P and Q working individually can finish a job in x and y days respectively. P starts the work and they both work on alternate days and the job is finished in 7 days. If Q starts the work and with P working on alternate days, the work gets over in $$6 \frac{3}{4}$$​days. Compare the efficiencies of P and Q.

P and Q can complete the job individually in x and y days, respectively.

Case 1: P starts and they alternate, finishing the job in 7 days.

Case 2: Q starts and they alternate, finishing the job in $$6\frac{3}{4}$$​ days.

If A person finishes a work in x days, then;

Work rate = $$\frac{1}{x},$$​​

For case 1;

P starts the work, so in 7 days

P works for 4 day and Q works for 3 days, So;

​$$\frac{4}{x} + \frac{3}{y} = 1 ……….(i)$$​​

For case 2;

Q starts the work, so in $$6\frac{3}{4}$$  or 6.75 days

Q works for 3.75 days and P works for 3 days, So;

​$$\frac{3.75}{y}+\frac{3}{x} = 1 ……..(ii)$$​​

Subtracting equation(i) and (ii);

​$$\frac{1}{x} - \frac{0.75}{y} = 0 \\ \ \\\frac{1}{x} = \frac{0.75}{y} \\ \ \\\frac{1}{x} = 0.75 \times\frac{1}{y} \\ \ \\\implies \text{Efficiency of P} = 0.75 \times \text{Efficiency of Q}$$​​

Thus, Efficiency of P is 75% of Efficiency of Q