If the three quantities P, Q, and R satisfy the common equality: (3P)/5 = (7Q)/2 = (7R)/5, determine the simplest ratio P : Q : R.

Introduction / Context:
This problem tests the standard “equal ratios” technique. When several expressions are equal to the same value, we introduce a common constant and express each variable in terms of that constant, then form the required ratio.

Given Data / Assumptions:

• (3P)/5 = (7Q)/2 = (7R)/5 • P, Q, R are real quantities (ratio requested in simplest whole numbers)

Concept / Approach:
Let the common value be k: 3P/5 = k, 7Q/2 = k, 7R/5 = k. Solve each for P, Q, R in terms of k. Then form P : Q : R and clear denominators by multiplying by the LCM of denominators.

Step-by-Step Solution:
1) Set each equal to k: 3P/5 = k, 7Q/2 = k, 7R/5 = k 2) Solve for P: P = (5k)/3 3) Solve for Q: Q = (2k)/7 4) Solve for R: R = (5k)/7 5) Ratio becomes: P : Q : R = (5/3) : (2/7) : (5/7) 6) Multiply all parts by 21 (LCM of 3 and 7): (5/3)*21 = 35, (2/7)*21 = 6, (5/7)*21 = 15 7) So the simplest ratio is 35 : 6 : 15.

Verification / Alternative check:
Check by plugging ratio values: Let P = 35, Q = 6, R = 15. Then 3P/5 = 3*35/5 = 21, 7Q/2 = 7*6/2 = 21, and 7R/5 = 7*15/5 = 21. All equal, so ratio is consistent.

Why Other Options Are Wrong:
• Any other ordering or numbers fail the “all three expressions equal” check.

Common Pitfalls:
• Forgetting to use a common constant k. • Clearing denominators incorrectly when converting to whole-number ratio.

Final Answer:
35 : 6 : 15