Time and Work

I. Three labourers A. B and C can individually build or break a wall in 30 days, 60 days and 20 days respectively.

Q1: In how many days can A and B together build the wall?

Step 1: A’s 1 day work = \(\frac{1}{30}\), B’s 1 day work = \(\frac{1}{60}\)
Step 2: A + B’s 1 day work = \(\frac{1}{30} + \frac{1}{60} = \frac{2+1}{60} = \frac{3}{60} = \frac{1}{20}\)
Step 3: Time taken together = \(\frac{1}{\frac{1}{20}} = 20\) days
Answer: b. 20 days.

Q2: In how many days will the wall be built, if A builds it and B breaks it simultaneously?

Step 1: A’s 1 day work = \(\frac{1}{30}\), B’s 1 day destruction = \(-\frac{1}{60}\)
Step 2: Net work/day = \(\frac{1}{30} – \frac{1}{60} = \frac{2 – 1}{60} = \frac{1}{60}\)
Step 3: Time = \(\frac{1}{\frac{1}{60}} = 60\) days
Answer: b. 60 days.

Q3: In how many days will the wall built, if all the three labourers A, B and C are employed to build it?

Step 1: A’s 1 day work = \(\frac{1}{30}\), B’s 1 day work = \(\frac{1}{60}\), C’s 1 day work = \(\frac{1}{20}\)
Step 2: Combined work/day = \(\frac{1}{30} + \frac{1}{60} + \frac{1}{20}\)
LCM of 30, 60, 20 = 60 \[ = \frac{2}{60} + \frac{1}{60} + \frac{3}{60} = \frac{6}{60} = \frac{1}{10} \] Step 3: Time = \(\frac{1}{\frac{1}{10}} = 10\) days
Answer: a. 10 days.

Q4: How many days will it take to build the wall, if A and B work together to build it while C breaks it simultaneously?

Step 1: A’s 1 day work = \(\frac{1}{30}\), B’s = \(\frac{1}{60}\), C’s breaking = \(-\frac{1}{20}\)
Step 2: Net work/day = \(\frac{1}{30} + \frac{1}{60} – \frac{1}{20}\)
LCM of 30, 60, 20 = 60 \[ = \frac{2}{60} + \frac{1}{60} – \frac{3}{60} = \frac{3 – 3}{60} = \frac{0}{60} = 0 \] Step 3: Net work done = 0 ⇒ Wall will never be built
Answer: d. The wall will never be built.