Person A completes a certain amount of work in one day that is half of what Person B can complete in a day. Person B's daily work rate is half of Person C's daily work rate. If Person C can finish the entire work in 7 days working alone, how many days will it take for all three persons working together to complete the work?

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Accepted Answer

Given:  
 Person C can complete the work in 7 days. Therefore, C's 1 day's work = $$ \frac{1}{7} $$.  
Person B's work rate is half of C's, so B's 1 day's work = $$ \frac{1}{2} 	imes \frac{1}{7} = \frac{1}{14} $$.  
  - Person A's work rate is half of B's, so A's 1 day's work = $$ \frac{1}{2} 	imes \frac{1}{14} = \frac{1}{28} $$.

Combined daily work rate of A, B, and C:  
$$
	ext{Combined rate} = \frac{1}{28} + \frac{1}{14} + \frac{1}{7}
$$  
Converting all fractions to a common denominator:  
$$
\frac{1}{28} + \frac{2}{28} + \frac{4}{28} = \frac{7}{28} = \frac{1}{4}
$$

Therefore, the three together can complete $$ \frac{1}{4} $$ of the work in one day. Hence, they will finish the entire work in:  
$$
4 	ext{ days}.
$$

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