To solve the problem, we will first determine the work rates of the man, woman, and boy, and then find out how many days the boy takes to finish the work alone.
### Step 1: Determine the work rates of the man and woman.
- A man can complete the work in 10 days. Therefore, his work rate is:
\[
\text{Man's work rate} = \frac{1}{10} \text{ (work per day)}
\]
- A woman can complete the work in 24 days. Therefore, her work rate is:
\[
\text{Woman's work rate} = \frac{1}{24} \text{ (work per day)}
\]
### Step 2: Calculate the combined work rate of the man and woman.
To find the combined work rate of the man and woman, we add their individual work rates:
\[
\text{Combined work rate of man and woman} = \frac{1}{10} + \frac{1}{24}
\]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 10 and 24 is 120. Thus, we convert the fractions:
\[
\frac{1}{10} = \frac{12}{120}, \quad \frac{1}{24} = \frac{5}{120}
\]
Now, adding these:
\[
\text{Combined work rate} = \frac{12}{120} + \frac{5}{120} = \frac{17}{120}
\]
### Step 3: Determine the combined work rate of the man, woman, and boy.
According to the problem, the man, woman, and boy together can complete the work in 6 days. Therefore, their combined work rate is:
\[
\text{Combined work rate of man, woman, and boy} = \frac{1}{6}
\]
### Step 4: Set up the equation to find the boy's work rate.
Let the boy's work rate be \( \frac{1}{x} \) (where \( x \) is the number of days the boy takes to finish the work alone). We can set up the equation:
\[
\frac{17}{120} + \frac{1}{x} = \frac{1}{6}
\]
### Step 5: Solve for \( \frac{1}{x} \).
First, we need to express \( \frac{1}{6} \) with a common denominator of 120:
\[
\frac{1}{6} = \frac{20}{120}
\]
Now, substituting this into the equation:
\[
\frac{17}{120} + \frac{1}{x} = \frac{20}{120}
\]
Subtract \( \frac{17}{120} \) from both sides:
\[
\frac{1}{x} = \frac{20}{120} - \frac{17}{120} = \frac{3}{120}
\]
### Step 6: Solve for \( x \).
To find \( x \), we take the reciprocal of \( \frac{3}{120} \):
\[
x = \frac{120}{3} = 40
\]
### Conclusion:
The boy alone can finish the work in **40 days**.
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