To determine how long it takes for the two crews to paint the apartment together, we can use the concept of work rates.
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Calculate the work rate for each crew:
- The new crew can paint the apartment in 12 hours, so their work rate is \( \frac{1}{12} \) of the apartment per hour.
- The experienced crew can paint the apartment in 6 hours, so their work rate is \( \frac{1}{6} \) of the apartment per hour.
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Combine their work rates: \[ \text{Combined work rate} = \frac{1}{12} + \frac{1}{6} \]
To add these fractions, we need a common denominator. The common denominator of 12 and 6 is 12. Thus:
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Convert \( \frac{1}{6} \) to a fraction with a denominator of 12: \[ \frac{1}{6} = \frac{2}{12} \]
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Now add the rates together: \[ \frac{1}{12} + \frac{2}{12} = \frac{3}{12} = \frac{1}{4} \]
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Determine the time to complete the job together: If their combined work rate is \( \frac{1}{4} \) of the apartment per hour, it means that together they can complete the apartment in 4 hours, since:
\[ \text{Time} = \frac{1 \text{ apartment}}{\frac{1}{4} \text{ apartment/hour}} = 4 \text{ hours} \]
Thus, when the two crews work together, it takes 4 hours to paint the apartment.