Final answer:
The correct approach to solve for time t in the work equation w = r1.t + r2.t is to combine both rates (r1 + r2) before dividing the total work w. Options b, c, and d fail to follow the correct algebraic steps, either through improper subtraction or incorrect treatment of the rates.
Step-by-step explanation:
The original equation provided depicts the total work w, done by two people working at different rates. The equation w = r1.t1 + r2.t2 is for calculating work when person 1 works at rate r1 for time t1 and person 2 works at rate r2 for time t2. When solving for time t, it is assumed that both people work for the same amount of time (t=t1=t2), hence the equation simplifies to w = (r1 + r2)t. To solve for t, we rearrange the equation to t = w / (r1 + r2). Therefore, options b and d incorrectly isolate t, as they fail to combine the rates r1 and r2 correctly, and option b incorrectly subtracts r1 multiplied by t1 from w before dividing by r2. Option c is also incorrect as it introduces a subscript on w (w1), which is not part of the original equation, and improperly tries to subtract the second term without properly isolating t first.
Option a is the only correct transformation mentioned, representing the proper initial setup: w = r1.t + r2.t (where, implicit in this context, t = t1 = t2).