To determine how long it will take Bill and Tom to mow the lawn together, we need to calculate their combined work rate and then find the time required for both of them to complete the task when working together.
1. Individual Work Rates:
- Bill can mow the lawn in 2 hours. Therefore, Bill's work rate is [tex]\(\frac{1}{2}\)[/tex] lawns per hour.
- Tom can mow the lawn in 3 hours. Therefore, Tom's work rate is [tex]\(\frac{1}{3}\)[/tex] lawns per hour.
2. Combined Work Rate:
To find their combined work rate, we add their individual work rates:
[tex]\[
\text{Combined work rate} = \frac{1}{2} + \frac{1}{3}
\][/tex]
To add these fractions, first find a common denominator which in this case is 6:
[tex]\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}
\][/tex]
Therefore,
[tex]\[
\text{Combined work rate} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \text{ lawns per hour}
\][/tex]
3. Time Taken Together:
If Bill and Tom have a combined work rate of [tex]\(\frac{5}{6}\)[/tex] lawns per hour, the time it takes for them to mow one lawn together can be found by taking the reciprocal of their combined work rate:
[tex]\[
\text{Time taken together} = \frac{1}{\frac{5}{6}} = \frac{6}{5} \text{ hours}
\][/tex]
Therefore, if both brothers mow the lawn together, it will take them [tex]\( \boxed{1.2} \)[/tex] hours.
