To determine how long it will take Ted and Jacob to clear the field together, we can use the concept of combined work rates.
1. Work Rates:
- Ted's work rate is [tex]\(\frac{1}{3}\)[/tex] (since he can clear the field in 3 hours).
- Jacob's work rate is [tex]\(\frac{1}{2}\)[/tex] (since he can clear the field in 2 hours).
2. Combined Work Rate:
- When the two work together, their combined work rate is the sum of their individual work rates:
[tex]\[
\text{Combined work rate} = \frac{1}{3} + \frac{1}{2}
\][/tex]
3. Finding a Common Denominator:
- To add the fractions, we find a common denominator. The common denominator for 3 and 2 is 6:
[tex]\[
\frac{1}{3} = \frac{2}{6}, \quad \frac{1}{2} = \frac{3}{6}
\][/tex]
Therefore:
[tex]\[
\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}
\][/tex]
4. Time Taken Together:
- The combined work rate of [tex]\(\frac{5}{6}\)[/tex] represents the fraction of the field they can clear per hour. We need to find the total time [tex]\(t\)[/tex] it takes for them to clear the entire field working together:
[tex]\[
\frac{1}{t} = \frac{5}{6}
\][/tex]
Solving for [tex]\(t\)[/tex], we get:
[tex]\[
t = \frac{6}{5}
\][/tex]
Thus, the time taken [tex]\( t \)[/tex] is [tex]\(\frac{6}{5}\)[/tex] hours.
5. Convert Time to Minutes:
- We can convert this time into minutes to make it easier to compare with the given options. Since 1 hour = 60 minutes:
[tex]\[
t = \frac{6}{5} \text{ hours} \times 60 \text{ minutes/hour} = 72 \text{ minutes}
\][/tex]
6. Breaking Down the Minutes:
- 72 minutes can be expressed as 1 hour and 12 minutes:
[tex]\[
72 \text{ minutes} = 1 \text{ hour} + 12 \text{ minutes}
\][/tex]
Therefore, the closest and accurate answer is:
1 hour 12 minutes
