K is 4 times as fast as L in doing a certain job. Working together, they can complete the work in 24 days. In how many days can L alone complete the entire work if he works by himself from the beginning?

Introduction / Context:
This Time and Work problem involves two workers, K and L, whose efficiencies are given in a simple ratio. K is 4 times as fast as L, meaning K does four times as much work as L in the same time. We are also told how long they take together to complete the work. From this, we need to determine how long L alone would take to complete the same work.

Given Data / Assumptions:

• K is 4 times as fast as L (efficiency ratio K : L = 4 : 1).
• Working together, K and L complete the work in 24 days.
• Both work at constant rates.
• Total work is regarded as 1 unit.

Concept / Approach:
If K is 4 times as fast as L, then K's time to complete the work alone is one-fourth of L's time. Let L's time be T days, making K's time T / 4 days. Their daily work rates are therefore 1 / T and 4 / T respectively. The combined rate is the sum of these rates, and since together they take 24 days, the combined rate must be 1 / 24. We set up an equation in T and solve to get L's time.

Step-by-Step Solution:
Let L alone take T days to complete the work. Then K alone, being 4 times as fast, takes T / 4 days to complete the work. L's daily rate = 1 / T units per day. K's daily rate = 1 / (T / 4) = 4 / T units per day. Combined daily rate = 1 / T + 4 / T = 5 / T units per day. Working together, they complete the work in 24 days, so combined rate = 1 / 24 units per day. Thus, 5 / T = 1 / 24. Cross-multiplying: 5 * 24 = T * 1 ⇒ T = 120 days. Therefore, L alone takes 120 days to finish the work.

Verification / Alternative check:
If L takes 120 days alone, his rate is 1 / 120 units per day. K, being 4 times as fast, has time 120 / 4 = 30 days, so his rate is 1 / 30 units per day, which equals 4 / 120 units per day. Combined rate = 1 / 120 + 1 / 30 = 1 / 120 + 4 / 120 = 5 / 120 = 1 / 24. Thus, together they take 24 days, which matches the given condition and confirms our answer.

Why Other Options Are Wrong:
If L took 30 or 40 or 80 days, then K's corresponding times would not yield a combined rate of 1 / 24 when added. For example, if L took 80 days, K would take 20 days, and combined rate would be 1 / 80 + 1 / 20 = 5 / 80 = 1 / 16, implying a joint time of 16 days instead of 24. Only 120 days for L gives a consistent solution.

Common Pitfalls:
A common mistake is to misinterpret "K is 4 times as fast as L" as meaning K takes 4 times as much time as L, which is the opposite of the correct relationship. Always remember that if someone is faster, they take less time, and efficiencies are inversely proportional to time. Another error is to forget to convert K's time correctly to T / 4 before computing the rate 4 / T.

Final Answer:
L alone will complete the work in 120 days.