**Points to remember:**

1. Uniform rate of doing work is assumed

2. If a person can complete a work in n days, he will do (1/n) part of the work in 1 day

3. If tap A can do a work in m time units and B can complete the same work in n time units then, they can together finish the work (mn)/(m+n) time units

4. If the number of persons involved in doing a work changes in a particular ratio (Eg., 2:3) then, the time taken to complete the same work will change in the reverse ratio (Eg., 3:2)

5.. If A can work twice as fast as B then, A takes half of the time taken by B to complete the same work

6. If tap A can fill a tank in m time units and B can fill the same tank in n time units then, they can together fill the tank in (mn)/(m+n) time units

7. If tap A can fill a tank in m time units and B can empty the same tank in n time units then, they can together fill the tank in (mn)/(n-m) time units … (n>m)

#### Note:

We will use the following notations.

Work done by A in one time unit = A

Number of time units required by A to complete the work = a

**Example 1:**

A can do a piece of work in 40 days and B can complete the same work in 60 days. Working together, in how many days the work can be completed?

#### Solution:

a = 40, b = 60

A and B together can complete the work in (ab)/(a+b) days = (40×60)/(40+60) = 24 days

**Example 2:**

If one tap can fill a tank in 60 minutes and the other can empty it in 40 minutes, find the time taken to fill the tank if both the tanks are opened simultaneously.

#### Solution:

m = 40, n = 60 … (n to be > m)

Time taken to fill the tank equals (mn)/(n-m) minutes = (40×60)/(60-40) = 120 minutes

**Example 3:**

50 men together take ten hours to construct a wall. If 120 men are employed, how much time it will take to construct the same wall?

#### Solution:

Total man hours required to construct the wall = 50 x 10 =500

If 120 men take x hours then, 120x = 500

x = (500/120) = (25/6) hours = 60x(25/6) = 250 minutes

**Example 4:**

20 men and 30 women can do a piece of work in 80 days and 30 men and 20 women can do it in 70 days. If a woman is paid Rs. 30, what will be the pay for a man?

#### Solution:

Work done by one woman in one day = W

Work done by one man in one day = M

Number of human days required for the job = 80(20M+30W) = 70(30M+20W)

1600M + 2400W = 2100M + 1400W

1000W = 500M (or) 2W=1M

Therefore, if a woman is paid Rs. 30, a man will get (2×30) = Rs. 60.

**Example 5:**

A can do a work in 40 hours. He works at it for 5 hours and then B finishes the remaining work in 30 hours. In how many hours both A and B can finish the work together?

#### Solution:

Work done by A in one day = A

Work done by B in one day = B

Number of man hours required for the work = 40A

A works for 5 hours and B works for 30 hours.

Therefore, 40A=(5A+30B)

35A=30B (or) 7A=6B (or) A = B(6/7)

40A = Bx40(6/7) = 240B/7

That is, B requires (240/7) days to complete the same work.

If A and B work together then, the work can be completed in (mn)/(m+n) hours

= (40×240/7)/(40+(240/7))

= (40×240/7)/((280+240)/7)

= (40×240)/(520) = (960/52) = 18 (6/13) hours

**Example 6:**

A is twice as fast as B, and they together can finish a work in 140 hours. In how many hours A can do it alone?

#### Solution:

Work done by A in one day = A

Work done by B in one day = B

A=2B and B = (A/2)

Number of man hours required for the work = 140(A+B) = 140(A+A/2) = 140A+70A = 210A

That is, A alone can complete the same work in 210 hours.

**Example 7:**

A and B have promised to complete a job for Rs. 2000. A alone can complete the work in 24 hours, B alone can do it in 30 hours. To complete the work earlier, they involved C also. They all three together completed the job in 12 hours. Evaluate individual shares of amount.

#### Solution:

Speeds are directly proportional to distances covered

A can alone complete the job in 24 days.

That is, work completed by A in one day = (1/24)

A along with B and C work for 12 days

Hence, A could have completed 12x(1/24) = (1/2) of the total work.

Therefore, he will get half of Rs 2000 = Rs 1000.

B can alone complete the job in 30 days.

That is, work completed by B in one day = (1/30)

B along with A and C work for 12 days

Hence, B could have completed 12x(1/30) = (2/5) of the total work.

Therefore, B will get (2/5)th of Rs 2000 = Rs 800.

Therefore C will get Rs. (2000-1000-800) = Rs. 200

**Example 8:**

A alone can complete a work in 24 hours, B alone can do it in 30 hours. To complete the work earlier, they involved C also. They all three together completed the job in 12 hours. Evaluate the number of hours required to complete the job by C alone.

#### Solution:

A can alone complete the work in 24 hours

That is, work completed by A in one hour = (1/24)

B can alone complete the work in 30 hours

That is, work completed by B in one hour = (1/30)

Let, the number of hours required by C to complete the job = c

A, B and C can complete the work in 12 hours.

That is, in one hour, (1/12) part of the work will be completed

(1/24) + (1/30) + (1/c) = (1/12)

[30c+24c+(24×30)]/(24x30c) = (1/12)

(12x54c)+(12x24x30) = 24x30c

72c=12x24x30

c=120 hours

**Example 9:**

Rajan and Shankar together can finish a piece of work in 80 minutes. Both together started the work. After 30 minutes, Rajan left due to some emergency. Shankar continued and completed the remaining work in 150 minutes. How many minutes would Rajan alone take to finish the work?

#### Solution:

Number of minutes required by Rajan to complete the work = r

Number of minutes required by Shankar to complete the work = s

Together they take 80 minutes to complete the work

Therefore, (1/r) + (1/s) = (1/80)

Part of work completed by Rajan and Shankar in 30 minutes = 30/80 = 3/8

Remaining 1-(3/8) =(5/8) part of the work is completed by Shankar in 150 minutes.

Shankar alone can take 150x(8/5) = 240 minutes = r

(1/240) + (1/s) = (1/80)

(1/s) = (1/80) – (1/240) = (3-1)/240 = 1/120

s = 120 minutes.

**Example 10:**

P and Q can do a work in 24 days. Q and R can do the same work in 30 days. P and R can complete in 40 days. If all three work together, how long will they take to finish the work?

#### Solution:

Number of days required by P to complete the work = p

Number of days required by Q to complete the work = q

Number of days required by R to complete the work = r

(1/p) + (1/q) = (1/24) … (i)

(1/q) + (1/r) = (1/30) … (ii)

(1/p) + (1/r) = (1/40) … (iii)

Adding equations (i), (ii) and (iii),

2[(1/p) + (1/q) + (1/r)] = (1/24) + (1/30) + (1/40)

= [(30×40) + (24×40) + (24×30)]/(24x30x40)

We have to evaluate, inverse of [(1/p) + (1/q) + (1/r)]
[(1/p) + (1/q) + (1/r)] = (1200+960+720)/2×28800

Inverse of [(1/p) + (1/q) + (1/r)] = (2×28800)/2880

= 20 days.

**About Author**

#### Dr. BASKAR .A,

a former Research Scientist Government of India, presently working as a Professor in the Dept of Mechanical Engineering, Authored books on Kinematics and Dynamics of Machinery, Also Published Numerous Research papers moreover a Geek in Mathematics