P works thrice as fast as Q, whereas P and Q together can work four times as fast as R. If P, Q and R together work on a job, in what ratio should they share the earnings?

Correct Option (A)

The distribution of earnings for a job is directly proportional to the individual efficiencies or the rate at which each person works. To determine the sharing ratio, we establish the relative efficiencies of P, Q, and R.

• Let the efficiency of Q be represented by 'e' units per unit time.
• According to the problem statement, P works thrice as fast as Q. Therefore, P's efficiency is 3e.
• The combined efficiency of P and Q is the sum of their individual efficiencies: Efficiency(P+Q) = 3e + e = 4e.
• It is also stated that P and Q together work four times as fast as R. This implies Efficiency(P+Q) = 4 × Efficiency(R).
• Substituting the combined efficiency: 4e = 4 × Efficiency(R).
• From this equation, R's efficiency is derived as e.
• Thus, the ratio of efficiencies for P : Q : R is 3e : e : e.
• Simplifying this ratio yields 3 : 1 : 1.

Consequently, P, Q, and R should share the earnings in the ratio 3 : 1 : 1.

Incorrect Options:

• Option (B) 3 : 2 : 4: This ratio implies that Q's efficiency is twice that of R (2:4 simplified to 1:2), which contradicts the derived equal efficiencies for Q and R (1:1). It also suggests P's efficiency is 3/2 times Q's, not 3 times.
• Option (C) 4 : 3 : 4: This ratio suggests P's efficiency is 4/3 times Q's, which is inconsistent with the condition that P works thrice as fast as Q.
• Option (D) 3 : 1 : 4: This ratio indicates that R's efficiency is four times that of Q (1:4), which is incorrect as the derived efficiencies show Q and R have equal efficiencies (1:1).