A man, a woman or a boy can do a job in 20 days, 30 days or 60 days respectively. How many boys must assist 2 men and 8 women to do the work in 2 days?

Options

A. 15 boys
B. 8 boys
C. 10 boys
D. None of these

Answer

To solve the problem step by step, we will first determine the work done by each individual (man, woman, and boy) in one day and then calculate how many boys are needed to assist 2 men and 8 women to complete the work in 2 days.

### Step 1: Determine the work done by each individual in one day.

- A man can complete the job in 20 days. Therefore, the work done by one man in one day is:
\[ \text{Work by 1 man in 1 day} = \frac{1}{20} \]

- A woman can complete the job in 30 days. Therefore, the work done by one woman in one day is:
\[ \text{Work by 1 woman in 1 day} = \frac{1}{30} \]

- A boy can complete the job in 60 days. Therefore, the work done by one boy in one day is:
\[ \text{Work by 1 boy in 1 day} = \frac{1}{60} \]

### Step 2: Calculate the total work done by 2 men and 8 women in one day.

- Work done by 2 men in one day:
\[ \text{Work by 2 men} = 2 \times \frac{1}{20} = \frac{2}{20} = \frac{1}{10} \]

- Work done by 8 women in one day:
\[ \text{Work by 8 women} = 8 \times \frac{1}{30} = \frac{8}{30} = \frac{4}{15} \]

### Step 3: Combine the work done by men and women.

To find the total work done by 2 men and 8 women in one day, we need to add the two fractions:
\[ \text{Total work by 2 men and 8 women} = \frac{1}{10} + \frac{4}{15} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 10 and 15 is 30.

Converting both fractions:
- \(\frac{1}{10} = \frac{3}{30}\)
- \(\frac{4}{15} = \frac{8}{30}\)

Now add them:
\[ \text{Total work} = \frac{3}{30} + \frac{8}{30} = \frac{11}{30} \]

### Step 4: Determine the total work needed to be done in 2 days.

Since the total work is 1 (the whole job), the amount of work that needs to be done in 2 days is:
\[ \text{Work needed in 2 days} = 1 \]
Thus, the work needed in one day is:
\[ \text{Work needed in 1 day} = \frac{1}{2} \]

### Step 5: Set up the equation to find the number of boys (X).

Let \(X\) be the number of boys. The total work done in one day by 2 men, 8 women, and \(X\) boys is:
\[ \frac{11}{30} + X \times \frac{1}{60} = \frac{1}{2} \]

### Step 6: Solve for \(X\).

Convert \(\frac{1}{2}\) to a fraction with a denominator of 30:
\[ \frac{1}{2} = \frac{15}{30} \]

Now we have the equation:
\[ \frac{11}{30} + \frac{X}{60} = \frac{15}{30} \]

Multiply through by 60 to eliminate the denominators:
\[ 60 \times \frac{11}{30} + X = 60 \times \frac{15}{30} \]
\[ 2 \times 11 + X = 2 \times 15 \]
\[ 22 + X = 30 \]
\[ X = 30 - 22 \]
\[ X = 8 \]

### Conclusion

Thus, the number of boys required to assist 2 men and 8 women to complete the work in 2 days is **8 boys**.

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