Once you choose $qr$ as the common denominator, rewrite each fraction with denominator $qr$ and add their numerators.

After you combine the fractions, you will have something like a single fraction equal to $1/s$. How can you turn an equation of the form $\text{fraction} = 1/s$ into an expression for $s$?

If you know $1/s$ equals some fraction in $q$ and $r$, think about what happens when you take the reciprocal of both sides of the equation.

In Desmos, choose simple positive numbers for $q$ and $r$, such as typing q = 2 and r = 3 on separate lines so Desmos assigns those values.

Use the relationship $\frac{1}{q} + \frac{1}{r} = \frac{1}{s}$. In Desmos, type 1/(1/q + 1/r) on a new line; the numeric output is the value that $s$ must have for your chosen $q$ and $r$.

On new lines, type each choice’s right-hand side using your $q$ and $r$: (q + r)/(q*r), q*r/(q + r), q*r/(q - r), and (q - r)/(q*r). Compare each numeric result with the value from step 2 and note which expression matches it exactly.

Change $q$ and $r$ to different positive values (for example, q = 4, r = 5) and repeat steps 2–3. The correct formula for $s$ will match the computed value of $s$ for every choice of positive $q$ and $r$, while the others will not.

You are given

and asked to rewrite this so that $s$ is alone on one side in terms of $q$ and $r$ only. To do this, first turn the sum on the left into a single fraction.

Use the common denominator $qr$ to add the two fractions:

So the left side simplifies to a single fraction with numerator $q+r$ and denominator $qr$.

Replace the left side of the original equation with the simplified fraction:

Now you have an equation directly relating $1/s$ to $q$ and $r$.

Because $q$, $r$, and $s$ are positive, you can safely take reciprocals of both sides of the equation:

Taking reciprocals gives

So the correct expression is $s = \frac{qr}{q + r}$, which matches choice B.