All Data Interpretation and Logical Reasoning (DILR) Practice

All Data Interpretation and Logical Reasoning (DILR)

A traveler plans a journey from City P to City V through a network of one-way roads. Each road segment has an associated travel time (in hours) and a toll fee (in units of currency). The network and the details for each road are provided in the table below and can be visualized as a directed graph. $$ \begin{array}{|c|c|c|c|} \hline \textbf{From} & \textbf{To} & \textbf{Time (hours)} & \textbf{Toll (currency units)} \\ \hline \text{P} & \text{Q} & 2 & 10 \\ \text{P} & \text{R} & 3 & 15 \\ \text{Q} & \text{S} & 4 & 12 \\ \text{Q} & \text{T} & 3 & 8 \\ \text{R} & \text{S} & 2 & 18 \\ \text{R} & \text{U} & 5 & 10 \\ \text{S} & \text{T} & 1 & 5 \\ \text{S} & \text{V} & 6 & 20 \\ \text{T} & \text{U} & 2 & 7 \\ \text{T} & \text{V} & 4 & 13 \\ \text{U} & \text{V} & 3 & 15 \\ \hline \end{array} $$ Cities S and T are designated as 'Trading Posts'. The traveler must adhere to the following conditions: 1. The journey must start at City P and end at City V. 2. The traveler must visit at least one 'Trading Post' (i.e., City S or City T, or both). 3. The total travel time for the entire journey must not exceed $15$ hours. 4. The traveler aims to minimize the total toll fee for the journey. What is the minimum possible total toll fee (in currency units) for a journey from City P to City V that satisfies all the given conditions? A. $31$ B. $40$ C. $42$ D. $49$

Correct Option: A To find the minimum possible total toll fee, we need to systematically list all possible simple paths from City P to City V, calculate their total time and total toll, and then check if they satisfy the given conditions. We will keep track of the path with the minimum toll among all valid paths. Let's denote each edge as (Start City, End City, Time, Toll). 1. **Path: P $\to$ Q $\to$ S $\to$ V** * Segments: (P,Q,2,10), (Q,S,4,12), (S,V,6,20) * Total Time: $2 + 4 + 6 = 12$ hours * Total Toll: $10 + 12 + 20 = 42$ currency units * Trading Post Visited: S (Satisfies Condition 2) * Time Constraint: $12 \le 15$ hours (Satisfies Condition 3) * **Status: Valid. Toll = 42.** 2. **Path: P $\to$ Q $\to$ S $\to$ T $\to$ V** * Segments: (P,Q,2,10), (Q,S,4,12), (S,T,1,5), (T,V,4,13) * Total Time: $2 + 4 + 1 + 4 = 11$ hours * Total Toll: $10 + 12 + 5 + 13 = 40$ currency units * Trading Post Visited: S, T (Satisfies Condition 2) * Time Constraint: $11 \le 15$ hours (Satisfies Condition 3) * **Status: Valid. Toll = 40.** 3. **Path: P $\to$ Q $\to$ S $\to$ T $\to$ U $\to$ V** * Segments: (P,Q,2,10), (Q,S,4,12), (S,T,1,5), (T,U,2,7), (U,V,3,15) * Total Time: $2 + 4 + 1 + 2 + 3 = 12$ hours * Total Toll: $10 + 12 + 5 + 7 + 15 = 49$ currency units * Trading Post Visited: S, T (Satisfies Condition 2) * Time Constraint: $12 \le 15$ hours (Satisfies Condition 3) * **Status: Valid. Toll = 49.** 4. **Path: P $\to$ Q $\to$ T $\to$ V** * Segments: (P,Q,2,10), (Q,T,3,8), (T,V,4,13) * Total Time: $2 + 3 + 4 = 9$ hours * Total Toll: $10 + 8 + 13 = 31$ currency units * Trading Post Visited: T (Satisfies Condition 2) * Time Constraint: $9 \le 15$ hours (Satisfies Condition 3) * **Status: Valid. Toll = 31.** (Current minimum toll) 5. **Path: P $\to$ Q $\to$ T $\to$ U $\to$ V** * Segments: (P,Q,2,10), (Q,T,3,8), (T,U,2,7), (U,V,3,15) * Total Time: $2 + 3 + 2 + 3 = 10$ hours * Total Toll: $10 + 8 + 7 + 15 = 40$ currency units * Trading Post Visited: T (Satisfies Condition 2) * Time Constraint: $10 \le 15$ hours (Satisfies Condition 3) * **Status: Valid. Toll = 40.** 6. **Path: P $\to$ R $\to$ S $\to$ V** * Segments: (P,R,3,15), (R,S,2,18), (S,V,6,20) * Total Time: $3 + 2 + 6 = 11$ hours * Total Toll: $15 + 18 + 20 = 53$ currency units * Trading Post Visited: S (Satisfies Condition 2) * Time Constraint: $11 \le 15$ hours (Satisfies Condition 3) * **Status: Valid. Toll = 53.** 7. **Path: P $\to$ R $\to$ S $\to$ T $\to$ V** * Segments: (P,R,3,15), (R,S,2,18), (S,T,1,5), (T,V,4,13) * Total Time: $3 + 2 + 1 + 4 = 10$ hours * Total Toll: $15 + 18 + 5 + 13 = 51$ currency units * Trading Post Visited: S, T (Satisfies Condition 2) * Time Constraint: $10 \le 15$ hours (Satisfies Condition 3) * **Status: Valid. Toll = 51.** 8. **Path: P $\to$ R $\to$ S $\to$ T $\to$ U $\to$ V** * Segments: (P,R,3,15), (R,S,2,18), (S,T,1,5), (T,U,2,7), (U,V,3,15) * Total Time: $3 + 2 + 1 + 2 + 3 = 11$ hours * Total Toll: $15 + 18 + 5 + 7 + 15 = 60$ currency units * Trading Post Visited: S, T (Satisfies Condition 2) * Time Constraint: $11 \le 15$ hours (Satisfies Condition 3) * **Status: Valid. Toll = 60.** 9. **Path: P $\to$ R $\to$ U $\to$ V** * Segments: (P,R,3,15), (R,U,5,10), (U,V,3,15) * Total Time: $3 + 5 + 3 = 11$ hours * Total Toll: $15 + 10 + 15 = 40$ currency units * Trading Post Visited: None (Does NOT satisfy Condition 2) * **Status: Invalid.** Comparing the total tolls of all valid paths: * Path 1: $42$ * Path 2: $40$ * Path 3: $49$ * Path 4: $31$ * Path 5: $40$ * Path 6: $53$ * Path 7: $51$ * Path 8: $60$ The minimum toll among the valid paths is $31$ currency units, achieved by the path P $\to$ Q $\to$ T $\to$ V. Option A is the correct answer. The other options represent: B. $40$: This toll is achievable by valid paths like P $\to$ Q $\to$ S $\to$ T $\to$ V and P $\to$ Q $\to$ T $\to$ U $\to$ V. It is also the toll for an invalid path P $\to$ R $\to$ U $\to$ V (due to not visiting a Trading Post), making it a strong distractor. C. $42$: This toll is achievable by the valid path P $\to$ Q $\to$ S $\to$ V. D. $49$: This toll is achievable by the valid path P $\to$ Q $\to$ S $\to$ T $\to$ U $\to$ V. The final answer is $\boxed{\text{31}}$

100%

← →

Keep Learning?

Quick Edit Card #