Step 1: Understanding the Question:
The topic for this problem is Time and Work. In such problems, work is considered a constant whole (usually represented as 1). The time taken to complete the work is inversely proportional to the efficiency or the "rate" of the person. We are given the combined rate of two people (A and B) and the individual rate of one (A). We need to isolate and find the individual rate of the second person (B) to determine how long it takes them to finish the task alone.
Step 2 : Key Formulas and approach:
1. Work Rate Formula: $\text{Rate} = \frac{1}{\text{Time taken}}$.
2. Individual Rate Isolation: $\text{Rate of B} = \text{Combined Rate (A+B)} - \text{Rate of A}$.
3. Time Recovery: $\text{Time for B} = \frac{1}{\text{Rate of B}}$.
The approach involves subtracting fractions to find the individual daily work capacity of person B.
Step 3 : Detailed Explanation:
First, we find the daily work done by A and B together. If they finish the work in 12 days, they complete $\frac{1}{12}$ of the work in a single day.
Next, we find the daily work done by A alone. Since A takes 20 days, A completes $\frac{1}{20}$ of the work in a single day.
To find B's daily work, we subtract A's contribution from the total daily progress: $\text{B's daily work} = \frac{1}{12} - \frac{1}{20}$.
To perform the subtraction, we need a Least Common Multiple (LCM) for 12 and 20. The LCM of 12 and 20 is 60.
We convert the fractions: $\frac{1}{12} = \frac{5}{60}$ and $\frac{1}{20} = \frac{3}{60}$.
Now subtract the numerators: $\frac{5}{60} - \frac{3}{60} = \frac{2}{60}$.
Simplifying $\frac{2}{60}$, we get $\frac{1}{30}$. This means B completes $\frac{1}{30}$ of the total task in one day.
To find the total number of days B needs to complete the whole task (1 unit), we take the reciprocal of the daily rate. So, $1 \div \frac{1}{30} = 30$ days.
This confirms that B is slightly slower than A, as 30 days is longer than A's 20 days, which makes sense given that their combined speed is 12 days.
Step 4 : Final Answer:
B alone can complete the work in 30 days, which matches option (C).