Time & Work, Chain Rule

Question: Rahim and Karim can finish a work together in 6 hours. If Rahim takes 3 times as long as Karim to finish the job alone, how long will Karim take to finish the job alone?

Solution:
ধরি, করিম একা কাজটি শেষ করতে সময় নেয় = x ঘন্টা
রহিম একা কাজটি শেষ করতে সময় নেয় = 3x ঘন্টা

এখন,
করিমের 1 ঘন্টার কাজের পরিমাণ = 1/x অংশ
রহিমের 1 ঘন্টার কাজের পরিমাণ = 1/(3x) অংশ

তারা একসাথে 1 ঘন্টায় কাজ করতে পারে = (1/x) + 1/(3x)
= (3+1)/(3x)
= 4/(3x) অংশ

একসাথে কাজটি শেষ করতে সময় লাগে = 6 ঘন্টা

অর্থাৎ, 6 ঘন্টায় তারা পুরো 1টি কাজ শেষ করে

তাহলে,
(4/3x) × 6 = 1
⇒ 24/(3x) = 1
⇒ 8/x = 1
⇒ x = 8

∴ করিমের কাজটি একা শেষ করতে 8 ঘন্টা সময় লাগবে।

Question: Karim and Bablu can build a wooden table in 3 days. Bablu can do it alone in 5 days. How many days would it take Karim to do this job alone?

Solution:
Let's say Karim's work rate per day = 1/x, where x is the days he takes alone
bablu's work rate per day = 1/5 (as he takes 5 days alone)
Together they take 3 days, so their combined rate = 1/3

ATQ,
(1/x) + (1/5) = 1/3 
⇒ (5 + x)/5x = 1/3
⇒ 15 + 3x = 5x
⇒ 15 + 3x - 5x = 0
⇒ 15 - 2x = 0
⇒ 2x = 15
∴ x = 7.5

Therefore, Karim would take 7.5 days to build the table alone.

Time taken by the tap to make the tank half full= 3 hrs.
Remaining part = 1/2
Part filled by 4 taps in 1 hour= (4×1/6) = 2/3
2/3 part is filled in 1 hour.
1/2 part is filled in (3/2×1/2) hr = 3/4 hr = 45 min.
Required time = 3hrs 45 min.

Question: A garrison of 600 men has provisions to last 30 days. After 3 days, 300 additional men arrive. How many days will the remaining food now last?

Solution:
After 3 days, food having for = (30 - 3) = 27 days
After arriving 300 men, total men = (600 + 300) = 900 men

600 men can eat the food for 27 days
∴ 1 man can eat the food for (27 × 600) days
∴ 900 men can eat the food for (27 × 600)/900 days
= 18 days

Take the LCM of days = LCM of (20, 30, and 60) = 60
Let the total work = 60

1 Man's one day work = 60/20 = 3 unit
1 Woman's one day work = 60/30 = 2 unit
1 The boy's one day work = 60/60 = 1 unit........................ (i)

ATQ, (2Men + 8Women + x Boy) can finish a work in 2 days.

Or, (2Men + 8women)'s 1 day work = (2 × 3) + (8 × 2) = 22

Or, (2Men + 8women)'s 2 day work = 22 × 2 = 44
Remaining work = 60 - 44 = 16
That means the remaining 16 work is done by X boys in 2 days
Or, X boys' one day work = 16/2 = 8

By equation 1, 1 boy completes 1 unit of work per day
So, 8 boys are required to complete 8 units of work per day.

Question: A group of 8 men can paint a house in 20 days. If they want to complete the painting in 16 days, how many more men should they add to the group?

Solution:
২০ দিনে কাজ শেষ করতে পারে ৮ জন
১ দিনে কাজ শেষ করতে পারে (২০ × ৮) জন
১৬ দিনে কাজ শেষ করতে পারে (২০ × ৮)/১৬ জন
= ১০ জন

∴ অতিরিক্ত (১০ - ৮) বা ২ জন যোগ করতে হবে।

In 18 minutes a car washer can wash 8 cars
In 1 minutes the car washer can wash 8/18 cars
So, in 3 hrs (180 minutes) car washer can wash = (8×180)/18 = 80 cars

Question: A barrack has enough food for 150 soldiers or 300 sailors. If 120 sailors have already taken the food, how many soldiers can be fed with the remaining food?

Solution:
এখানে,
300 জন নাবিকের খাবার = 150 জন সৈন্যের খাবার।

মোট নাবিক যাদের জন্য খাবার ছিল = 300 জন।
খাবার গ্রহণ করেছে = 120 জন।
অবশিষ্ট খাবার = (300 - 120) জন নাবিকের খাবার
= 180 জন নাবিকের খাবার।

এখন,
300 জন নাবিকের খাবার = 150 জন সৈন্যের খাবার।
∴ 1 জন নাবিকের খাবার = (150/300) জন সৈন্যের খাবার।
∴ 180 জন নাবিকের খাবার = (150 × 180)/300
= (1/2) × 180
= 90 জন সৈন্যের খাবার।

সুতরাং, অবশিষ্ট খাবার দিয়ে 90 জন সৈন্যকে দেওয়া যাবে।

Question: Rahim finishes 40% of a task in 10 days. He then asks Karim to help and together they complete the remaining work in 10 days. How many days would Karim take to complete the whole task alone?

Solution:
রহিম 40% বা 2/5 অংশ কাজ করে 10 দিনে।
∴ রহিম 1 দিনে কাজ করে = (2/5) ÷ 10 = 1/25 অংশ
∴ রহিম শেষ 10 দিনেও কাজ করেছে = 10 × (1/25) = 2/5 অংশ

মোট কাজ বাকি ছিল = 1 - 2/5 = 3/5 অংশ
এই 3/5 অংশ কাজের মধ্যে রহিম করেছে 2/5 অংশ
বাকি কাজ টুকু করিম করেছে = 3/5 - 2/5 = 1/5 অংশ

করিম 1/5 অংশ কাজ করে = 10 দিনে
∴ করিম সম্পূর্ণ (1 অংশ) কাজ একা করতে সময় নেবে = 10 × 5 দিন
= 50 দিন

Question: If 4 carpet weavers can complete 4 carpets in 4 days, how many carpets can 8 weavers produce in 8 days, assuming they work at the same constant rate?

Solution:
4 carpet-weavers in 4 days weave 4 carpets
∴ 1 carpet-weaver in 1 day weaves 4/(4 × 4) = 1/4 carpet

∴ 8 carpet-weavers in 8 days weave = (1/4 × 8 × 8) carpets
= 16 carpets

Question: 48 men can complete a piece of work in 18 days. In how many days can 36 men complete the same piece of work?

Solution:
Let the required number of days be x
Less men, More days (Indirect proportion)
∴ 36 : 48 : : 18 : x
⇔ 36/48 = 18/x
⇔ 36x = 18 × 48
⇔ x = (18 × 48)/36
∴ x = 24

Question: A can finish a work in 15 days, and B can do it in 25 days. They work together for 5 days. What fraction of the work remains unfinished?

Solution:
Work done by A in 1 day = 1/15
Work done by B in 1 day = 1/25

Combined work in 1 day = 1/15 + 1/25
= (5 + 3) / 75
= 8/75

Work done in 5 days = 5 × (8/75) = 40/75 = 8/15

Fraction of work left = 1 - 8/15 = 7/15 

Let,
The required number of days be x.
Less Cows, More days[Indirect proportion]
Less Bags, Less days [Direct proportion]
Cows(1 : 40), bags(40 : 1)}::40 : x
∴ 1 × 40 × x = 40 × 1 × 40
= 40.

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, 11/14 part of the work is done in = (11/14) × (140/11) = 10 days

(A +B)'s 20 day's work = 20 × 1/30 = 2/3
Remaining work = 1 − 2/3 = 1/3
Now,1/ 3 work is done by A in 20 days
∴ The whole work will be done by A
in 20×3 = 60days

Let work done by 1 man in 1 day = m and work done by 1 woman in 1 day = b
Work is done by 6 men and 8 women in 1 day = 1/10
⇒ 6m + 8b = 1/10
⇒ 60m + 80b = 1 --- (1)
Work done by 26 men and 48 women in 1 day = 1/2
⇒ 26m + 48b = 1/2
⇒ 52m + 96b = 1--- (2)
Solving equation 1 and equation 2.
We get m = 1/100 and b = 1/200
Work is done by 15 men and 20 women in 1 day
= 15/100 + 20/200 =1/4
⇒ Time is taken by 15 men and 20 women in doing the work = 4 days.

A's 2 day's work = (1/20) × 2
= 1/10
(A + B + C)'s 1 day's work = (1/20) + (1/30) + (1/60)
= 6/60
= 1/10
(A + B + C) work in 3 days = (1/10) + (1/10)
= 1/5.
Now, 1/5 work is done in 3 days.
∴ Whole work is done in (3 × 5)
= 15 days.

Question: A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 2 days. B can do the work alone in:

Solution:
Let
A, B and C take x, x/2 and x/3 days respectively to finish the work.

Now
(1/x) + (2/x) + (3/x) = 1/2
(1 + 2 + 3)/x = 1/2
⇒ 6/x = 1/2
⇒ x = 12.

So, B takes to finish the work = 12/2 days

Question: A worker’s regular pay is Tk. 15 per hour up to 40 hours. Overtime is paid at twice the regular rate. If he was paid Tk. 1,050, how many hours of overtime did he work?

Solution:

ধরি,
তিনি x ঘণ্টা overtime কাজ করেছেন।

সাধারণ বেতন (40 ঘণ্টার জন্য) = 15 × 40 = 600 টাকা
Overtime বেতন = 15 × 2 × x = 30x টাকা

মোট বেতন = সাধারণ বেতন + overtime বেতন

অর্থাৎ,
600 + 30x = 1,050
⇒ 30x = 1,050 - 600
⇒ 30x = 450
⇒ x = 450/30
∴ x = 15

∴ তিনি 15 ঘন্টা overtime করেছেন।

Question: A man, a woman, and a child together receive 84 Taka for 6 days' work. A woman's daily wage is twice a child's wage, and a man earns twice as much as a woman's. How much does a woman earn per day?

Solution: 
Let, 
C = Daily wage of the child (in Taka).
W = Daily wage of the woman (in Taka).
M = Daily wage of the man (in Taka).

A.T.Q, 
W = 2C
M = 2(2C) = 4C

Total payment of 6 days = 6 (C + W + M)
84 = 6(C + 2C + 4C)
42C = 84
∴ C = 2

So, the woman earns = 2 × 2 = 4 taka

Question: Since A is three times as efficient as B, he completes a task 40 days sooner. Find the time required for both to complete it working together.

Solution:
Let A alone takes x days 
and B alone takes 3x days to complete the job.

ATQ,
3x - x = 40
⇒ 2x = 40
⇒ x = 20

So, A alone takes 20 days and B alone takes 3 × 20 = 60 days to complete the job.

(A + B)'s 1 days work = (1/20 + 1/60) = 1/15 part

∴ A and B together can do the work in 15 days.

Ratio of times taken by A and B = 100 : 130 = 10 : 13.
Let, B takes x days to do the work.
ATQ, x : 23 = 13 : 10
Or, x = (23×13 / 10)
Or, x = 299/10
A’s 1 day’s work = 1/23
B’s 1 day’s work = 10/299
(A + B)’s 1 day’s work = ( 1/23 + 10/299 ) = 23/299 = 1/13 part
So, A and B together can complete the work in 13 days.

Question: Five machines working together can process 1200 items in 4 days. If one machine breaks down, how many items can the remaining machines process in 6 days?

Solution: 
Total items processed by 5 machines in 4 days = 1200
Items processed by 1 machine in 4 days = (1200/5) items
Items processed by 1 machine in 1 day = {1200/(5 × 4)} items
= 60 items

Therefore, 
Items processed by 4 machines in 6 days = 60 × 6 × 4
= 1440 items

After 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 × x = 150 × 35
⇒ x = (150 × 35)/125
= 42

Question: A worker earns Tk. 250 on the first day and spends Tk. 200 on the second day, earns Tk. 250 on the third day and again spends Tk. 200 on the fourth day and so on. On which day would he have had Tk. 1000?

Solution:
১ম দিনে আয় করে = ২৫০ টাকা
২য় দিনে ব্যয় করে = ২০০ টাকা
∴ প্রতি ২ দিনে জমা হয় = ২৫০ - ২০০ = ৫০ টাকা

শুধু ১ম দিনে আয় করে তার হাতে থাকে ২৫০ টাকা
তাহলে, ১০০০ - ২৫০ = ৭৫০ টাকা আরও জমা করতে হবে।

৫০ টাকা জমা হয় ২ দিনে
∴ ৭৫০ টাকা জমা হয় = (২ × ৭৫০)/৫০ = ৩০ দিনে

অর্থাৎ, ৩০ দিনের শেষে তার হাতে জমা থাকে = ৭৫০ টাকা
৩১তম দিনে সে আবার আয় করে = ২৫০ টাকা
⇒ মোট = ৭৫০ + ২৫০ = ১০০০ টাকা

∴ ৩১তম দিনে তার কাছে ১০০০ টাকা ছিল।

12 men can do a piece of work in 24 days
⇒ M₁ = 12 and D₂ = 24
8 men can do this work in D₂ days
⇒ M₂ = 8
M₁D₁ = M₂D₂
⇒ 12 × 24 = 8 × D₂
⇒ D₂ = (12 × 24)/8
= 36 days.

Question: If two men 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 men = 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

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 & Women 40:10} :: 10:x
∴ 100×40×x = 600×10×10
⇔ x = (600×10×10)(100×40)
⇔ x = 15

Question: Taposh and Sumon can finish a work together in 10 hours. If Taposh takes 4 times as long as Sumon to finish the job alone, how long will Sumon take to finish the job alone?

Solution:
ধরি, সুমন একা কাজটি শেষ করতে সময় নেয় = x ঘণ্টা
তাপস একা কাজটি শেষ করতে সময় নেয় = 4x ঘণ্টা

সুমনের 1 ঘণ্টার কাজের পরিমাণ = 1/x অংশ
তাপসের 1 ঘণ্টার কাজের পরিমাণ = 1/(4x) অংশ

তারা একসাথে 1 ঘণ্টায় কাজ করতে পারে = (1/x) + 1/(4x)
= (4 + 1) / (4x)
= 5 / (4x) অংশ

একসাথে কাজটি শেষ করতে সময় লাগে = 10 ঘণ্টা
অর্থাৎ, 10 ঘণ্টায় তারা পুরো 1টি কাজ শেষ করে

শর্তমতে,
(5/4x) × 10 = 1
⇒ 50/4x = 1
⇒ 4x = 50
⇒ x = 50 / 4
⇒ x = 12.5

∴ সুমনের কাজটি একা শেষ করতে 12.5 ঘণ্টা সময় লাগবে।

Question: Three workers can do a job in 12 days. Two of the workers work thrice as fast as the third. How long would it take one of the faster workers to do the job himself?

Solution:
Let,
The first and second worker need P days.
Third worker needs 3P days.

ATQ,
(1/P) + (1/P) + (1/3P) = 1/12
⇒ (3 + 3 + 1)/3P = 1/12
⇒ 7/3P = 1/12
⇒ 3P = 84
⇒ P = 84/3
∴ P = 28

So the one faster worker alone will take 28 days to complete the job.

A : B : C
Ratio of efficiency - 3 : 1 : 2
Ratio of No. of days - 1/3 : 1/1 : 1/2
or 2 : 6 : 3

Question: Six identical machines can produce 540 articles in 12 hours. How many articles would 8 such machines produce in 15 hours?

Solution: 
Total articles produced by 6 machines in 12 hours = 540.
Articles produced by 1 machine in 12 hours = 540/6
Articles produced by 1 machine in 1 hour = 540/(6×12) = 7.5 articles

So, Articles produced by 8 machines in 15 hours = 7.5 × 8 × 15 
= 900 articles

Let the required no of hours be x. Then
Less men , More hours (Indirect Proportion)
Therefore, 15:36 :: 25:x
=> (15 × x) = (36 × 25)
=> x = 60
Hence, 15 men can do it in 60 hours.

Question: A contractor undertook to complete a road project in 40 days. However, on the first day, 20 workers were absent. The remaining workers finished the project in 60 days. How many workers were originally supposed to work on the project?

Solution:
ধরি, শুরুতে শ্রমিকের সংখ্যা ছিল = x জন
মোট কাজ = x × 40 

যেহেতু 20 জন শ্রমিক অনুপস্থিত ছিল, তাই বর্তমান শ্রমিক সংখ্যা = (x - 20) জন।
তারা কাজটি শেষ করে 60 দিনে।
∴ বর্তমান মোট কাজ = (x - 20) × 60

শর্তমতে,
40x = 60(x - 20)
⇒ 40x = 60x - 1200
⇒ 60x - 40x = 1200
⇒ 20x = 1200
⇒ x = 1200/20
⇒ x = 60

∴ শুরুতে 60 জন শ্রমিক নিয়োগ করা হয়েছিল।

B's 10day's work = 1/15 × 10 = 2/3
Remaining work =
(1−2)/3 = 1/3
Now,
1/18 work is done by A in 1 day
∴ 1/3 work is done by A in 18 × 1/3 = 6 days

Question: Working 4 hours a day, P can complete a work in 8 days and working 8 hours a day, Q can complete the same work in 4 days. Working 8 hours a day, they can jointly complete the work in-

Solution:
Working 4 hours a day, P can complete the work in 8 days
∴ P can complete the work in = 4 × 8 = 32 hours
∴ P can complete in 1 hour = 1/32 part

Working 8 hours a day, Q can complete the work in 4 days
∴ Q can complete the work in = 8 × 4 = 32 hours
∴ Q can complete in 1 hour = 1/32 part

(P + Q)'s 1 hour's work,
(1/32) + (1/32) = 2/32 = 1/16

∴ P and Q can complete the work in 16 hours
So, working 8 hours a day they require = 16/8
= 2 days