Problem Solving

Mikrislac wrote: ↑

Tue Aug 11, 2020 4:46 pm

Three workers are hired to build a wall. Worker A can build the wall in 10 days if they work alone. Worker B can demolish the wall in 15 days and worker C can demolish the wall in 20 days. Worker A starts work on the first day and worker B starts work on the second day. On the third day, worker A works again, followed by worker C who works on the fourth day. This work routine is repeated until the wall is fully built. How many days are needed for the wall to be fully built?

(A) 35 and 1/3 days

(B) 38 and 1/2 days

(C) 40 and 7/8 days

(D) 42 and 2/3 days

(E) 44 and 5/6 days

Solution:

Let’s determine how much of the wall is built in one cycle of 4 days, i.e., A-B-A-C:

1/10 - 1/15 + 1/10 - 1/20 = 6/60 - 4/60 + 6/60 - 3/60 = 5/60 = 1/12

One might assume that since it takes 4 days to complete 1/12 of the wall, then it will take 4 x 12 = 48 days to complete the entire wall. However, this thinking is not correct. Since whenever A completes the remaining portion of the wall that is less than 1/10 of the wall, then the entire wall is completed and there is no need for B or C to be counterproductive (recall that they are demolishing the wall until it’s fully built). Therefore, let’s say after 11 cycles, or 44 days, 1/12 x 11 = 11/12 has been built. Then:

On the 45th day, A builds another 1/10: 11/12 + 1/10 = 55/60 + 6/60 = 61/60

We see that this is already greater than 1, therefore, the wall can be completed in 44 days and a fraction of the 45th day, which means choice E is the correct answer. (Note: We will leave the readers to show that the fraction of the 45th day needed to complete the wall is 5/6.)

Answer: E