Final answer:
P, Q, and R should share their earnings according to their individual work rates, leading to a sharing ratio of 3:1:1 when P is thrice as fast as Q and P and Q together are four times as fast as R.
Step-by-step explanation:
When P, Q, and R collaborate on a job, the equitable distribution of earnings is contingent on the proportional amount of work each individual can perform independently. Given that P works three times as fast as Q (P = 3Q) and P and Q combined operate four times as fast as R (P + Q = 4R), a system of equations can be devised to ascertain their individual work rates.
Let's denote the work rate of Q as Q, then the work rate of P would be 3Q, and the work rate of R is R. The equation representing their combined work rate is P + Q = 4R.
Substituting P = 3Q into the equation, we get 3Q + Q = 4R, simplifying to 4Q = 4R. This leads to the conclusion that Q equals R. By substituting P = 3Q into the equation, we find that P equals 3R. Hence, the individual work rates are in the ratio of 3:1:1 for P, Q, and R, respectively.
This means that P works three times as fast as R, Q works at the same rate as R, and together, P and Q contribute to the job four times faster than R alone. Consequently, the fair distribution of earnings should align with this ratio, reflecting the proportion of work each individual contributes to the collaborative effort.