It takes $1.5$ hours for Tim to mow the lawn. Linda can mow the same lawn in $2$ hours. How long will it take Tim and Linda, work together, to mow the lawn?

Hint:
Here, we will assume the total work done to be some variable. Then we will find the work done by both Tim and Linda in one hour. We will equate the sum of work done by both Linda to the total work done. We will solve it further to get the total time taken when both Linda and John work together.

Complete step by step solution:
Let total work done = $W$
It’s given that time taken by Tim to mow a lawn $ = 1.5$ hours
Time taken by Linda to mow a lawn $ = 2$ hours
Now, we can write it as
Work done by Tim in $1.5$ hours $=W$
Work done by Linda in 2 hours $=W$
Next, we will find work done by them in 1 hour
Work done by Tim in 1 hour $ = \dfrac{W}{{1.5}}$…………………..$\left( 1 \right)$
Work done by Linda in 1 hour $ = \dfrac{W}{2}$………………………$\left( 2 \right)$
Let work done when they both do it together is $ = \dfrac{W}{T}$ ………………………$\left( 3 \right)$
Where T is the time taken by both to complete the work.
Now, if we add equation $\left( 1 \right)$ and $\left( 2 \right)$ it will be equal to equation $\left( 3 \right)$
$\dfrac{W}{{1.5}} + \dfrac{W}{2} = \dfrac{W}{T}$
As $W$ is common on both sides we will remove it and get the equation as
$ \Rightarrow \dfrac{1}{{1.5}} + \dfrac{1}{2} = \dfrac{1}{T}$
Now taking L.C.M on left side, we get
$\ \Rightarrow \dfrac{{2 + 1.5}}{{2 \times 1.5}} = \dfrac{1}{T} \\
   \Rightarrow \dfrac{{3.5}}{{3.0}} = \dfrac{1}{T} \\ $
Now taking reciprocal, we get
$ \Rightarrow \dfrac{3}{{3.5}} = T$
Removing decimal from the denominator and simplifying it, we get
 $\ T = \dfrac{{3 \times 10}}{{35}} \\
   \Rightarrow T = \dfrac{6}{7} \\ $

Hence, Time taken when they both did the work together is $\dfrac{6}{7}$ hours.

Note:
The mistake that we can make in this question is that we consider the work done by a different person in their respective hours instead of work done by both of them in a specific hour. So our first step is to calculate the work done by both of them in 1 hour. The rate of work done is directly proportional to the time taken.