The work done by 2 men in a day is equal to the work done by 3 children in a day. The work done by 3 men in a day is equal to the work done by 5 women in a day. It takes 10 days for a man, a woman and a child to complete a job working together. How many days will 2 children working together take to complete the same job?

Options

A. 30
B. 15
C. 17
D. 34

Answer

To solve the problem step by step, we will use the information provided to set up equations based on the work done by men, women, and children.

### Step 1: Define Variables
Let:
- The work done by 1 man in a day = \( m \)
- The work done by 1 woman in a day = \( w \)
- The work done by 1 child in a day = \( c \)

### Step 2: Establish Ratios from Given Information
From the problem statement:
1. The work done by 2 men in a day is equal to the work done by 3 children in a day:
\[ 2m = 3c \quad \Rightarrow \quad \frac{c}{m} = \frac{2}{3} \quad \Rightarrow \quad c = \frac{2}{3}m \]

2. The work done by 3 men in a day is equal to the work done by 5 women in a day:
\[ 3m = 5w \quad \Rightarrow \quad \frac{m}{w} = \frac{5}{3} \quad \Rightarrow \quad m = \frac{5}{3}w \]

### Step 3: Substitute to Find Relationships
Now, substitute \( m \) in terms of \( w \) into the equation for \( c \):
\[ c = \frac{2}{3}m = \frac{2}{3} \left(\frac{5}{3}w\right) = \frac{10}{9}w \]

### Step 4: Combine the Work Done
Now we have:
- Work done by 1 man in a day = \( m \)
- Work done by 1 woman in a day = \( w \)
- Work done by 1 child in a day = \( \frac{10}{9}w \)

### Step 5: Total Work Calculation
It takes 10 days for a man, a woman, and a child to complete a job together. The total work done can be calculated as follows:
\[ \text{Total work} = \text{(Work by 1 man + Work by 1 woman + Work by 1 child)} \times \text{Time} \]
\[ = (m + w + c) \times 10 \]
Substituting the values:
\[ = \left(m + w + \frac{10}{9}w\right) \times 10 \]
Substituting \( m = \frac{5}{3}w \):
\[ = \left(\frac{5}{3}w + w + \frac{10}{9}w\right) \times 10 \]
Finding a common denominator (which is 9):
\[ = \left(\frac{15}{9}w + \frac{9}{9}w + \frac{10}{9}w\right) \times 10 \]
\[ = \left(\frac{34}{9}w\right) \times 10 = \frac{340}{9}w \]

### Step 6: Work Done by 2 Children
Now, we need to find out how many days 2 children will take to complete the same job. The work done by 2 children in a day is:
\[ 2c = 2 \left(\frac{10}{9}w\right) = \frac{20}{9}w \]

### Step 7: Calculate Days Required
Let \( D \) be the number of days required for 2 children to complete the job:
\[ \text{Total work} = \text{Work done by 2 children in a day} \times D \]
\[ \frac{340}{9}w = \left(\frac{20}{9}w\right) \times D \]
Cancelling \( w \) from both sides:
\[ \frac{340}{9} = \frac{20}{9}D \]
Multiplying both sides by 9:
\[ 340 = 20D \]
Dividing both sides by 20:
\[ D = \frac{340}{20} = 17 \]

### Final Answer
Thus, 2 children working together will take **17 days** to complete the same job.

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