Let's solve this step by step.
1. Understanding the problem:
- Kaityn can build a shed in 6 days. Therefore, her rate of work is [tex]\( \frac{1}{6} \)[/tex] sheds per day.
- Mark can build a shed in 8 days. Therefore, his rate of work is [tex]\( \frac{1}{8} \)[/tex] sheds per day.
2. Combined Work Rate:
- When Kaityn and Mark work together, their combined work rate is the sum of their individual rates.
[tex]\[
\text{Combined Rate} = \frac{1}{6} + \frac{1}{8}
\][/tex]
3. Finding a common denominator:
- The least common multiple (LCM) of 6 and 8 is 24.
[tex]\[
\frac{1}{6} = \frac{4}{24}
\][/tex]
[tex]\[
\frac{1}{8} = \frac{3}{24}
\][/tex]
- So, their combined rate is:
[tex]\[
\frac{4}{24} + \frac{3}{24} = \frac{7}{24} \text{ sheds per day}
\][/tex]
4. Setting up the equation:
- Let [tex]\( d \)[/tex] be the number of days it takes for both Kaityn and Mark to build the shed together.
- The combined rate multiplied by the time [tex]\( d \)[/tex] should equal 1 shed.
[tex]\[
\left(\frac{7}{24}\right)d = 1
\][/tex]
5. Solving for [tex]\( d \)[/tex]:
- To isolate [tex]\( d \)[/tex], multiply both sides by the reciprocal of [tex]\( \frac{7}{24} \)[/tex]:
[tex]\[
d = \frac{24}{7}
\][/tex]
6. Checking the result:
- Simplify [tex]\( \frac{24}{7} \)[/tex]:
[tex]\[
d \approx 3.43 \text{ days}
\][/tex]
Therefore, the correct equation to find [tex]\( d \)[/tex] is:
[tex]\[
\left( \frac{7}{24} \right) d = 1
\][/tex]
Or in another form:
[tex]\[
d = \frac{24}{7}
\][/tex]
This means that Kaityn and Mark together would take approximately 3.43 days to build the shed.
