Make S the subject of the relation p = s + $\frac{sm^2}{nr}$

Analysis

The goal is to isolate the variable s in the given equation: $p = s + \frac{sm^2}{nr}$

Concept

To make a variable the subject of a formula, we need to group all terms containing that variable on one side of the equation and factor it out. If the variable appears in a fraction, we must find a common denominator or multiply through to clear the fraction.

Step-by-Step Solution

Step 1: Identify the terms with 's' The equation is $p = s + \frac{sm^2}{nr}$. Both terms on the right side contain $s$.

Step 2: Factor out 's' Since $s$ is common to both terms, we can factor it out: $p = s(1 + \frac{m^2}{nr})$

Step 3: Simplify the expression inside the bracket To add $1$ and $\frac{m^2}{nr}$, find a common denominator, which is $nr$: $1 + \frac{m^2}{nr} = \frac{nr}{nr} + \frac{m^2}{nr} = \frac{nr + m^2}{nr}$

Now substitute this back into the equation: $p = s \left( \frac{nr + m^2}{nr} \right)$

Step 4: Solve for 's' To isolate $s$, multiply both sides by the reciprocal of the fraction $\left( \frac{nr + m^2}{nr} \right)$, which is $\left( \frac{nr}{nr + m^2} \right)$: $s = p \times \left( \frac{nr}{nr + m^2} \right)$

$s = \frac{nrp}{nr + m^2}$

Conclusion

Comparing our result to the given options, we find that it matches option A.

Exam Tip: When you see a variable appearing in multiple terms, always look to factor it out immediately. If you are ever unsure about your algebraic manipulation, try picking simple numbers for $p, n, r,$ and $m$, calculate $s$, and plug those values into the options to see which one gives you the same $s$ value!