To determine how long it will take Charla to complete the entire picture, we can first find the rate at which she is working and then use that to find the total time required for the full area.
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Find the area covered rate: Charla completed \( 23\frac{2}{3} \) of the area in \( 12\frac{1}{2} \) hours.
Convert to improper fractions:
- \( 23\frac{2}{3} = \frac{23 \times 3 + 2}{3} = \frac{69 + 2}{3} = \frac{71}{3} \)
- \( 12\frac{1}{2} = \frac{12 \times 2 + 1}{2} = \frac{24 + 1}{2} = \frac{25}{2} \)
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Find the rate of area covered per hour: To find the rate, we divide the area covered by the time taken: \[ \text{Rate} = \frac{\text{Area completed}}{\text{Time taken}} = \frac{\frac{71}{3}}{\frac{25}{2}} = \frac{71}{3} \times \frac{2}{25} = \frac{71 \times 2}{3 \times 25} = \frac{142}{75} \text{ of the area per hour} \]
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Determine time to complete the entire area: To find the total time required to complete the whole area (1 full area), we can set up the equation: \[ \text{Time} = \frac{\text{Total Area}}{\text{Rate}} = \frac{1}{\frac{142}{75}} = \frac{75}{142} \text{ hours} \]
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Calculate the total time: The total time in hours to complete the entire area is: \[ \text{Time to complete} = \frac{75}{\frac{142}{75}} = \frac{75 \times 75}{142} = \frac{5625}{142} \approx 39.6 \text{ hours (if needed)} \]
Thus, the time it will take her to complete the entire picture is \( \frac{5625}{142} \) hours, or approximately 39.6 hours.