Introduction / Context:
This is a classic system-of-equations problem with three unknowns (P, Q, R). Converting each sentence to an equation allows direct solving by elimination or substitution.
Given Data / Assumptions:
- P + R + 2Q = 59.
- 3P + Q + R = 68.
- P + 3Q + 3R = 108.
Concept / Approach:
Use elimination. Subtract equations smartly to isolate variables and solve stepwise. Any consistent method (matrix, substitution) works.
Step-by-Step Solution:
From (1): P + R + 2Q = 59.From (2): 3P + Q + R = 68.Subtract (1) from (2): 2P − Q = 9 ⇒ Q = 2P − 9.Use (3): P + 3Q + 3R = 108 and (1): P + R + 2Q = 59.Multiply (1) by 3 ⇒ 3P + 3R + 6Q = 177.Subtract (3): (3P − P) + (3R − 3R) + (6Q − 3Q) = 177 − 108 ⇒ 2P + 3Q = 69.Substitute Q = 2P − 9: 2P + 3(2P − 9) = 69 ⇒ 2P + 6P − 27 = 69 ⇒ 8P = 96 ⇒ P = 12.
Verification / Alternative check:
With P = 12 ⇒ Q = 15 and R = 20 satisfy all three equations (check quickly).
Why Other Options Are Wrong:
- 15, 19, 17 do not satisfy the three simultaneous equations for P.
Common Pitfalls:
- Arithmetic slips while subtracting equations.
- Forgetting to back-substitute to confirm consistency.
Final Answer:
12 years
Discussion & Comments