
    ক) 1
  2. খ
    খ) 1(1/4)
  3. গ
    গ) 1(3/4)
  4. ঘ
    ঘ) 1(1/2)
ব্যাখ্যা

After 25 days, 35 men complete the work in 12 days.
Thus, 35 men can finish the remaining work in 12 days.
∴ 30 men can do it in (12 × 35)/30
= 14 days.
which is 1 day behind.

৩৭.
M and N can do a piece of work in 20 days and 12 days respectively. M started the work alone and then after 4 days, N joined him till the completion of the work. How long did the work last?
  1. ক
    8 days
  2. খ
    10 days
  3. গ
    12 days
  4. ঘ
    15 days
ব্যাখ্যা
Question: M and N can do a piece of work in 20 days and 12 days respectively. M started the work alone and then after 4 days, N joined him till the completion of the work. How long did the work last?

Solution:
work done by m in 4 days = (1/20) × 4 = 1/5
∴ Remaining work = 1 - (1/5) = 4/5

(M + N)'s 1day's work = (1/20) + (1/12)
= 8/60 = 2/15

Now, 2/15 work is done by M and N in 1 day.
So, 4/5 work will be done by M and N in = (15/2) × (4/5)
= 6 days

∴ Total time taken = (6 + 4) days = 10 days
৩৮.
A man takes 6 hours 15 minutes walking a distance and riding back to starting place. He could walk both ways in 7 hours 45 minutes. The time taken by him to ride back both ways is -
  1. ক
    ক) 4 hours
  2. খ
    খ) 4 hours 30 minutes
  3. গ
    গ) 4 hours 45 minutes
  4. ঘ
    ঘ) 5 hours
ব্যাখ্যা

Time is taken in walking both the ways = 7 hours 45 minutes -------- (i)

Time is taken in walking one way and riding back = 6 hours 15 minutes ----------- (ii)
By the equation (ii) × 2 - (i), we have,

Time is taken by the man in riding both ways,
= 12 hours 30 minutes - 7 hours 45 minutes
= 4 hours 45 minutes.

৩৯.
Pavel is twice as good as workman as Tanveer. When they work together they can finish a task in 16 days. If Tanveer works alone, in how many days he will complete the task?
  1. ক
    46 days
  2. খ
    48 days
  3. গ
    50 days
  4. ঘ
    52 days
ব্যাখ্যা
Question: Pavel is twice as good as workman as Tanveer. When they work together they can finish a task in 16 days. If Tanveer works alone, in how many days he will complete the task?

Solution:
ধরি,
তানভীর কাজটি করতে পারে ২ক দিনে
পাভেল কাজটি করতে পারে ক দিনে

তাহলে,
তানভীর ১ দিনে করে ১/২ক অংশ
∴ তানভীর ১৬ দিনে করে ১৬/২ক অংশ = ৮/ক অংশ

পাভেল ১ দিনে করে ১/ক অংশ
∴ পাভেল ১ দিনে করে ১৬/ক অংশ

৮/ক + ১৬/ক = ১
বা, (৮ + ১৬)/ক = ১
বা, ২৪/ক = ১
∴ ক = ২৪

তানভীর ১ দিনে করে = ১/(২ × ২৪) = ১/৪৮ অংশ
∴ তানভীর একা ৪৮ দিনে কাজটি সম্পন্ন করবে। 
৪০.
If 7 workers can assemble a car in 8 hours, how long would it take 12 workers to assemble the same car?
  1. ক
    280 minutes
  2. খ
    290 minutes
  3. গ
    310 minutes
  4. ঘ
    330 minutes
  5. ঙ
    None
ব্যাখ্যা
Question: If 7 workers can assemble a car in 8 hours, how long would it take 12 workers to assemble the same car?

Solution:
Given,
7 workers can assemble a car in 8 hours
∴ 1 workers can assemble a car in (8 × 7) hours
∴ 12 workers can assemble a car in (56/12) hours
= (56 × 60)/12 minutes
= 280 minutes
৪১.
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
    4 days
  2. খ
    6 days
  3. গ
    8 days
  4. ঘ
    10 days
ব্যাখ্যা
Question: 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?

Solution:
(A + B), 1days works = 1/12,
(B + C), 1days work = 1/15,
(C + A), 1 days work = 1/20

Now,
2(A + B + C) 1days work = (1/12) + (1/20) + (1/15)
= (5 + 3 + 4)/60
=12/60
=1/5

 (A + B + C) 1 days works = 1/10
∴ A, B and C together can do it in = 10 days
৪২.
Rafi alone can complete a work in 10 days and Tareq alone can complete it in 15 days. Rafi and Tareq undertook to complete the work for Tk. 7500. With the help of Salman, they finished the work in 5 days. How much should Salman be paid?
  1. ক
    Tk. 1200
  2. খ
    Tk. 1250
  3. গ
    Tk. 1300
  4. ঘ
    Tk. 1380
ব্যাখ্যা

Question: Rafi alone can complete a work in 10 days and Tareq alone can complete it in 15 days. Rafi and Tareq undertook to complete the work for Tk. 7500. With the help of Salman, they finished the work in 5 days. How much should Salman be paid?

Solution:
Rafi's 1 day work = 1/10
Tareq's 1 day work = 1/15
Rafi + Tareq + Salman's 1 day work = 1/5

∴ Salman's 1 day work = 1/5 - (1/10 + 1/15)
= (6 - 3 - 2)/30
= 1/30
Salman's 5 days work = 5 × 1/30 = 1/6

Salman completed 1/6 of the total work.
∴ He should be paid 1/6 of Tk. 7500
∴ Salman’s payment = 7500 × 1/6 = Tk. 1250

৪৩.
Rubel can do a job alone in 12 days. He works for 8 days and then leaves, and Rajib finishes the remaining work in 7 days. How long would it take Rajib to do 3/7 of the work alone?
  1. ক
    3 days
  2. খ
    6 days
  3. গ
    9 days
  4. ঘ
    12 days
ব্যাখ্যা

Question: Rubel can do a job alone in 12 days. He works for 8 days and then leaves, and Rajib finishes the remaining work in 7 days. How long would it take Rajib to do 3/7 of the work alone?

Solution:
রুবেল একা,
12 দিনে করতে পারে = 1 অংশ
∴ 8 দিনে করতে পারে = 8/12 = 2/3 অংশ

∴ অবশিষ্ট কাজ = 1 - (2/3) = 1/3 অংশ

হাবিব একা,
1/3 অংশ কাজ করে = 7 দিনে
∴ হাবিব 1 অংশ কাজ করে = 3 × 7 দিনে 
∴ হাবিব 3/7 অংশ কাজ করে = (3 × 7 × 3)/7 দিনে
= 9 দিনে

৪৪.
A company employs 12 persons working 55 hours a week to complete a project. If the working hours turned into 44 hours a week. How many additional persons will be required to complete the project in time?
  1. ক
    ক) 3
  2. খ
    খ) 4
  3. গ
    গ) 5
  4. ঘ
    ঘ) 6
ব্যাখ্যা
Question: A company employs 12 persons working 55 hours a week to complete a project. If the working hours turned into 44 hours a week. How many additional persons will be required to complete the project in time?

Solution:
Total working hours is (55 × 12) = 660 hours

Total persons needed for 44 hours a week = 660/44 = 15

∴ Additional persons will be required = 15 - 12 = 3
৪৫.
A team started a work where the work ratio of a day is twice the previous day. If the work is fully finished in 10 days, how many days it took to finish 1/8th of the work?
  1. ক
    ক) 2 days
  2. খ
    খ) 3 days
  3. গ
    গ) 5 days
  4. ঘ
    ঘ) 7 days
ব্যাখ্যা
Question: A team started a work where the work ratio of a day is twice the previous day. If the work is fully finished in 10 days, how many days it took to finish 1/8th of the work?

Solution: 
প্রতিদিনের কাজ পূর্বের দিনের দ্বিগুণ হয়।

১০ দিনে যদি সম্পূর্ণ শেষ হয় তাহলে,
৯ দিনে শেষ হবে ১/২ কাজ
৮ দিনে শেষ হবে (১/২)/২ বা, ১/৪ কাজ 
৭ দিনে শেষ হবে (১/৪)/২ বা, ১/৮ কাজ

shortcut: 
১ অংশ কাজ হয় ১০ দিনে,
১/৮ অংশ বা ১/২৩ অংশ কাজ হয় = (১০ - ৩) দিনে = ৭ 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 = (১০ -৪) = ৬ দিনে।
৪৬.
An employee may claim Tk. 8 for each kilometer when he travels by taxi and Tk. 7 for each kilometer when he drives his own car. If in one week he claimed Tk. 1075 for travelling 140 km, how many kilometers did he travel by taxi?
  1. ক
    90 km
  2. খ
    95 km
  3. গ
    97 km
  4. ঘ
    100 km
ব্যাখ্যা
Question: An employee may claim Tk. 8 for each kilometer when he travels by taxi and Tk. 7 for each kilometer when he drives his own car. If in one week he claimed Tk. 1075 for travelling 140 km, how many kilometers did he travel by taxi?

Solution:
Let the distance travelled by taxi x km
Let the distance travelled by own car 140 - x km

Now
8x + 7(140 - x) = 1075
⇒ 8x + 980 - 7x = 1075
⇒ x + 980 = 1075
⇒ x = 1075 - 980
∴ x = 95
৪৭.
A wheel that has 12 cogs is meshed with a larger wheel of 24 cogs. If the smaller wheel has made 44 revolutions, then find the number of revolutions made by the larger wheel.
  1. ক
    20
  2. খ
    16
  3. গ
    24
  4. ঘ
    22
ব্যাখ্যা
Question: A wheel that has 12 cogs is meshed with a larger wheel of 24 cogs. If the smaller wheel has made 44 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 (↓)

24 : 12 : : 44 : x
⇒ 24 × x = 12 × 44
⇒ x = (12 × 44)/24
∴ x = 22
৪৮.
The R students in a class agree to contribute equally to buy their teacher a birthday present that costs y dollars. If x of the students later fail to contribute their share, which of the following represents the additional number of dollars that each of the remaining students must contribute in order to pay for the present?
  1. ক
    y/R
  2. খ
    y/(R - x)
  3. গ
    xy/(R - x)
  4. ঘ
    xy/{R(R - x)}
ব্যাখ্যা
Question: The R students in a class agree to contribute equally to buy their teacher a birthday present that costs y dollars. If x of the students later fail to contribute their share, which of the following represents the additional number of dollars that each of the remaining students must contribute in order to pay for the present?

Solution:
যদি R সংখ্যক শিক্ষার্থী সমান সমান টাকা দেয় তাহলে টাকা উঠে = y টাকা 
∴ একজন শিক্ষার্থী দেয় y/R টাকা

x জন শিক্ষার্থী টাকা না দেয়ায় মোট টাকা দেয় (R - x) জন
∴ একজন শিক্ষার্থী দেয় y/(R - x) টাকা

বাড়তি দিতে হয় = y/(R - x) - y/R
= (yR - yR + xy)/{R(R - x)}
= xy/{R(R - x)}
৪৯.
X can do a piece of work in 20 days and Y can do the 1/7th of the same work in 5 days. In how many days together can they complete the 11/20th of the total work?
  1. ক
    ক) 7 days
  2. খ
    খ) 14 days
  3. গ
    গ) 10 days
  4. ঘ
    ঘ) 12 days
ব্যাখ্যা
X can do in 1 day = 1/20 part 
Y can do in 1 day = 1/(7×5) = 1/35 part
X & Y together can do in 1 day = 1/20 + 1/35 = 11/140
11/140 part of the work is done in 1 day
So, 11/20 part of the work is done in = (11/20) × (140/11) = 7 days
৫০.
Worker P is 50% as efficient as worker Q. Worker R does half of the work done by P and Q together. If R alone does the work in 40 days, then P, Q and R together can do the work in -
  1. ক
    ক) 20(1/3) days
  2. খ
    খ) 25 days
  3. গ
    গ) 15 days
  4. ঘ
    ঘ) 13(1/3) days
ব্যাখ্যা

As, R takes 40 days to complete the work, (P + Q) will take = 40/2 = 20 Days
R can do 1/40 part of the work in one day and (P + Q) can do 1/20 part of the work in a day.
So, in 1 day (P + Q + R) can do = (1/20 + 1/40) = (3/40) part of the work
∴ Days required to complete 1 or Total work is = 40/3 = 13(1/3) days

৫১.
A, B and C each working alone can complete a job in 6, 8 and 12 days respectively. If all three of them work together to complete the job and earn $2340, what will be C's share of the earnings?
  1. ক
    ক) $1,100
  2. খ
    খ) $1,080
  3. গ
    গ) $630
  4. ঘ
    ঘ) $520
ব্যাখ্যা
Question: A, B and C each working alone can complete a job in 6, 8 and 12 days respectively. If all three of them work together to complete the job and earn $2340, what will be C's share of the earnings?

Solution: 
A, B এবং C 1 দিনে করতে পারে কাজটির 1/6,1/8, এবং 1/12 অংশ 

A, B এবং C কাজের অনুপাত = 1/6 : 1/8 : 1/12
= (1/6) × 24  : (1/8) × 24 : (1/12) × 24 
= 4 : 3 : 2
অনুপাতের রাশিগুলোর যোগফল = 4 + 3 + 2 = 9

C এর অংশ = 2340 এর 2/9
                   = 520 
৫২.
If three workers can mow a lawn in 4 hours, how many workers are needed to mow the lawn in 2 hours?
  1. ক
    3
  2. খ
    4
  3. গ
    6
  4. ঘ
    12
ব্যাখ্যা
Problem: If three workers can mow a lawn in 4 hours, how many workers are needed to mow the lawn in 2 hours?

Solution: 
৪ ঘণ্টায় শেষ করতে লোক লাগে ৩ জন 
২ ঘণ্টায় শেষ করতে লোক লাগে (৪ × ৩/২) জন 
= ৬ জন 
৫৩.
A, B, and C can do a piece of work in 8 days. B and C together do it in 24 days. B alone can do it in 40 days. In what time will it be done by C working alone?
  1. ক
    25 days
  2. খ
    24 days
  3. গ
    60 days
  4. ঘ
    20 days
  5. ঙ
    30 days
ব্যাখ্যা
Question: A, B, and C can do a piece of work in 8 days. B and C together do it in 24 days. B alone can do it in 40 days. In what time will it be done by C working alone?

Solution:
B & C do this work in 24 days.
B alone does this work in 40 days.

C alone can do work in 1 day 1/24 - 1/40 = 2/120 = 1/60 of the work
∴ C will take 60 days to finish the work.
৫৪.
5 men or 10 women can do a task in 20 days both. If 5 men and 5 women is assigned for the task, how much time will it take to complete it?
  1. ক
    12 days
  2. খ
    14 days
  3. গ
    20/3 days
  4. ঘ
    40/3 days
ব্যাখ্যা
Question:  5 men or 10 women can do a task in 20 days both. If 5 men and 5 women is assigned for the task, how much time will it take to complete it?

Solution: 
5 men can do in one day = 1/20

10 women can do in one day = 1/20
∴ 5 women can do in one day = 1/40

so in one day 5 men and 5 women can do = 1/20 + 1/40
= 3/40

∴ time required to do the full task = 40/3 days
৫৫.
Anik alone can do a piece of work in 6 days and Bishal alone in 8 days. Anik and Bishal undertook to do it for Tk. 4800. With the help of Dinesh, they completed the work in 3 days. How much is to be paid to Dinesh?
  1. ক
    Tk. 1375
  2. খ
    Tk. 1400
  3. গ
    Tk. 1600
  4. ঘ
    Tk. 600
ব্যাখ্যা
Question: Anik alone can do a piece of work in 6 days and Bishal alone in 8 days. Anik and Bishal undertook to do it for Tk. 4800. With the help of Dinesh, they completed the work in 3 days. How much is to be paid to Dinesh?

Solution:
Anik’s 1day work = 1/6
Bishal’s 1 day work = 1/8
(Anik + Bishal + Dinesh)’s 1 day work =1/3

Dinesh’s 1 day work = 1/3 - (1/6 + 1/8) = 1/24
So, Dinesh’s 3 day work = 3 × 1/24 = 1/8

If Dinesh contributed 8th part of work then he will receive 8th part of total payment
∴ 4800 × 1/8 = 600
৫৬.
Parking spaces are rented at Tk. 10 per week and Tk. 30 per month. Calculate the yearly savings from choosing the monthly rate rather than the weekly rate.
  1. ক
    Tk. 160
  2. খ
    Tk. 140
  3. গ
    Tk. 190
  4. ঘ
    Tk. 200
ব্যাখ্যা
Question: Parking spaces are rented at Tk. 10 per week and Tk. 30 per month. Calculate the yearly savings from choosing the monthly rate rather than the weekly rate.

Solution:
Tk. 10 per week
A year has 52 weeks.
Annual charges per year at Tk. 10 per week = 52 × 10 = 520

Tk. 30 per month
A year has 12 months.
Annual charges per year at Tk. 30 per month = 12 × 30 = 360

∴ Save = 520 - 360 = 160
৫৭.
A television service center can process 8 televisions in 18 minutes. How many televisions can it process in 3 hours at this rate? 
  1. ক
    150
  2. খ
    45
  3. গ
    80
  4. ঘ
    60
  5. ঙ
    None of these
ব্যাখ্যা

Question: A television service center can process 8 televisions in 18 minutes. How many televisions can it process in 3 hours at this rate? 

Solution:
3 hours = 3 × 60 = 180 minute

In 18 minutes, the service center can process = 8 tv
In 1 minute, the service center can process = 8/18 tv
In 180 minutes, the service center can process = (8 × 180)/18 tv
= 80 tv

৫৮.
A can do a certain work in 15 days. B is 25% more efficient than A. Both worked together for 4 days. C alone completed the remaining work in 8 days. A, B and C together will complete the same work in ?
  1. ক
    ক) 7 days
  2. খ
    খ) 5 days
  3. গ
    গ) 8 days
  4. ঘ
    ঘ) 9 days
ব্যাখ্যা
Work done by A = 15 days
B is 25% more efficient than A
A and B worked together = 4 days
C alone completed the remaining work = 8 days

Formula used:
Work done = Time × Efficiency

B is 25% more efficient than A
⇒ (15 × 100/125) 
⇒ 12 days

B can do the work on 12 days

Now,
LCM of 15 and 12 is 60

A's 1 day work = 4 units
B's 1 day work = 5 units

Work done by A and B work together for 4 days = (4 + 5) × 4 = 36 work

 Work left  = 60 – 36 = 24 units

C can do the remaining work = 24/8 units = 3 units

Now, A, B and C completed the whole work = 60/(4 + 5 + 3)

⇒ 60/12 days

⇒ 5 days

∴ A, B, and C completed the same work together in 5 days
৫৯.
A and B together complete a piece of work in x days. If A alone completes the work in x + 3 days and B alone completes the piece of work in x + 12 days, what is the value of "x"?
  1. ক
    3 days
  2. খ
    5 days
  3. গ
    6 days
  4. ঘ
    9 days
  5. ঙ
    Cannot be determined
ব্যাখ্যা
Question: A and B together complete a piece of work in x days. If A alone completes the work in x + 3 days and B alone completes the piece of work in x + 12 days, what is the value of "x"?

Solution:
A's 1 day's work = 1/(x + 3) part
B's 1 day's work = 1/(x + 12) part
and (A + B)'s 1 day's work = 1/x

ATQ,
1/(x + 3) + 1/(x + 12) = 1/x
⇒ (x + 12 + x + 3)/(x + 3)(x + 12) = 1/x
⇒ (2x + 15)/(x2 + 15x + 36) = 1/x
⇒ 2x2 + 15x = x2 + 15x + 36
⇒ 2x2 + 15x - x2 - 15x = 36
⇒ x2 = 36
∴ x = 6
৬০.
A does double the work of B in the same time. If they work together, they can dig a canal in 16 days. How many days would B take if he had to dig the same canal working alone?
  1. ক
    ক) 36 days
  2. খ
    খ) 24 days
  3. গ
    গ) 18 days
  4. ঘ
    ঘ) 48 days
ব্যাখ্যা
A does double the work of B in the same time

Number of days to dig canal = 16 days

Formula used:
Efficiency = Work done/Time


According to the question
The ratio of efficiency of A to B = 2 ∶ 1
Total work = (2 + 1) × 16 = 48

Number of days B alone takes to dig the canal = 48/1
⇒ 48 days

∴ The number of days B alone takes to dig the canal is 48 days.
৬১.
A monkey climbs a 10 meters-high slippery pillar. In his first minute, he climbs 2 meters, and in the next minute, he slip one meter down. In this way, how much time will he take to reach the top of the pillar?
  1. ক
    20 min
  2. খ
    21 min
  3. গ
    17 min
  4. ঘ
    19 min
ব্যাখ্যা
Question: A monkey climbs a 10 meters-high slippery pillar. In his first minute, he climbs 2 meters, and in the next minute, he slip one meter down. In this way, how much time will he take to reach the top of the pillar?

Solution: 
On first minute monkey climb = 2 m
On the second minute it slips = 1 m
For every two minute, it climbs 1 m
So, average speed = 1 m/2 min For 8 m,
time is taken = 16 min
For the last 2 m jump add 1 min
So time taken = (16 + 1) min
= 17 min

∴ Monkey takes 17 minutes to reach the top of the pole.
৬২.
A contractor undertakes to do a piece of work in 40 days. He engages 100 men at the beginning and 100 more  after 35 days and completes the work in stipulated time. If he had not  engaged  the additional men, how many days  behind schedule would it be finished?
  1. ক
    3
  2. খ
    5
  3. গ
    6
  4. ঘ
    9
  5. ঙ
    10
ব্যাখ্যা
Question: A contractor undertakes to do a piece of work in 40 days. He engages 100 men at the beginning and 100 more  after 35 days and completes the work in stipulated time. If he had not  engaged  the additional men, how many days  behind schedule would it be finished?

Solution:
[(100 × 35) + (200 × 5)]men can finish the work in 1 day
Therefore,
4500 men can finish the work in 1 day.
100 men can finish it in 4500/100 = 45 days.

∴ This is 5 days behind Schedule
৬৩.
If the cost of p metres of wire is Tk. d, then what is the cost of q metres of wire at the same rate (in Tk)?
  1. ক
    pq/d
  2. খ
    dp/q
  3. গ
    q/pd
  4. ঘ
    dq/p
ব্যাখ্যা
If the cost of p metres of wire is Tk. d, then what is the cost of q metres of wire at the same rate (in Tk)?

Solution:
Cost of p metres = Tk. d
Cost of 1 metre = Tk. d/p
Cost of q metres = Tk. dq/p
৬৪.
Working alone, pump A can empty a pool in 3 hours. Working alone, pump B can empty the same pool in 2 hours. Working together, how many minutes will it take pump A and pump B to empty the pool?
  1. ক
    ক) 72
  2. খ
    খ) 75
  3. গ
    গ) 84
  4. ঘ
    ঘ) 96
ব্যাখ্যা

Pump A can empty the pool in 3 hours therefore the rate at which it empties is 1/3 pool/hour
Pump b can empty the pool in 2 hours therefore the rate at which it empties is 1/2 pool/hour.
If they work together, the resulting rate is the addition of both rates (1/3 +1/2)pool/hour = 5/6 pool/hour

Now we have the following: (5/6pool)/60min = 1pool/x
Or, x = 72minutes

৬৫.
Running at the same constant rate, 10 identical machines can produce a total of 180 bottles per hour. How many bottles could 15 such machines produce in 30 minutes?
  1. ক
    ক) 120
  2. খ
    খ) 135
  3. গ
    গ) 150
  4. ঘ
    ঘ) 160
ব্যাখ্যা
Question: Running at the same constant rate, 10 identical machines can produce a total of 180 bottles per hour. How many bottles could 15 such machines produce in 30 minutes?

Solution:
In 60 minutes 10 machines can produce = 180 bottles
In 1 minute 1 machine can produce = 180/(60 × 10) bottles
In 30 minutes 15 machines can produce = (180 × 15 × 30)/(60 × 10) bottles
= 135 bottles
৬৬.
A is 30% more efficient than B. How much time will they, working together, take to complete a job which A alone could have done in 23 days?
  1. ক
    ক) 11 days
  2. খ
    খ) 13 days
  3. গ
    গ) 20(3/17) days
  4. ঘ
    ঘ) 15 days
  5. ঙ
    ঙ) None of these
ব্যাখ্যা

Ratio of times taken by A and B = 100:130 = 10:13
Suppose B takes x days to do the work.
Then,
10:13 :: 23:x
⇒ x = (23 × 13)/10
⇒ x = 299/10
A's 1 day's work = 1/23
B's 1 day's work = 10/299
(A + B)'s 1day's work = 1/23 + 10/299
= 23/299
= 1/13
∴ A and B together can complete the work in 13 days.

৬৭.
41 girls can complete a piece of work in 40 days. If x girls started the work and after 30 days 64 more girls joined them so that the whole work gets finished in the desired time, find the value of x.
  1. ক
    ক) 21
  2. খ
    খ) 25
  3. গ
    গ) 23
  4. ঘ
    ঘ) 27
ব্যাখ্যা
41 girls can complete a piece of work in 40 days.
x girls started the work 
And after 30 days 64 more girls joined them

Concept Used:
Total work = Efficiency × Time 
Here, we will take Efficiency = Number of workers
i.e Total work = Number of workers × Time 

Calculation:
Total work = 41 × 40 = 1640 units

According to the question

For the second case 
Total work = (x × 30) + (x + 64) × 10

So, we can write 
(x × 30) + (x + 64) × 10 = 1640 
⇒ 30x + 10x + 640 = 1640 
⇒ 40x = 1000
⇒ x = 25

∴ The value of x is 25.
৬৮.
A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last, is-
  1. ক
    29.5
  2. খ
    37.25
  3. গ
    42
  4. ঘ
    54
  5. ঙ
    None of these
ব্যাখ্যা
Question: A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last, is-

Solution:
fter 10 days : 150 men had food for 35 days.
Suppose 125 men had food for x days.

Now, Less men, More days (Indirect Proportion)
125 : 150 : : 35 : x
⇒ 125/150 = 35/x
⇒ 125 x = 150 × 35
⇒ x = (150 × 35)/125
∴ x = 42.
৬৯.
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
    320 Tk.
  2. খ
    400 Tk.
  3. গ
    450 Tk.
  4. ঘ
    560 Tk.
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

Now,
A's wages : B's wages : C's wages = (1/6) : (1/8) : (1/24)
= (1/6) × 24 : (1/8) × 24 : (1/24) × 24
= 4 : 3 : 1

∴ C's share (for 3 days) = 3 × (1/24) × 3200
= 400 Tk.

৭০.
A can do a work in 8 days and B in 12 days. If they work on it together for 4 days, then the fraction of the work that is left is:
  1. ক
    1/12 part
  2. খ
    5/8 part
  3. গ
    3/8 part
  4. ঘ
    1/6 part
ব্যাখ্যা

Question: A can do a work in 8 days and B in 12 days. If they work on it together for 4 days, then the fraction of the work that is left is-

Solution:
A's 1 day's work = 1/8
B's 1 day's work = 1/12

∴ (A + B)'s 1 day's work = (1/8 + 1/12) part
= (3 + 2)/24 part
= 5/24 part

∴ (A + B)'s 4 day's work = (5/24 × 4) part
= 5/6 part

Therefore, Remaining work = (1 - 5/6) part
= 1/6 part

৭১.
A certain number of men can finish a piece of work in 80 days. If there were 15 men less, it would take 20 days more for the work to be finished. How many men were there originally?
  1. ক
    75
  2. খ
    82
  3. গ
    100
  4. ঘ
    110
  5. ঙ
    None of these
ব্যাখ্যা
Question: A certain number of men can finish a piece of work in 80 days. If there were 15 men less, it would take 20 days more for the work to be finished. How many men were there originally?

Solution:
Originally let there be x men.
Less men, More days (Indirect Proportion)
Therefore, (x - 15) : x : : 80 : 100
⇒ (x - 15)/x = 80/100
⇒ (x - 15) × 100 = x × 80
⇒ 100x - 1500 = 80x
⇒ 20x = 1500
∴ x = 75
৭২.
A can complete a work in 24 days and B in 16 days. They work together for 6 days. How many more days will A take alone to finish the remaining work?
  1. ক
    15 days
  2. খ
    9 days
  3. গ
    12 days
  4. ঘ
    10 days
ব্যাখ্যা

Question: A can complete a work in 24 days and B in 16 days. They work together for 6 days. How many more days will A take alone to finish the remaining work?

Solution:
A একা কাজটি করতে পারে = 24 দিনে
∴ A এর একদিনের কাজ = 1/24 অংশ
এবং, 
   B একা কাজটি করতে পারে = 16 দিনে
∴ B এর একদিনের কাজ = 1/16 অংশ

∴ A ও B একসাথে একদিনের কাজ = (1/24) + (1/16) = (2 + 3)/48 = 5/48 অংশ
তারা 6 দিনে একসাথে কাজ করে = 6 × (5/48) = 5/8 অংশ

বাকি কাজ = 1 - (5/8) = 3/8 অংশ

অতএব,
A, 1/24 অংশ কাজ করে 1 দিনে 
∴ 3/8  অংশ কাজ করে = (24 × 3)/8 = 9 দিনে 

অতএব, A একা বাকি কাজ শেষ করতে ৯ দিন লাগবে।

৭৩.
A works twice as fast as B. If B can complete a work in 18 days independently, the number of days in which A and B can together finish the work is - 
  1. ক
    ক) 4 days 
  2. খ
    খ) 6 days 
  3. গ
    গ) 8 days 
  4. ঘ
    ঘ) 9 days 
ব্যাখ্যা
Question: A works twice as fast as B. If B can complete a work in 18 days independently, the number of days in which A and B can together finish the work is - 

Solution:
Ratio of rates of working of A and B = 2 : 1 
So, the ratio of time taken = 1 : 2

Since, B takes 18 days, A takes 9 days 

∴ (A + B)'s 1 day's work = (1/9) + (1/18) 
= 3/18
= 1/6 

∴ A + B can finish the work in (6/1) = 6 days
৭৪.
In a factory, 10 machines can produce 10 toys in 10 minutes. How many minutes will it take for one machine to produce one toy?
  1. ক
    1 minute
  2. খ
    10 minutes
  3. গ
    100 minutes
  4. ঘ
    1/10 minutes
ব্যাখ্যা

Question: In a factory, 10 machines can produce 10 toys in 10 minutes. How many minutes will it take for one machine to produce one toy?

Solution:
10টি মেশিন 10টি খেলনা তৈরি করে 10 মিনিটে।
1টি মেশিন 10টি খেলনা তৈরি করে = 10 × 10 মিনিটে 
1টি মেশিন 1টি খেলনা তৈরি করে = (10 × 10)/10 মিনিটে 
= 10 মিনিটে।

৭৫.
A and B together can complete a work in 10 days. A alone can complete it in 30 days. If B works only for half a day daily, then in how many days will A and B together complete the work?
  1. ক
    10 days
  2. খ
    12 days
  3. গ
    15 days
  4. ঘ
    18 days
ব্যাখ্যা

Question: A and B together can complete a work in 10 days. A alone can complete it in 30 days. If B works only for half a day daily, then in how many days will A and B together complete the work?

Solution:
দেওয়া আছে,
A ও B একসাথে কাজটি সম্পন্ন করে = 10 দিনে
সুতরাং, তাদের এক দিনের কাজ = 1/10 অংশ

A একা কাজটি সম্পন্ন করে = 30 দিনে
সুতরাং, A এর এক দিনের কাজ = 1/30 অংশ

অতএব, B এর এক দিনের কাজ = (A ও B এর একসাথে কাজ) - (A এর একা কাজ)
= 1/10 - 1/30 অংশ
= (3 - 1)/30 অংশ
= 2/30 অংশ
= 1/15 অংশ

B প্রতিদিন অর্ধেক দিন কাজ করলে, তার প্রতিদিনের কাজ হবে,
= (1/15)/2 অংশ
= 1/30 অংশ

এখন, A (পুরো দিন) এবং B (অর্ধেক দিন) একসাথে কাজ করলে তাদের প্রতিদিনের মোট কাজ হবে:
= (A এর এক দিনের কাজ) + (B এর প্রতিদিনের কাজ)
= 1/30 + 1/30 অংশ
= 2/30 = 1/15 অংশ

যেহেতু তারা প্রতিদিন কাজের 1/15 অংশ সম্পন্ন করে, তাই সম্পূর্ণ কাজটি সম্পন্ন করতে তাদের সময় লাগবে 15 দিন।

সুতরাং, তারা একসাথে 15 দিনে কাজটি সম্পন্ন করবে।

৭৬.
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 meters ?
  1. ক
    20 days
  2. খ
    12 days
  3. গ
    18 days
  4. ঘ
    15 days
ব্যাখ্যা
Question: 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 meters ?

Solution: 
7 men in 10 days can do 100m
1 man in 1 day can do (100/70)m 
14 men in 1 days can do (100 × 14)/70 m
= 20m 

10 women in 10 days can do 100m
1 women in 1 days can do (100/100)m
20 women in 1 days can do (100 × 20)/100m
= 20m

14 men and 20 women in one day can do = 20 + 20 m
= 40m

40m can be done in 1 days
1m can be done = (1/40) days
600m can be done = (600/40) days
= 15 days
৭৭.
If half kg of potatoes costs 80 paise, how many paise will 300 gm cost?
  1. ক
    40 paise
  2. খ
    48 paise
  3. গ
    86 paise
  4. ঘ
    94 paise
ব্যাখ্যা
Question: If half kg of potatoes costs 80 paise, how many paise will 300 gm cost?

Solution:
Let the required weight be x kg. Less weight, less cost (Direct Proportion)

500 : 300 : : 80 : x
⇒ 500/300 = 80/x
⇒ 500x = 24000
⇒ x = 24000/500
∴ x = 48.
The cost of 300 gm of potatoes will be 48 paise.
৭৮.
A pipe can fill a tank in 6 hours and another pipe can empty the tank in 12 hours. If both the pipes are opened at the same time,the tank can be filled in-
  1. ক
    10 hours
  2. খ
    16 hours
  3. গ
    12 hours
  4. ঘ
    14 hours
ব্যাখ্যা
Question: A pipe can fill a tank in 6 hours and another pipe can empty the tank in 12 hours. If both the pipes are opened at the same time,the tank can be filled in-

Solution:
1st pipe can fill  in 1 hour 1/6 of the tank
2nd pipe can empty in 1 hour 1/12 of the tank

∴  Both pipe can fill in 1 hour (1/6 - 1/12) of the tank
= (2 - 1)/12 of the tank
= 1/12 of the tank

∴ the tank can be filled in 12 hours
৭৯.
A ship was stocked with food to last for 40 days for 2500 sailors. However, some sailors could not board the ship and the food could last for 50 days. How many sailors could not board the ship?
  1. ক
    ক) 400
  2. খ
    খ) 500
  3. গ
    গ) 700
  4. ঘ
    ঘ) 1000
ব্যাখ্যা

Take work done = 1
Let number of sailors who could not board = S.
So sailors who boarded = 2500 - S
∴ 2500 sailors x 40 days x 1 = (2500 - S) sailors x 50 days x 1
∴ S = 500 = Sailors who could not board the ship.

[Men = M; Days = D; Time/Hours = T; Work = W
M1D1T1W2 = M2D2T2W1
Note that - W2 is on left side and W1 is on right side]

৮০.
A contractor employs 40 persons for doing a job in 60 days. After 20 days it was found that only one-fourth of work was finished. How many more persons are to be employed to finish the job as per schedule?
  1. ক
    ক) 10
  2. খ
    খ) 20
  3. গ
    গ) 40
  4. ঘ
    ঘ) 60
ব্যাখ্যা
Question: A contractor employs 40 persons for doing a job in 60 days. After 20 days it was found that only one-fourth of work was finished. How many more persons are to be employed to finish the job as per schedule?

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

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

∴ He needs = (60 - 40) = 20 more persons.
৮১.
A building contractor undertook to finish a certain work in 162 days and employed 150 men. After 72 days, he found that he already done 2/3 of the work. How many men can be discharged now, so that the work finish in time?
  1. ক
    ক) 70
  2. খ
    খ) 75
  3. গ
    গ) 80
  4. ঘ
    ঘ) 90
ব্যাখ্যা
Question: A building contractor undertook to finish a certain work in 162 days and employed 150 men. After 72 days, he found that he already done 2/3 of the work. How many men can be discharged now, so that the work finish in time?

Solution:
2/3 of the work done in 72 days by 150 men
∴ Day remaining = 162 - 72 = 90 days

Work remaining = 1- (2/3) = 1/3

In 72 days 2/3 of the work can be done by 150 men
In 1 day 1 of the work can be done by (150 × 3 × 72)/2 men
In 90 days 1/3 of the work can be done by = (150 × 3 × 72)/(2 × 3 × 90) men
= 60 men

∴ The number of men can be discharged = 150 - 60 = 90 men
৮২.
At a stationary shop, it costs Tk. 185 for 4 gel-pens, 8ball-point pens and 1 marker pen and Tk. 315 for 7 gel-pens, 15 ball-point pens and 1 marker pen. What would be the cost of 3 gel-pen, 3 ball-point pen and 3 marker pen?
  1. ক
    Tk. 55
  2. খ
    Tk. 120
  3. গ
    Tk. 155
  4. ঘ
    Tk. 165
ব্যাখ্যা
Question: At a stationary shop, it costs Tk. 185 for 4 gel-pens, 8 ball-point pens and 1 marker pen and Tk. 315 for 7 gel-pens, 15 ball-point pens and 1 marker pen. What would be the cost of 3 gel-pen, 3 ball-point pen and 3 marker pen?

Solution:
Let,
the cost of 1 gel-pen, 1 ball-point pen and 1 marker pen respectively x, y, z.
 
ATQ, 
7x + 15y + z = 315 .............. (1)
4x + 8y + z = 185 ............... (2)

From (1) - (2) we get,
3x + 7y = 130  ............ (3)
⇒ 6x + 14y = 260 .............. (4) [multiplied with 2]
 
From (1) - (4) we get,
x + y + z = 55
⇒ 3x + 3y + 3z = 55 × 3 = 165.

∴ The cost of 3 gel-pen, 3 ball-point pen and 3 marker pen is 165 taka
৮৩.
Rahim's regular pay is Tk 40 per hour up to 40 hours. Overtime is paid at twice the regular rate. If he was paid Tk 2400 in total, how many hours of overtime did he work? 
  1. ক
    5 hours
  2. খ
    10 hours
  3. গ
    7 hours
  4. ঘ
    9 hours
  5. ঙ
    11 hours
ব্যাখ্যা

Question: Rahim's regular pay is Tk 40 per hour up to 40 hours. Overtime is paid at twice the regular rate. If he was paid Tk 2400 in total, how many hours of overtime did he work?

Solution:
Rahim’s regular wage for 40 hours = (40 × 40) = 1600 Taka.
Amount earned from overtime = (2400 - 1600) Taka = 800 Taka.
Since the overtime rate is twice the regular hourly wage,
Total overtime hours worked = 800 ÷ (40 × 2) hours
= 800 ÷ 80
= 10 hours

৮৪.
A contractor undertakes to do a piece of work in 40 days. He engages 100 men at the begining and 100 more after 35 days and completes the work in stipulated time. If he had not engaged the additional men, how many days behind schedule would it be finished?
  1. ক
    4 days
  2. খ
    5 days
  3. গ
    3 days
  4. ঘ
    6 days
  5. ঙ
    7 days
ব্যাখ্যা
Question: A contractor undertakes to do a piece of work in 40 days. He engages 100 men at the begining and 100 more after 35 days and completes the work in stipulated time. If he had not engaged the additional men, how many days behind schedule would it be finished?

Solution:
Remaining days = 40 - 35 = 5 days
After engaging 100 men total men = 100 + 100 = 200

200 men can finish the work in = 5 days
1 men can finish the work in = 5 × 200 days
So, 100 men can do it in (5 × 200)/100 days
=10 days.

Now, he will be behind schedule by = 10 - 5 days
= 5 days 
৮৫.
If a team of 5 workers can assemble a motorcycle in 6 hours, how many hours would it take a team of 10 workers to assemble the same motorcycle, working at the same constant rate?
  1. ক
    180 minutes
  2. খ
    160 minutes
  3. গ
    150 minutes
  4. ঘ
    100 minutes
  5. ঙ
    90 minutes
ব্যাখ্যা

Question: If a team of 5 workers can assemble a motorcycle in 6 hours, how many hours would it take a team of 10 workers to assemble the same motorcycle, working at the same constant rate?

Solution:
Given,
5 workers can assemble a motorcycle in 6 hours
∴ 1 worker can assemble a motorcycle in (6 × 5) = 30 hours
∴ 10 workers can assemble a motorcycle in 30/10 = 3 hours
= 3 × 60 minutes
= 180 minutes

৮৬.
Nipa is 25% more efficient than Tima. Tima alone can build a craft in 25 days. Find the number of days taken by Nipa to finish the same piece of work?
  1. ক
    25 days
  2. খ
    16 days
  3. গ
    20 days
  4. ঘ
    22 days
ব্যাখ্যা
The ratio of times taken by Tima and Nipa
= 125 : 100
= 5: 4
Suppose Nipa takes x days to do the work.
5 : 4 = 25 : x
so, 5x= (4 x 25)
or, 5x = 100
x = 20 days
৮৭.
A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last is-
  1. ক
    ক) 50 days
  2. খ
    খ) 38 days
  3. গ
    গ) 48 days
  4. ঘ
    ঘ) 42 days
ব্যাখ্যা
প্রশ্ন: A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last is-

সমাধান: 
দিন বাকি আছে (৪৫ - ১০) দিন
= ৩৫ দিন

সৈন্য বাকি থাকে (১৫০ - ২৫) জন
= ১২৫ জন

১৫০ জনের চলবে ৩৫ দিন
∴ ১ জনের চলবে ৩৫ × ১৫০ দিন
∴ ১২৫ জনের চলবে (৩৫ × ১৫০)/১২৫ দিন
= ৪২ দিন
৮৮.
Mr. Ronaldo can finish a Task in 20 days and Mr. Messi can do in 30 days. With help of Mr. Salah, they did the job in 10 days only. Then how many days are necessary to complete the task by Mr. Salah?
  1. ক
    ক) 80 days
  2. খ
    খ) 60 days
  3. গ
    গ) 50 days
  4. ঘ
    ঘ) 70 days
ব্যাখ্যা
মনে করি, মি সালেহ x দিনে করতে পারে।
প্রশ্নমতে,
1/20 + 1/30 + 1/x = 1/10
=> 1/x = 1/10 - 1/20 - 1/30
=> 1/x = 1/60
=> x = 60
৮৯.
An employee pays Tk.100 for each day a worker and forfeits Tk.20 for each day he idle. At the end of 50 days, if the worker got Tk.2600, for how many days did the worker remain idle?
  1. ক
    ক) 18 days
  2. খ
    খ) 20 days
  3. গ
    গ) 21 days
  4. ঘ
    ঘ) 23 days
ব্যাখ্যা
Question: An employee pays Tk.100 for each day a worker and forfeits Tk.20 for each day he idle. At the end of 50 days, if the worker got Tk.2600, for how many days did the worker remain idle?

Solution: 
প্রতিদিন উপস্থিত থাকার জন্য পায় ১০০ টাকা এবং একদিন অনুপস্থিত থাকলে জরিমানা হয় ২০ টাকা।

যদি ৫০ দিন উপস্থিত থাকত তাহলে মোট পেত = ৫০ × ১০০ = ৫০০০ টাকা
তাহলে, কম পায় = ৫০০০ - ২৬০০ = ২৪০০ টাকা

একদিন অনুপস্থিত থাকলে মোট ক্ষতি হয় = ১০০ + ২০ = ১২০ টাকা

∴ মোট অনুপস্থিত ছিল = ২৪০০/১২০ = ২০ দিন
৯০.
‘A’ can complete 1/3rd of a work in 4 days. He worked alone for 6 days and left. The remaining work is completed by ‘B’ alone in 10 days. In how many days can ‘A’ and ‘B’ together complete the whole work?
  1. ক
    10.5 days
  2. খ
    4.5 days
  3. গ
    8 days
  4. ঘ
    7.5 days
ব্যাখ্যা
Question: ‘A’ can complete 1/3rd of a work in 4 days. He worked alone for 6 days and left. The remaining work is completed by ‘B’ alone in 10 days. In how many days can ‘A’ and ‘B’ together complete the whole work?

Solution:
A can complete 1/3 of the work in 4 days
A xan complete full work in 12 days
∴ In 6 days A complete 1/2 of work.

∴ B can complete 1/2 of the work in 10 days
∴ B can complete full work in 20 days

∴ In 1 day A and B together can work 1/12 + 1/20 = (5 + 3)/60 = 8/60 = 2/15 of the work

∴ A and B together can complete the full work in 15/2 = 7.5 days
৯১.
A contract is to be finished in 46 days and 117 men involved in it, each working 8 hours per day. After 33 days, 4/7 of the work is finished, how many additional men may be employed so that it may be completed in time, each man now working 9 hours a day?
  1. ক
    81
  2. খ
    85
  3. গ
    123
  4. ঘ
    196
ব্যাখ্যা
Question: A contract is to be finished in 46 days and 117 men involved in it, each working 8 hours per day. After 33 days, 4/7 of the work is finished, how many additional men may be employed so that it may be completed in time, each man now working 9 hours a day?

Solution:
Remaining work after 33 days = 1 - 4/7 = 3/7
Remaining period = 46 - 33 = 13 days
Now, we have
Less work, less man (directly proportion)
Less days, more men (inverse proportion)
More hours/days, less man (inverse proportion)
Now,
we can say that work = 4/7 : 3/7
Therefore, days = 13 : 33
And, hours/day = 9 : 8

(4/7) × 13 × 9 : (3/7) × 33 × 8 = 117 : x   [Where, x is total number of men after 33 days.]
⇒ (4/7) × 13 × 9 × x = (3/7) × 33 × 8 × 117
⇒ x = (3 × 33 × 8 × 117)/(4 × 13 × 9)
∴ x = 198

Therefore, extra men to be employed = 198 - 117 = 81
৯২.
If 10 men can dig a pit in 8 days, how many men are required to dig half the pit in 4 days?
  1. ক
    20 men
  2. খ
    10 men
  3. গ
    8 men
  4. ঘ
    5 men
ব্যাখ্যা
Question: If 10 men can dig a pit in 8 days, how many men are required to dig half the pit in 4 days?

Solution:
৮ দিনে ১ বা সম্পূর্ণ অংশ করে ১০ জন
১ দিনে ১ বা সম্পূর্ণ অংশ করে ১০ × ৮ জন
৪ দিনে ১ বা সম্পূর্ণ অংশ করে (১০ × ৮)/৪ জন
৪ দিনে ১/২ অংশ করে (১০ × ৮)/(৪ × ২) জন = ১০ জন
৯৩.
5 men undertook a piece of work and finished half the work in 18 days if two men drop out, in how many days the remaining work will be completed?
  1. ক
    30 days
  2. খ
    32 days
  3. গ
    34 days
  4. ঘ
    36 days
ব্যাখ্যা
Question: 5 men undertook a piece of work and finished half the work in 18 days if two men drop out, in how many days the remaining work will be completed?

Solution:
5 men completed half of the work in 18 days so the entire work will be completed in 36 days.
5 men' one day work will be = 1/36
One man's one day work = 1/(36 × 5) = 1/180

Two men drop out, so the three men have to complete the remaining work.
Three men's one day work will be = (1/180) × 3 = 1/60

1/60 part of the work is completed by three men in one day
1 or full part of the work is completed by three men in 60 day
1/2 part of the work is completed by three men in 60/2 = 30 day
৯৪.
An examiner checks 4 scripts in 5/3 hours. How many scripts can be check in 50 minutes?
  1. ক
    2
  2. খ
    3
  3. গ
    22/3
  4. ঘ
    25/6
  5. ঙ
    None of these
ব্যাখ্যা
প্রশ্ন: An examiner checks 4 scripts in 5/3 hours. How many scripts can be check in 50 minutes?

সমাধান:
একজন পরীক্ষক 5/3 ঘণ্টায় চেক করে = 4 টি স্ক্রিপ্ট
∴ একজন পরীক্ষক 1 ঘণ্টায় চেক করে = (4 × 3)/5 টি স্ক্রিপ্ট
∴ একজন পরীক্ষক 50/60 ঘণ্টায় চেক করে = (4 × 3 × 50)/(5 × 60) টি স্ক্রিপ্ট
= 2 টি স্ক্রিপ্ট 
৯৫.
A cricket team has won 40 games out of 60 played. It has 32 more games to play. How many of these must the team win to make it record 70% win for the season?
  1. ক
    ক) 20
  2. খ
    খ) 25
  3. গ
    গ) 23
  4. ঘ
    ঘ) 32
ব্যাখ্যা
মনেকরি,
অবশিষ্ট খেলারগুলোর মধ্যে x টিতে জিততে হবে। 

প্রশ্নমতে, 
40 + x = (60 + 32) এর 70%
40 + x = 92 এর 70/100
40 + x = 64.4 
x = 64.4 - 40 
x = 24.4  ≈ 25
৯৬.
A, B, and C completed a work costing Tk 2250. A worked for 6 days, B for 4 days, and C for 9 days. If their daily wages are in the ratio of 5 : 6 : 4, how much will A receive?
  1. ক
    ক) 600 Tk
  2. খ
    খ) 620 Tk
  3. গ
    গ) 750 Tk
  4. ঘ
    ঘ) 800 Tk
ব্যাখ্যা
Question: A, B, and C completed a work costing Tk 2250. A worked for 6 days, B for 4 days, and C for 9 days. If their daily wages are in the ratio of 5 : 6 : 4, how much will A receive?

Solution:
Let the daily wages of A, B, and C be Tk 5x, 6x, and 4x respectively.

Then, ratio of their amounts
=(5x × 6) : (6x × 4) : (4x × 9)
= 30 : 24 : 36
= 5 : 4 : 6

∴A's amount = 2250 × (5/15) = 750 Tk
৯৭.
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 alone?
  1. ক
    30
  2. খ
    35
  3. গ
    40
  4. ঘ
    42
  5. ঙ
    None of these
ব্যাখ্যা

Let, the first and 2nd worker need X days.
Third worker needs 2X days
ATQ, 1/X + 1/X + 1/2x = 1/12
or, (2 + 2 + 1)/2x = 1/12
or, x = 30

৯৮.
64 persons can dig a trench 50m long, 2m wide, and 2m deep in 5 days, working 12 hours daily. In how many days, working 8 hours daily, will 80 persons dig another trench 75m long, 4m wide, and 3m deep?
  1. ক
    18 days
  2. খ
    45 days
  3. গ
    27 days
  4. ঘ
    36 days
ব্যাখ্যা
Question: 64 persons can dig a trench 50m long, 2m wide, and 2m deep in 5 days, working 12 hours daily. In how many days, working 8 hours daily, will 80 persons dig another trench 75m long, 4m wide, and 3m deep?

Solution:
The volume of 50m long, 2m wide, and 2m deep is (50 × 2 × 2) m3 = 200 m3
The volume of 75m long, 4m wide, and 3m deep is (75 × 4 × 3) m3 = 900 m3

64 persons by working 12 hours daily can dig 200m3 in = 5 days.
∴ 1 person by working 1 hour daily can dig 1 m3 in = (5 × 64 × 12)/200 days.
∴ 80 persons by working 8 hour daily can dig 900 m3 in = (5 × 64 × 12 × 900)/(200 × 80 × 8) days.
= 27 days
৯৯.
Karim can complete 2/5th of a certain work in 4 hours. Rahim works three times as fast as Karim and finishes the remaining work. How many hours did Rahim work?
  1. ক
    2 hours
  2. খ
    3 hours
  3. গ
    4 hours
  4. ঘ
    6.5 hours
ব্যাখ্যা

Question: Karim can complete 2/5th of a certain work in 4 hours. Rahim works three times as fast as Karim and finishes the remaining work. How many hours did Rahim work?

Solution:
ধরি, সম্পূর্ণ কাজ = 1 অংশ

করিমের কাজের পরিমাণ = 2/5 অংশ
সময় = 4 ঘণ্টা

∴ পুরো কাজ (1 অংশ) করতে করিমের সময় লাগবে = 4 ÷ (2/5)
= 4 × 5/2
= 10 ঘণ্টা।

এখন, রহিম যেহেতু করিমের চেয়ে তিনগুণ দ্রুত কাজ করেন, তাই পুরো কাজটি সম্পন্ন করতে তার এক-তৃতীয়াংশ সময় লাগবে।

∴ পুরো কাজ করতে রহিমের সময় = 10 ÷ 3 = 10/3 ঘণ্টা।

বাকি কাজের পরিমাণ = 1 - (2/5) = 3/5 অংশ।

সুতরাং, রহিমের 3/5 অংশ কাজ করতে সময় লাগবে = (10/3) × (3/5) ঘণ্টা
= (10 × 3)/(3 × 5)
= 30/15
= 2 ঘণ্টা।

১০০.
A car with a 12-gallon gas tank used 1/2 of a full tank of gas to make a 150-mile trip. How many miles per gallon did the car average on the trip?
  1. ক
    30
  2. খ
    25
  3. গ
    12.5
  4. ঘ
    8.33
  5. ঙ
    6
ব্যাখ্যা
Question: A car with a 12-gallon gas tank used 1/2 of a full tank of gas to make a 150-mile trip. How many miles per gallon did the car average on the trip?

Solution:
For 150 mile trip the car used (12/2) = 6 gallon gas

Using 6 gallon the car travels 150 mile
∴ Using 1 gallon the car travels (150/6) mile
= 25 mile