To find how long it would take Renee and Genera to sew a 25-square-foot quilt together, we need to determine their individual rates of sewing and then combine those rates.
1. Determine the individual rates:
- Renee can sew the quilt in 16 hours. Her rate is therefore [tex]\( \frac{1}{16} \)[/tex] quilts per hour.
- Genera can sew the quilt in 12 hours. Her rate is therefore [tex]\( \frac{1}{12} \)[/tex] quilts per hour.
2. Determine the combined rate when working together:
- To find their combined rate, we simply add their individual rates:
[tex]\[
\text{Combined rate} = \frac{1}{16} + \frac{1}{12}
\][/tex]
3. Calculate the combined rate:
- To add these fractions, we need a common denominator. The least common multiple of 16 and 12 is 48.
- Convert each rate to have this common denominator:
[tex]\[
\frac{1}{16} = \frac{3}{48} \quad \text{(since } 16 \times 3 = 48)
\][/tex]
[tex]\[
\frac{1}{12} = \frac{4}{48} \quad \text{(since } 12 \times 4 = 48)
\][/tex]
- Now add these fractions:
[tex]\[
\frac{3}{48} + \frac{4}{48} = \frac{7}{48}
\][/tex]
4. Determine the time to complete one quilt working together:
- The combined rate is [tex]\( \frac{7}{48} \)[/tex] quilts per hour. To find the time it takes to complete one quilt, we take the reciprocal of this rate:
[tex]\[
\text{Time to complete one quilt} = \frac{1}{\frac{7}{48}} = \frac{48}{7} \approx 6.857 \text{ hours}
\][/tex]
Therefore, working together, Renee and Genera would take approximately [tex]\( \frac{48}{7} \)[/tex] hours, or approximately 6 hours and [tex]\( \frac{6}{7} \)[/tex] of an hour, to sew the 25-square-foot quilt. Thus, the correct answer is:
[tex]\[
\boxed{6 \frac{6}{7} \text{ hours}}
\][/tex]
