Word Problems

Kaitlyn can build a shed in 6 days. Mark can build the same shed in 8 days. Which equation can be used to find [tex]$d$[/tex], the number of days it would take Kaitlyn and Mark to build the shed together?

\begin{tabular}{|c|c|c|c|}
\hline
& \begin{tabular}{c}
Rate \\
(Sheds per Day)
\end{tabular}
& \begin{tabular}{c}
Time \\
(Days)
\end{tabular}
& Fraction Completed \\
\hline Kaitlyn
& [tex]$\frac{1}{6}$[/tex]
& [tex]$d$[/tex]
& [tex]$\frac{1}{6} d$[/tex] \\
\hline Mark
& [tex]$\frac{1}{8}$[/tex]
& [tex]$d$[/tex]
& [tex]$\frac{1}{8} d$[/tex] \\
\hline
\end{tabular}

A. [tex]$\frac{1}{6} d + \frac{1}{8} d = 1$[/tex]

B. [tex]$\frac{1}{6} d - \frac{1}{8} d = 1$[/tex]

C. [tex]$\frac{1}{6} d + \frac{1}{8} d = \frac{1}{d}$[/tex]

D. [tex]$\frac{1}{6} d - \frac{1}{8} d = \frac{1}{d}$[/tex]



Answer :

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.