Question

5 women can complete a work in 8 days, while 8 children take 10 days to complete the same work. How many days will 2 women and 4 children take to complete the work?

• A. 10
• B. 12
• C. 40
• D. 8

Hint:

1. In time and work problems, always convert days into daily rates (fractions of work per day) to make addition straightforward.
2. Always verify the units: 1 woman-day rate is the reciprocal of the total days multiplied by the number of women.
3. Working with simplified fractions early on helps prevent calculation errors in the final steps.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
1. In this problem, we need to find the number of days required by a combined group of 2 women and 4 children to complete a specific task.
2. We are given the separate rates of completion for a group of 5 women and a group of 8 children.
3. To solve this, we must first calculate the individual work efficiency of a single woman and a single child per day.
4. Once the individual daily work rates are established, we can sum their rates for the new group composition and find the reciprocal of this combined daily work rate to get the final number of days.

Step 2: Key Formula or Approach:
1. Let \( W \) be the work rate (efficiency) of one woman per day, and \( C \) be the work rate of one child per day.
2. If \( N \) workers of the same efficiency complete a job in \( D \) days, then their combined rate is given by:
\[ N \times \text{Rate} \times D = 1 \text{ (complete work)} \]
3. The individual rate is therefore:
\[ \text{Rate} = \frac{1}{N \times D} \]
4. The combined rate of a new group is:
\[ \text{Combined Rate} = N_1 \cdot W + N_2 \cdot C \]
5. The total days required by the new group is:
\[ \text{Total Days} = \frac{1}{\text{Combined Rate}} \]

Step 3: Detailed Explanation:
1. Firstly, let us determine the daily work rate of one woman.
2. We know that 5 women can complete the work in 8 days.
3. This implies that the total work required can be represented in terms of woman-days as:
\[ 5 \text{ women} \times 8 \text{ days} = 40 \text{ woman-days} \]
4. Therefore, the efficiency or daily work done by a single woman is:
\[ W = \frac{1}{40} \text{ of the total work per day} \]
5. Secondly, let us find the daily work rate of one child.
6. We are told that 8 children can complete the same work in 10 days.
7. This gives the total work required in child-days as:
\[ 8 \text{ children} \times 10 \text{ days} = 80 \text{ child-days} \]
8. Therefore, the daily work done by a single child is:
\[ C = \frac{1}{80} \text{ of the total work per day} \]
9. Thirdly, we need to find the joint rate of 2 women and 4 children working together for one day.
10. The combined rate of 2 women and 4 children is:
\[ \text{Combined daily work} = 2W + 4C \]
11. Substituting the values of \( W \) and \( C \) into this equation:
\[ 2W + 4C = 2\left(\frac{1}{40}\right) + 4\left(\frac{1}{80}\right) \]
12. Simplifying each fraction individually gives:
\[ \frac{2}{40} = \frac{1}{20} \]
\[ \frac{4}{80} = \frac{1}{20} \]
13. Now, summing these two simplified rates together:
\[ \text{Combined daily work} = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10} \]
14. This means that 2 women and 4 children together can complete \( \frac{1}{10} \) of the entire work in a single day.
15. Fourthly, the total number of days required to finish the complete work is the reciprocal of their daily rate:
\[ \text{Number of days} = \frac{1}{\text{Combined daily work}} = \frac{1}{\frac{1}{10}} = 10 \text{ days} \]

Step 4: Final Answer:
1. The combined group consisting of 2 women and 4 children will take exactly 10 days to complete the given task.
2. Thus, the correct option matching this calculated result is Option (A).