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Time & Work

There are three techniques to solve the fundamental TIME and WORK calculation

1. Fraction Method

2. Percentage Method

3. LCM Method

Explaining the three methods using a common example – 

Suppose three persons A, B, and C are working on a project. If A, B, and C take 10, 20, and 40 days to complete the project individually, then how much does A, B, and C together take to complete the project? How much A and B will take? How much for B and C? Same for C and A?

Fraction Method

Let the whole work be 1

For A

Too many steps!!!!

Percentage Method

Calculating the number of days,

The number of steps is less than that of the Fraction Method!!!!

LCM Method

Calculating Efficiencies

A, B, C → 4 + 2 + 1 = 7

A, B → 4 + 2 = 6

B, C → 2 + 1 = 3

C, A → 1 + 4 = 5

Calculating the Number of days,

Least calculation, so Time and Energy are saved!!!!

If we find the reciprocal of time ratio, we will get an efficiency ratio because A, B, and C all are doing the same work.

In actual terms, work is also constant.

So, Efficiency X Time = Total Work (very important)

1. A can do a work in 18 days and B in 16 days. If they work on it together for 6 days then what fraction of work is left?

Correct option is:  Option a

2. A can do 50% of the job in 16 days, B can do ¼th of the job in 24 days. In how many days they can do 3/4th of the job working together?

a) 24

b) 16

c) 21

d) 18

Correct option is: Option d

3. P → 1/ 4 → 10 days

Q → 40% → 15 days

R → 1/3 → 13 days

S → 37.5 → 17 days

T → 6.66% → 3 days

Who is less efficient?

S is less efficient here.

4. A can dig (1/x)th part of the field in 20 hours, while A and B together can dig complete field in 60 hours, then find how many parts of the field B can dig alone in 20 hours?

Correct option is: Option d

5. A can paint a house in 55 days and B can do it in 66 days. Along with C, they did the job in 12 days only. Then C alone can do the job in how many days?

a) 15

b) 20

c) 25

d) 30

Correct option is: Option b

6. A man has three daughters. The man can do twice the work of his three daughters. The first and second daughters can do the work in 12 days and 18 days respectively. 

Then find out the time by the third daughter to finish the work?

Correct option is: Option a

7. Ajay is thrice as good a workman as Vijay and therefore is able to finish the job in 60 days less than Vijay. In how many days will they finish the job working together?

Correct option is: Option a

Aman and Biman can do 40% faster than Dhiman. If all of them complete the work in 28 days. How long would Biman take to complete the same work last?

a) 192 days

b) 190 days

c) 188 days

d) 196 days

Correct option is: Option a

9. Akshay takes 4 hours to plough as much field as Bhushan in 3 hours, while Chirag takes 6 hours to plough as much Bhushan in 5 hours. They all can plough the field in 22 hours working together, in what time Bhushan and Chirag together can plough the field?

a) 94

b) 31

c) 62

d) 72

Correct option is: Option b

10. In the beginning, Sam works at the rate such that he can finish a piece of work in 30 hours, but he actually works at the same rate for 24 hours. After that, he works at a rate such that he can complete the whole work in 20 hours. If Sam is to finish this work at a stretch, then how many hours will he take to complete the whole work?

a) 27

b) 28

c) 29

d) 30

Correct option is: Option b

11. Arjun and Bhim can complete a piece of work in 60 days. While Bhim and Nakul can complete the same amount of work in 40 days. The efficiency of Bhim is 25% more than that of Nakul. In how many days will Arjun and Nakul together complete the same work?

a) 36

b) 75

c) 72

d) 90

Correct option is: Option c

12. 22 boys and 19 girls can complete a piece of work in 12 hours, while 13 boys and 15 girls can complete the same work in 18 hours. Then, in how many hours 9 boys and 4 girls can complete the same work?

a) 30

b) 27

c) 36

d) 40

Correct option is: Option c

TIME – AND – WORK SOLUTIONS

1. A does the work in 18 days. B does it in 16 days. So, 

A → 18 days     

B → 16 days 

LCM of (18, 16) is 144 which is also the TOTAL WORK 

Efficiency of

That means, A can do 8 units of work in 1 day and B can do 9 units of work in 1 day. (Efficiency is the amount of work done by a person in 1 day).

If they do the work together, then they can complete (8 + 9) = 17 units of work in a day.

In 1 day, they together complete 17 units.

In 6 days, they together complete (17 X 6) units.

Answer: Option a is the correct one.

2. A can do 50% of work in 16 days.

B completes 1/4th of work (or 25%) in 24 days.

Let the total work be (the LCM of 24 and 72) 72.

Then the efficiencies (i.e. the amount of work done in 1 day) of A and B are 3 and 1 units respectively.

The efficiency of both A and B if they work together is 3 + 1 = 4 units.

Therefore, No. of days required by them to complete 75% of the work = 72/4 = 18 days.

So, S takes the maximum number of days to do the work. Therefore, he is less efficient.

Answer: ‘S’ is the required answer.

A and B (AB) digs the whole field in 60 hours.

Taking the LCM of 20x and 60 = 60x as the total work.

The efficiency of A is 3 and that of (AB) is x.

So, the efficiency of B is (x – 3).

Answer: Option d

5. A → 55 days

B → 66 days

ABC → 12 days

LCM (55, 66, 12) = 5 X 11 X 6 X 2 = 660

10, 12, 55 are the efficiencies of A, B and ABC respectively.

So, C’s efficiency = E(C)

E(C)

= E(ABC) – [E(A) + E(B)]

= 55 – (10 + 12) = 33

Therefore, C alone can do the work in =

Answer: Option b is correct.

D1 → 12 days

D2 → 18 days

LCM of total work = 36

22, 3, 2 are the efficiencies of M, D1 and D2 respectively.

Given:

M = 2(D1 + D2 + D3)

So, 22 = 2(3 + 2 + D3)

Or, D3 = 6 units (Efficiency)

Answer: Option a is the correct one.

Let Ajay took 1 day and Vijay took 3 days to complete the work. Ajay took 2 days less than Vijay. But in the problem, it is given that Ajay took 60 days less than Vijay.

2 days → 60 days

So, 1 day → 30 days

3 days → 90 days.

Therefore, Ajay and Vijay take 30 and 90 days respectively.

LCM (30, 90) = 90 is the total work.

3, 1 are the efficiency of A and V respectively.

Answer: Option a is the correct one.

8. Efficiency ratio

E(ABD) = 48

ABD takes 28 days.

We know that Total work can also be calculated as Efficiency X Time. 

So, Total Work = (48 X 28) units

E(B) = E(BD) – E(D) = 27 – 20 = 7

Answer: Option a is the correct one.

9. Time ratio and Efficiency ratio are inverse of each other.

So, directly calculating the E-ratio.

E(ABC) = 9 + 12 + 10 = 31

ABC takes 22 hours.

So, Total Work = E(ABC) X T(ABC) = 31 X 22

Answer: Option b is the correct one.

10. Let the efficiency of SAM be S and the new efficiency be S (new).

S X 30 days = S(new) X 20 day = Total Work.

According to the problem,

So, total numbers of days required = 24 + 4 = 28 days

Answer: Option b is the correct one.

11. A & B → 60 days

B & N → 40 days

From here, E(BN) can be 5 + 4 = 9 units

So, total work = E(BN) X T(BN) = (9 X 40) units.

E(AN) = E(AB) + E(BN) – 2 X E(B) = 6 + 9 – 10 = 5 units

Answer: Option c is the correct one.

12. Let the efficiencies of Boys and Girls be B and G respectively.

(22B + 19G) X 12 = (13B + 15G) X 18 = Total Work (W)

On Eq(i) – Eq(ii),

Answer: Option c is correct one.