Time & Work, Chain Rule

A's 5 days work = 50%
B's 5 days work = 33.33%
C's 2 days work = 16.66% [100 - (50 + 33.33)]

Ratio of contribution of work of A, B and C = 50: 33(1/3) : 16(2/3)
= 3 : 2 : 1

A's total share = Tk. 1500
B's total share = Tk. 1000
C's total share = Tk. 500

A's one day's earning = Tk.300
B's one day's earning = Tk.200
C's one day's earning = Tk.250

When the hands of the clock are in the same straight line but not together, they are 30 minute spaces apart.
At 7 o'clock, they are 25 min. spaces apart.
∴ Minute hand will have to gain only 5 min. spaces.
55 min. spaces are gained in 60 min.
5 min. spaces are gained in
=(60/55)×5min.
= 5(5/11) min.
∴ Required time = 5 (5/11) min. past 7

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

Question: X can complete a work in 12 days, and Y alone can do it in 18 days. They work together for 6 days, and Z completes the remaining work in 3 days. If the total payment for the work is Tk. 600, how much should Z get?

সমাধান:
X-এর একদিনের কাজ = 1/12
Y-এর একদিনের কাজ = 1/18

X ও Y একসাথে ৬ দিন কাজ করে:
= 6 × (1/12 + 1/18)
= 6 × {(3 + 2)/36} = 6 × (5/36) = 30/36 = 5/6

∴ বাকি কাজ = 1 − 5/6 = 1/6

Z এই 1/6 কাজ ৩ দিনে করেছে, অর্থাৎ Z-এর কাজ = 1/6

X-এর কাজ = 6 × 1/12 = 1/2
Y-এর কাজ = 6 × 1/18 = 1/3
Z-এর কাজ = 1/6

তাহলে অনুপাত = 1/2 : 1/3 : 1/6
= 3 : 2 : 1

মোট টাকা = 600
Z-এর অংশ = 1/(3+2+1) = 1/6
∴ Z পাবে = 600 × (1/6) = 100 টাকা

Working 5 hours a day, A can complete the work in 8 days i.e.
= 5 × 8 = 40 hours
Working 6 hours a day, B can complete the work in 10 days i.e.
= 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 i.e. they require 3 days to complete the work.

Question: A tank is filled using 20 buckets, each having a capacity of 10 liters. How many buckets would be required to fill the same tank if each bucket can hold 40 liters of water?

Solution:

Total capacity of the tank = 20 × 10 = 200 liters

∴ Number of buckets needed with 40-liter buckets = 200 ÷ 40
= 5 buckets

We know,
Indirect proportion: Less cows (↓) More days (↑)
Direct proportion: Less bags (↓) Less days (↓)

20 : x :: { 1 : 20 ------ (Cows); 20 : 1 -------(Bags)}
1 × 20 × x = 20 × 1 × 20
⇒ x = (20 × 1 × 20)/(1 × 20)
⇒ x = 20 Days.

A in 3 days can do = 1/4 of a work
∴ In 1 day = 1/12 of a work
B in 4 days can do = 1/6 of a work
∴ In 1 day = 1/24 of a work
In terms of ratio, A : B = 1/12 : 1/24 = 2 : 1
∴ A gets = 180 × 2/(2+1) = Tk. 120

A ও B একত্রে 1 দিনে করে = (1/20 + 1/30) = 1/12 অংশ
B একা 10 দিনে করে = 10/30 অংশ = 1/3 অংশ
অবশিষ্ট কাজ = (1 - 1/3) = 2/3 অংশ

এখন, A ও B একত্রে 1/12 অংশ করে 1 দিনে
A ও B একত্রে 1 অংশ করে = (12 × 1) দিনে
A ও B একত্রে 2/3 অংশ করে = (12 × 2) /3 = 8 দিনে
∴ মোট সময় লাগে = (10 + 8) দিন = 18 দিন

Question: A takes three times as much time as B or twice as much time as C to finish a piece of work. Working together, they can finish the work in 3 days. B can do the work alone in how many days?

Solution:
ধরা যাক,
A, B ও C যথাক্রমে কাজটি শেষ করতে সময় লাগে x, x/3, এবং x/2 দিন।

তারা একসাথে ৩ দিনে কাজ শেষ করে।
অর্থাৎ তাদের একদিনের কাজ হলো 1/3।

∴ A + B + C এর একদিনের কাজ = (1/x) + (3/x) + (2/x)
= (1 + 3 + 2)/x
= 6/x

এখন,
6/x = 1/3
∴ x = 18

∴ B একা কাজ শেষ করতে সময় নেবে = 18/3 = 6 দিন।

Question: Ali can type 60 pages in 20 minutes. Sara can type 12 pages in 12 minutes. Working together, how many pages can they type in 30 minutes?

Solution:
Ali can type in 1 min = 60/20 = 3 pages
Sara can type in 1 min = 12/12 = 1 page

∴ Working together they can type in 1 min = (3 + 1) pages = 4 pages

∴ They can type in 30 min = 4 × 30 pages = 120 pages

Let,
The required number of days be x.
Less pumps, More days[Indirect proportion]
Less weight, Less days [Direct proportion]
More hours/days, Less days [Indirect proportion]
{Pumps(16 : 18) , Weight (2170 : 1736), Hours/Day(9 :7)} :: 10 : x
∴ (16 × 2170 × 9 × x) = (18 × 1736 × 7 × 10)
⇒ x = (18 × 1736 × 7 × 10)/(16 × 2170 × 9)
= 7.

If 'w₁' work is done by 'm₁' men by working for 'h₁' hours per day in 'd₁' days & 'w₂' is work done by men 'm₂' working for 'h₂' hours per day in 'd₂' days,
then,
m₁d₁h₁/w₁ = m₂d₂h₂/w₂

Since we need to find 'd₂', we can rearrange the formula as
d2 = (m₁d₁h₁w₂)/(m₂d₂h₂w₁)
= (6 x 7 x 7 x 18)/(14 x 9 x 12)
= 3.5 days

Question: Robi and Bobi together can paint a wall in 16 days. Robi can do it alone in 20 days. How many days would it take Bobi to do this work alone?

Solution:
Robi's 1 day's work 1/20
Robi's and Bobi's 1 day's work 1/16
∴ Bobi's 1 day's work = (1/16) - (1/20)
= (5 - 4)/80
= 1/80

1/80 part of the job done by Bobi in 1 day
∴ Full work done by Bobi in (80/1) days
= 80 days

Question: Rakib can do a piece of work in 15 days, Rakib and Asif together can do in 12 days. If Asif does the work only for half a day daily then in how many days the work will be completed ?

Solution: 

Rakib's 1 day work = 1/15

Since, Rakib and Asif can together complete in 12 days
i.e. (Rakib + Asif)'s 1 day work = 1/12

Then,
Asif's 1 day work,
= (1/12) - (1/15) = (5 - 4)/60 = 1/60

If Asif Works only for half a day daily, then his 1 day work becomes (1/2) × (1/60)
  = (1/120)

Therefore, 1 day work of both Rakib and Asif,
=(1/15) + (1/120) = ( 8 + 1)/120 = 9/120

Hence, the work will be completed in 120/9 = 40/3 days

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

We are 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 x 3 = 24 hours

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

Question: Akib can do 1/6 of a work in 7 days. In how many days will he complete the work?

Solution:
Akib can do 1/6 of a work in 7 days
∴ he will complete the work in (6 × 7) days
= 42 days

∴ Akib will complete the work in 42 days.

Let each tree take T amount of water every day.
So 72 trees take 72T water in one day.

With 10% reduction each tree will consume 90T/100 amount each day.
So 90 trees take 90 (90T/100) amount in one day
Total water quantity is constant
∴ 72T × 54 = 90 × (90T/100) × D
∴ D = 48 days = Number of days 90 trees can use the water.

Wages of 1 woman for 1 day = 21600/(40 × 30)
Wages of 1 man for 1 day = (21600 × 2)/(40 × 30)
Wages of 1 man for 25 days = (21600 × 2 × 25)/(40 × 30)
Number of men = 14400/{(21600 × 2 × 25)/(40 × 30)}
= 144/{(216 × 50)/(40 × 30)}
= 144/9
= 16.

Question: A water pump fills a tank in 8 hours. If two identical pumps work together, how long will it take to fill the same tank?

Solution: 
One pump fills the tank in 8 hours.
Rate of one pump = 1/8 per hour

If two identical pumps work together, their rates add up.
So, the combined rate is = 1/8 + 1/8
= 1/4 per hour

∴ Time = 1/(1/4) = 4 hours

Question: If 40 workers can construct a road in 30 days, how many workers are needed to construct the same road in 20 days?

Solution:
Here,
M₁ = 40, M₂ = ?, D₁ = 30, D₂ = 20

∴ (M₁ × D₁) = (M₂ × D₂) 
⇒ (40 × 30) = (20 × M₂)
⇒ M₂ = (40 × 30)/20 
⇒ M₂ = 60

So 60 workers are needed to construct the same road in 20 days.

Question: 3 pumps, working 8 hours a day, can empty a tank in 2 days, How many hours a day must 4 pumps work to empty the tank in 1 day?

Solution:
3 pumps, working 8 hours a day, can empty a tank in 2 days
Formula used: M₁ × T₁ = M₂ × T₂
Where M₁ and M₂ is men and T₁ and T₂ is time

Calculation:
Let H hours be the number of hours required Applying the above formula
⇒ 3 × 8 × 2 = 4 × 1 × H
⇒ H = 48/4
⇒ H = 12 hours

∴ 4 pump need to work 12 hours to complete the work in 1 day.

Question: If 3 men or 9 boys can do a job in 60 days, how long will it take 11 men and 27 boys to finish the same job?

Solution:
9 জন বালক = 3 জন পুরুষ 

1 জন বালক = 3/9 = 1/3 জন পুরুষ 
∴ 27 জন বালক = 27/3 = 9 জন পুরুষ 

11 জন পুরুষ + 27 জন বালক = (11 + 9) = 20 জন পুরুষ 

3 জন পুরুষ কাজটি করতে সময় নেয় = 60 দিন
∴ 1 জন পুরুষ কাজটি করতে সময় নেয় = (60 × 3) দিন
∴ 20 জন পুরুষ কাজটি করতে সময় নেয় = (60 × 3)/20 = 9 দিন 

Question: 'A' can do a work in 12 days, and 'B' in 24 days. They work together for 6 days. How much of the work is left?

সমাধান:
মনে করি,
সম্পূর্ণ কাজ = 1 অংশ

∴ A একা একদিনে করে = 1/12 অংশ।
B একা একদিনে করে = 1/24 অংশ।

∴ A ও B একসাথে একদিনে করে = (1/12) + (1/24) অংশ
= (2 + 1)/24
= 3/24
= 1/8 অংশ

∴ A ও B একসাথে 6 দিনে করে = 6 × (1/8) অংশ
= 3/4 অংশ

∴ কাজ বাকি থাকে = 1 - (3/4) অংশ
= (4 - 3)/4
= 1/4 অংশ

Question: If X and Y can complete a work together in 12 days, Y and Z together in 20 days, and X and Z together in 15 days, then in how many days can Y alone complete the work?

Solution:
মনে করি, সম্পূর্ণ কাজ = 1 অংশ

∴ (X + Y) একদিনে করে = 1/12 অংশ ..........(1)
(Y + Z) একদিনে করে = 1/20 অংশ ..........(2)
(X + Z) একদিনে করে = 1/15 অংশ ..........(3)

(1), (2), (3) যোগ করে পাই,
2 × (X + Y + Z) = (1/12) + (1/20) + (1/15)
⇒ 2 × (X + Y + Z) = (5 + 3 + 4)/60 = 12/60
⇒ 2 × (X + Y + Z) = 1/5
⇒ (X + Y + Z) = 1/10

∴ X, Y এবং Z একসাথে একদিনে করে = 1/10 অংশ

(X + Z) একসাথে একদিনে করে = 1/15 অংশ
∴ Y-এর একদিনের কাজ = (1/10) - (1/15) অংশ
= (3 - 2)/30 
= 1/30

অর্থাৎ, Y সম্পূর্ণ কাজ করে = 1 ÷ (1/30) = 30 দিনে

Question: If 500 laborers can complete a work in 48 days, determine the number of extra laborers required to finish the same work in 40 days.

Solution:
48 দিনে কাজটি করতে শ্রমিক লাগে = 500 জন
∴ 1 দিনে কাজটি করতে শ্রমিক লাগে = (500 × 48) জন লোক
∴ 40 দিনে কাজটি করতে শ্রমিক লাগে = (500 × 48)/40 জন
= 600 জন

∴  অতিরিক্ত শ্রমিক লাগবে = (600 - 500) জন = 100 জন

To complete work in 18 days we need either 3 men or 6 boys.
∴ 1 man = 2 boys
Take work done = 1
∴ 3 men x 18 days x 1 = (4 men + 4 boys) x ? days x 1
∴ 6 boys x 18 days x 1 = (8 boys + 4 boys) x ? days x 1 [Convert either all men to boys or all boys to men.]
∴ ? = 9 days = they will need these many days.

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

Question: 12 workers can dig a 36-meter long trench working 8 hours per day. How many extra workers are required to dig a 45-meter long trench working 5 hours per day?

Solution:
8 ঘণ্টা কাজ করে 36 মিটার পরিখা খনন করে 12 জন শ্রমিক।
1 ঘণ্টা কাজ করে 1 মিটার পরিখা খনন করতে প্রয়োজন = (12 × 8)/36 জন শ্রমিক
5 ঘণ্টা কাজ করে 45 মিটার পরিখা খনন করতে প্রয়োজন = (12 × 8 × 45)/(36 × 5) জন শ্রমিক
= (12 × 8 × 45)/180 জন শ্রমিক
= 4320/180 জন শ্রমিক
= 24 জন শ্রমিক

∴ অতিরিক্ত শ্রমিকের সংখ্যা = 24 - 12 = 12 জন