If V = 12R / (r + R), express R in terms of V and r.

Introduction / Context:
This is a basic algebraic manipulation problem where you are asked to solve a formula for a different variable. Initially, the expression is given as V = 12R / (r + R), and you must rewrite it so that R is the subject. Such rearrangement tasks are fundamental in algebra and also appear in physics and engineering when you need to isolate a particular quantity in a formula.

Given Data / Assumptions:

• Original formula: V = 12R / (r + R).
• V, r and R are real variables, with r + R not equal to zero.
• The goal is to express R explicitly in terms of V and r.
• We assume R is non zero so that division by R is valid when we factor.

Concept / Approach:
To solve for R, we treat the formula like an equation in R. The main steps are to clear the fraction by multiplying both sides by the denominator, then group all terms involving R on one side. After that, factor out R and finally divide to isolate R. Careful handling of each algebraic step ensures that we do not introduce or lose any valid solutions. Throughout, we keep track of conditions such as 12 − V not being zero in the final expression.

Step-by-Step Solution:
Given: V = 12R / (r + R). Multiply both sides by (r + R): V(r + R) = 12R. Expand the left side: Vr + VR = 12R. Move all R terms to one side: Vr = 12R − VR. Factor R on the right: Vr = R(12 − V). Assuming 12 − V is not zero, divide both sides by (12 − V): R = Vr / (12 − V). Thus R is expressed purely in terms of V and r.

Verification / Alternative check:
We can verify the rearranged formula by substituting R = Vr / (12 − V) back into the original expression for V. If we start with R = Vr / (12 − V) and compute 12R / (r + R), the algebra simplifies back to V, confirming that the manipulation is correct. This reverse substitution is a powerful way to check rearrangements of formulas and ensure no step was performed incorrectly.

Why Other Options Are Wrong:

• (Vr + V) / 12: This treats 12 as if it were distributed differently and does not satisfy the original equation when substituted back.
• V: This would imply R is always equal to V, which clearly is not true for arbitrary r and V.
• V / (r - 12): This mixes constants and variables incorrectly and gives a dimensionally inconsistent result.
• 12r / (V - 12): This resembles the correct expression but uses r instead of R inside the factorised term, so it does not satisfy V = 12R / (r + R).

Common Pitfalls:
Students sometimes try to cross multiply too quickly and make sign errors when moving terms across the equals sign. Another frequent mistake is forgetting to factor R from all terms containing it, leading to incomplete or incorrect expressions. Always expand, collect like terms, factor the desired variable and only then divide to isolate it. This four step approach makes rearranging formulas more systematic and reliable.

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