A works twice as fast as B. If B can complete a work in 12 days independently, how many days can A and B together finish the work?
Understand the Problem
The question is asking us to determine how many days A and B can complete a task together given that A works twice as fast as B and B can independently finish the task in 12 days. We can solve this by first finding A's work rate and then combining it with B's rate to find the total time taken when working together.
Answer
4 days
Answer for screen readers
The total time taken for A and B to complete the task together is 4 days.
Steps to Solve
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Find B's work rate B can complete the task alone in 12 days. Therefore, B's work rate is given by the formula: $$ \text{Rate of B} = \frac{1 \text{ task}}{12 \text{ days}} = \frac{1}{12} \text{ tasks per day} $$
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Find A's work rate Since A works twice as fast as B, we can calculate A's work rate: $$ \text{Rate of A} = 2 \times \frac{1}{12} = \frac{2}{12} = \frac{1}{6} \text{ tasks per day} $$
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Combine A's and B's work rates Now, we add A's and B's work rates together to find their combined work rate: $$ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{6} + \frac{1}{12} $$
To add these fractions, we need a common denominator, which is 12: $$ \text{Combined Rate} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \text{ tasks per day} $$
- Calculate the time taken to complete the task together To find the total time taken when A and B work together, use the formula: $$ \text{Time} = \frac{1 \text{ task}}{\text{Combined Rate}} = \frac{1}{\frac{1}{4}} = 4 \text{ days} $$
The total time taken for A and B to complete the task together is 4 days.
More Information
Working with rates allows us to effectively determine how combined efforts can impact the total time required to complete a task. In this scenario, knowing the individual rates is essential for finding the combined rate.
Tips
- Forgetting to convert work rates to a common denominator when adding fractions.
- Miscalculating the work rate of A by incorrectly assuming its speed relative to B.
- Confusing the concept of rate (tasks per day) with total time taken to complete the task.
