Given the algebraic relation V = 12R / (r + R), express R explicitly in terms of V and r (solve the equation for R).

Introduction:
This is an algebraic manipulation problem where you must rearrange a given formula to isolate a particular variable. Such formula-transposition questions are common in both mathematics and physics, and they help you become comfortable with symbolic manipulation.

Given Data / Assumptions:

• The given formula is V = 12R / (r + R).
• V, r and R are variables; treat them as real quantities with r + R ≠ 0.
• We are required to solve this equation for R in terms of V and r.

Concept / Approach:
We will clear the denominator by multiplying both sides of the equation by (r + R), then collect all terms involving R on one side and factor R out. Finally, we will isolate R by dividing by the remaining coefficient.

Step-by-Step Solution:
Step 1: Start from the given equation.V = 12R / (r + R).Step 2: Multiply both sides by (r + R).V(r + R) = 12R.Step 3: Expand the left-hand side.Vr + VR = 12R.Step 4: Collect terms involving R on one side.Vr = 12R − VR = R(12 − V).Step 5: Solve for R, assuming 12 − V ≠ 0.R = Vr / (12 − V).

Verification / Alternative check:
We can substitute R = Vr / (12 − V) back into the original formula to see if it holds. Plugging in:12R / (r + R) = 12 * [Vr / (12 − V)] / [r + Vr / (12 − V)].With careful algebraic simplification, this expression reduces back to V, confirming that our rearrangement is correct.

Why Other Options Are Wrong:
Vr + (V / 12), V, V / (r − 12), or (12r) / (V − r) are not consistent with the stepwise algebraic isolation of R. Substituting any of these into the original equation does not satisfy it identically for all r and V.

Common Pitfalls:
Typical mistakes include incorrectly distributing V over (r + R), forgetting to factor R properly, or dividing by V instead of (12 − V). Keeping each step explicit and systematic prevents such errors.

Final Answer:
The correct expression for R in terms of V and r is R = Vr / (12 − V).