3 women and 18 children together take 2 days to complete a piece of work. How many days will 9 children alone take to complete the piece of work, if 6 women alone can complete the piece of work in 3days

Options

A. 9
B. 7
C. 5
D. 6

Answer

To solve the problem step by step, we will first determine the work done by women and children, and then calculate how long it will take for 9 children to complete the work alone.

### Step 1: Determine the total work done by 3 women and 18 children in 2 days.
Let the total work be represented as 1 unit of work.

Since 3 women and 18 children together complete the work in 2 days, we can express this as:
\[ \text{Work done in 1 day} = \frac{1 \text{ unit}}{2 \text{ days}} = \frac{1}{2} \text{ units of work per day} \]

### Step 2: Calculate the work done by 3 women and 18 children in terms of their individual work rates.
Let the work done by 1 woman in 1 day be \( W_w \) and the work done by 1 child in 1 day be \( W_c \).

Thus, the equation for the work done in 1 day can be expressed as:
\[ 3W_w + 18W_c = \frac{1}{2} \]

### Step 3: Determine the work done by 6 women in 3 days.
According to the problem, 6 women can complete the work in 3 days. Therefore, the work done by 6 women in 1 day is:
\[ \text{Work done by 6 women in 1 day} = \frac{1 \text{ unit}}{3 \text{ days}} = \frac{1}{3} \text{ units of work per day} \]
This gives us the equation:
\[ 6W_w = \frac{1}{3} \]

### Step 4: Solve for \( W_w \).
From the equation \( 6W_w = \frac{1}{3} \):
\[ W_w = \frac{1}{3 \times 6} = \frac{1}{18} \text{ units of work per day} \]

### Step 5: Substitute \( W_w \) back into the equation for work done by 3 women and 18 children.
Now we can substitute \( W_w \) back into the equation:
\[ 3 \left(\frac{1}{18}\right) + 18W_c = \frac{1}{2} \]
This simplifies to:
\[ \frac{1}{6} + 18W_c = \frac{1}{2} \]

### Step 6: Solve for \( W_c \).
Subtract \( \frac{1}{6} \) from both sides:
\[ 18W_c = \frac{1}{2} - \frac{1}{6} \]
To subtract these fractions, we need a common denominator (which is 6):
\[ \frac{1}{2} = \frac{3}{6} \]
Thus,
\[ 18W_c = \frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \]
Now, divide both sides by 18:
\[ W_c = \frac{1}{3 \times 18} = \frac{1}{54} \text{ units of work per day} \]

### Step 7: Calculate the total work done by 9 children in one day.
Now, we find the work done by 9 children in one day:
\[ \text{Work done by 9 children} = 9W_c = 9 \left(\frac{1}{54}\right) = \frac{9}{54} = \frac{1}{6} \text{ units of work per day} \]

### Step 8: Determine how many days it will take for 9 children to complete the work.
Since the total work is 1 unit, the number of days \( D \) taken by 9 children to complete the work is:
\[ D = \frac{1 \text{ unit}}{\frac{1}{6} \text{ units per day}} = 6 \text{ days} \]

### Final Answer:
Thus, 9 children alone will take **6 days** to complete the piece of work.

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