Link three ratios to get P : S: Given P : Q = 8 : 15, Q : R = 5 : 8, and R : S = 4 : 5, find the ratio P : S.

Introduction / Context:
When multiple linked ratios are given, proceed stepwise, keeping the middle terms consistent. This problem requires moving from P to S via Q and R.

Given Data / Assumptions:

• P : Q = 8 : 15.
• Q : R = 5 : 8.
• R : S = 4 : 5.

Concept / Approach:
Assign variables to maintain consistency. Start with P = 8k and Q = 15k. Use Q : R to derive R, then apply R : S to get S. Finally simplify P : S.

Step-by-Step Solution:
Let P = 8k, Q = 15k.From Q : R = 5 : 8 ⇒ if Q = 15k = 5 × 3k, then R = 8 × 3k = 24k.From R : S = 4 : 5 ⇒ S = (5/4) × R = (5/4) × 24k = 30k.Thus P : S = 8k : 30k = 4 : 15.

Verification / Alternative check:
Convert all to a single base k and inspect: P = 8k, S = 30k. The ratio reduces cleanly by 2 to 4 : 15.

Why Other Options Are Wrong:

• 2 : 15 halves P again unnecessarily.
• 3 : 19 and 7 : 15 do not match the chained calculations.

Common Pitfalls:

• Mismatching the intermediary term Q or R when scaling ratios.
• Not simplifying the final ratio to lowest terms.

Final Answer:
4 : 15