Let's break down the problem step by step.

Given:

1. P works thrice as fast as Q.

• Therefore, if Q's work rate is $x$, then P's work rate is $3x$.

2. P and Q together can work four times as fast as R.

• The combined work rate of P and Q is $3x + x = 4x$.
• This is given to be 4 times R's work rate, so $4x = 4r$, where $r$ is R's work rate.
• Therefore, $x = r$.

Work Rates:

• P's work rate = $3x$
• Q's work rate = $x$
• R's work rate = $r$

Since $x = r$, we can substitute:

• P's work rate = $3r$
• Q's work rate = $r$
• R's work rate = $r$

Total work rate of P, Q, and R together:

• Total work rate = $3r + r + r = 5r$

Earnings Ratio: The earnings should be distributed in the same ratio as their work rates. Thus, the ratio of P's, Q's, and R's work rates is:

• P : Q : R = $3r : r : r$

This simplifies to:

• P : Q : R = $3 : 1 : 1$

Answer: The ratio of earnings should be 3 : 1 : 1.

Thus, the