The time taken by 4 men to complete a job is double the time taken by 5 children to complete the same job. Each man is twice as fast as a woman. How long will 12 men, 10 children and 8 women take to complete a job, given that a child would finish the job in 20 days.

Options

A. 2 days
B. `2(1)/(8)` days
C. 4 days
D. 1 day

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the relationship between men, children, and women
From the problem, we know:
- The time taken by 4 men to complete a job is double the time taken by 5 children.
- Each man is twice as fast as a woman.

Let’s denote:
- The time taken by 5 children to complete the job as \( T_C \).
- Therefore, the time taken by 4 men to complete the job is \( 2T_C \).

### Step 2: Find the work done by men and children
The work done can be represented as:
- Work done by 5 children in \( T_C \) days = 1 job.
- Work done by 4 men in \( 2T_C \) days = 1 job.

Let's find the rates of work:
- Rate of work of 5 children = \( \frac{1}{T_C} \) jobs per day.
- Rate of work of 4 men = \( \frac{1}{2T_C} \) jobs per day.

Since 4 men complete the same job in double the time of 5 children:
\[ \text{Rate of 4 men} = \text{Rate of 5 children} \]
\[ \frac{4M}{2T_C} = \frac{5C}{T_C} \]
This simplifies to:
\[ 4M = 10C \implies M = \frac{5}{2}C \]

### Step 3: Establish the relationship between men and women
Since each man is twice as fast as a woman:
\[ M = 2W \implies W = \frac{M}{2} \]

### Step 4: Set up the efficiency ratios
From the relationships we have:
- \( M : C = 5 : 8 \)
- \( M : W = 2 : 1 \)

### Step 5: Find a common ratio for men, women, and children
Let’s express everything in terms of men:
- For children, if \( M = 5 \), then \( C = 8 \) (from the ratio \( M:C = 5:8 \)).
- For women, if \( M = 2 \), then \( W = 1 \) (from the ratio \( M:W = 2:1 \)).

### Step 6: Scale the ratios to find the number of workers
We can scale these ratios to find a common base:
- Let’s take \( M = 10 \) (to simplify calculations):
- Then \( C = 16 \) (since \( \frac{10}{5} \times 8 = 16 \)).
- And \( W = 5 \) (since \( \frac{10}{2} \times 1 = 5 \)).

### Step 7: Calculate the total work done by 12 men, 10 children, and 8 women
Now we have:
- 12 men, 10 children, and 8 women.

Calculating their work rates:
- Rate of 12 men = \( 12 \times \frac{1}{2T_C} = \frac{12}{2T_C} = \frac{6}{T_C} \)
- Rate of 10 children = \( 10 \times \frac{1}{T_C} = \frac{10}{T_C} \)
- Rate of 8 women = \( 8 \times \frac{1}{W} = 8 \times \frac{1}{\frac{M}{2}} = \frac{16}{M} = \frac{16}{10} = \frac{8}{5T_C} \)

### Step 8: Combine the rates
Total rate of work:
\[ \text{Total Rate} = \frac{6}{T_C} + \frac{10}{T_C} + \frac{8}{5T_C} \]
To combine these, find a common denominator (which is \( 5T_C \)):
\[ = \frac{30}{5T_C} + \frac{50}{5T_C} + \frac{8}{5T_C} = \frac{88}{5T_C} \]

### Step 9: Find the time to complete the job
The total work is 1 job, so:
\[ \text{Time} = \frac{1 \text{ job}}{\text{Total Rate}} = \frac{1}{\frac{88}{5T_C}} = \frac{5T_C}{88} \]

### Step 10: Substitute \( T_C \)
Given that a child would finish the job in 20 days, \( T_C = 20 \):
\[ \text{Time} = \frac{5 \times 20}{88} = \frac{100}{88} = \frac{25}{22} \text{ days} \]

### Final Answer
The time taken by 12 men, 10 children, and 8 women to complete the job is approximately 1.14 days.

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