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?

Given:

P works thrice as fast as Q.

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

We need to find the ratio of earnings of P, Q, and R when they work together.

Formula used:

Work done ∝ Efficiency

Let the efficiency of Q = x.

Then, efficiency of P = 3x (since P works 3 times as fast as Q).

Efficiency of P + Q = 4 × Efficiency of R.

Efficiency ratio = Earnings ratio (as earnings are proportional to work done).

Calculation:

Let efficiency of Q = x.

⇒ Efficiency of P = 3x.

⇒ Efficiency of P + Q = 3x + x = 4x.

Given, Efficiency of P + Q = 4 × Efficiency of R.

⇒ 4x = 4 × Efficiency of R.

⇒ Efficiency of R = x.

Now, total efficiency of P, Q, and R when working together:

⇒ Total Efficiency = Efficiency of P + Efficiency of Q + Efficiency of R.

⇒ Total Efficiency = 3x + x + x = 5x.

Ratio of earnings = Ratio of efficiencies:

⇒ Earnings Ratio of P : Q : R = 3x : x : x = 3 : 1 : 1

∴ The correct answer is option (1).

Alternate MethodCalculation:

P works 3 times as fast as Q.

P and Q together are 4 times as fast as R.

Q's 1 day work = 1 unit

⇒ P's 1 day work = 3 units

⇒ P + Q = 3 + 1 = 4 units/day

Given P + Q = 4 × R

⇒ 4 = 4 × R

⇒ R = 1 unit/day

Work Rates:

P = 3, Q = 1, R = 1

Total work = 3 + 1 + 1 = 5 units

∴ The earnings ratio = P : Q : R = 3 : 1 : 1