Puzzle | Completion of Task

A man is assigned a task and doubles the amount of work he does each day. If he completes the entire task in 18 days, how many days does it take him to complete 25% of the task?

Check if you were right - full answer with solution below.

Solution:

Let the man do x units of work on Day 1. Since the work doubles each day, the amount of work done on Day 2 is 2x, Day 3 is 4x and so on.

 So, the total work done by the end of Day 18 is:

S = x + 2x + 4x + \dots + 2^{17}x

This is a geometric series with a first term x and a common ratio 2. The sum of the first 18 terms is:

S = x(1 + 2 + 4 + \dots + 2^{17}) = x(2^{18} - 1)

We are asked to find the day n when the man completes 25\% of the total work, i.e. \frac{1}{4}S

Let the cumulative work done by the end of day n be:

W_n = x(1 + 2 + 4 + \dots + 2^{n-1}) = x(2^n - 1)

Now, equating this to \frac{1}{4}S:

x(2^n - 1) = \frac{1}{4}x(2^{18} - 1)

Cancel x from both sides:

2^n - 1 = \frac{2^{18} - 1}{4}

Solving for2^n :

2^n = \frac{2^{18} - 1}{4} + 1

Since,

\frac{2^{18} - 1}{4} \ 2^{16} - \frac{1}{4},

2^n \approx 2^{16} \Rightarrow n = 16

Therefore, the man completes 25% of the task by the end of Day 16.

Simple Method:

When 100% of the work is completed in 18 days, and the work doubles every day.

50% of the work would be completed on day 17.

Therefore, 25% of the work would have been completed in 16 days.