Rohit, Harsh, and Sanjeev are typists who together type 216 pages in 4 hours (working simultaneously). In one hour, Sanjeev exceeds Harsh by as many pages as Harsh exceeds Rohit. Over 5 hours, Sanjeev types as many pages as Rohit does in 7 hours. How many pages per hour do Rohit, Harsh, and Sanjeev type, respectively?

Introduction / Context:
This is a rates problem with two constraints: (i) total combined output, and (ii) an arithmetic progression (AP) relationship between the three individual rates. A further ratio links Sanjeev’s and Rohit’s outputs over different time spans.

Given Data / Assumptions:

• Combined in 4 h = 216 pages ⇒ combined rate = 54 pages/h.
• In 1 h: (S − H) = (H − R) ⇒ R, H, S are in AP.
• 5 hours of Sanjeev equals 7 hours of Rohit ⇒ 5S = 7R.

Concept / Approach:
Let R = r, S = (7/5)r from 5S = 7R. Then H, being the AP mean, is H = (R + S)/2. Sum the three rates to 54 and solve for r. This yields all three individual rates.

Step-by-Step Solution:

Verification / Alternative check:
Check AP: 18 − 15 = 3; 21 − 18 = 3; consistent. Check 5S vs 7R: 5*21 = 105 = 7*15; consistent. Check total: 15 + 18 + 21 = 54; consistent.

Why Other Options Are Wrong:
Each other triple fails either the AP condition, the 5S = 7R constraint, or the total 54 pages/h requirement.

Common Pitfalls:
Assuming equal rates or mixing geometric and arithmetic means; skipping the combined-rate check.

Final Answer:
15, 18, 21