KBAT Problems
KBAT Problem: Finding a Shortest Route
A KBAT-style problem using a weighted network, the kind of real-situation graph question that tests whether you can model, not just calculate.
Model the situation
A delivery must visit several towns with given distances between them. Draw it as a weighted network, with towns as vertices and roads as weighted edges.
Seeing the picture is half the problem.
Compare the routes
List the possible routes, total the weights along each, and choose the smallest. A full answer shows the totals for each route considered, so the marker can see the comparison, not just the final choice.
Understand the problem
A courier must travel from town P to town T. The roads and their lengths in km are P–Q = 8, P–R = 5, R–Q = 2, Q–T = 6, R–S = 7, S–T = 3.
Find the shortest route from P to T. What it really asks: model the roads as a weighted network, then compare complete routes, not just pick the shortest single road.
Plan and solve
- Draw the network: towns P, Q, R, S, T are vertices; each road is an edge labelled with its length.
- List every route from P to T. Route 1: P–Q–T. Route 2: P–R–Q–T. Route 3: P–R–S–T. Route 4: P–Q–R–S–T.
- Total the weights: Route 1 = 8 + 6 = 14 km. Route 2 = 5 + 2 + 6 = 13 km. Route 3 = 5 + 7 + 3 = 15 km. Route 4 = 8 + 2 + 7 + 3 = 20 km.
- The smallest total is 13 km, so the shortest route is P–R–Q–T at 13 km.
Check and a variant
Check that 13 km really is the least by scanning the four totals 14, 13, 15 and 20, no route beats 13. A variant the examiner could add: the road R–Q is closed for repairs.
Remove it and only P–Q–T = 14 km and P–R–S–T = 15 km remain, so the shortest becomes P–Q–T at 14 km. Removing one edge can change the whole answer, which is why you list routes rather than trust the first short-looking road.
Frequently asked questions
Do I have to list every route, or can I just spot the shortest?
List them. The marker gives credit for the comparison, so showing each route with its total protects you if one arithmetic slip happens.
In small networks the shortest is not always the one that looks direct, as the closed-road variant shows. Listing is both safer for marks and safer for accuracy.
What is the difference between a vertex and an edge here?
A vertex is a place, here a town, drawn as a dot. An edge is a connection between two vertices, here a road, drawn as a line and labelled with its length or weight.
Getting this vocabulary right matters because the exam question and its marks are written in exactly these terms.
Are weights always distances?
No. A weight can be distance, time, cost or capacity, depending on the situation.
Always read what the numbers stand for before totalling them, because a shortest-distance route and a cheapest route can differ. Answering the wrong quantity is a common KBAT slip, so pin down what the weight means first.