Question

In 15 days, 9 women and 15 girls can reap a field. In what time could 15 women and 16 girls reap the same field, assuming 3 women can do as much work as 4 girls?

• A. 10.5 days
• B. 25 days
• C. 11.25 days
• D. 20 days

Hint:

In work problems with different types of workers, always start by converting all worker types into a single equivalent unit based on their given efficiency ratio. This transforms the problem into a simpler one with a single type of worker.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This is a time and work problem where we have two types of workers (women and girls) with a given efficiency relationship. We need to calculate the time required for a different combination of workers to complete the same task. 
Step 2: Key Formula or Approach: 
The core concept is that Total Work = (Number of Workers \( \times \) Efficiency of one worker) \( \times \) Time. A simpler form is Total Work = Total Work Rate \( \times \) Time. We will first establish the efficiency ratio, calculate the total work done, and then find the time for the new group. 
Step 3: Detailed Explanation: 
Part 1: Establish the efficiency ratio. We are given that 3 women can do as much work as 4 girls in the same amount of time. Let W be the work rate of one woman and G be the work rate of one girl. \[ 3 \times W = 4 \times G \] \[ W = \frac{4}{3} G \] This means one woman is as efficient as 4/3 of a girl. We will convert all workers into a single equivalent unit (e.g., girls). 
Part 2: Calculate the total work. The first group consists of 9 women and 15 girls, and they take 15 days. Let's find the total work rate of this group in terms of 'G'. \[ \text{Rate}_1 = (9 \times W) + (15 \times G) \] Substitute \(W = \frac{4}{3} G\): \[ \text{Rate}_1 = \left(9 \times \frac{4}{3} G\right) + (15 \times G) = (3 \times 4G) + 15G = 12G + 15G = 27G \] So, the combined work rate is equivalent to that of 27 girls. Total Work = Rate \( \times \) Time \[ \text{Total Work} = 27G \times 15 \text{ days} = 405G \text{ (girl-days)} \] Part 3: Calculate the time for the second group. The second group has 15 women and 16 girls. Let's find the total work rate of this new group in terms of 'G'. \[ \text{Rate}_2 = (15 \times W) + (16 \times G) \] Substitute \(W = \frac{4}{3} G\): \[ \text{Rate}_2 = \left(15 \times \frac{4}{3} G\right) + (16 \times G) = (5 \times 4G) + 16G = 20G + 16G = 36G \] The second group's work rate is equivalent to that of 36 girls. Now, we can find the time they will take to complete the same total work. \[ \text{Time}_2 = \frac{\text{Total Work}}{\text{Rate}_2} = \frac{405G}{36G} = \frac{405}{36} \] \[ \text{Time}_2 = \frac{45}{4} = 11.25 \text{ days} \] 
Step 4: Final Answer: 
It would take 15 women and 16 girls 11.25 days to reap the same field.