Bus Route Profitability and Rider Thresholds
A transit authority reports that a bus route is considered profitable only if it carries at least 40 riders per hour on average. Route 12 currently carries 45 riders per hour. The authority's planning director concludes that Route 12 is therefore profitable. However, the route's operating cost is $120 per hour, and the average fare is $2.50. Additionally, the 45 riders per hour figure includes 10 riders who use free transfer passes, which generate no direct fare revenue. The director's conclusion is being challenged by a city auditor.
1. Identify the main logical flaw in the planning director's reasoning.
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The director commits a conditional-reasoning error: treating a necessary condition as sufficient. The stated rule is 'profitable → at least 40 riders per hour.' From Route 12 having 45 riders, the director concludes 'profitable,' which would require the reverse: 'at least 40 riders → profitable.' That reverse is not given. Moreover, the quantitative details undercut the conclusion: 45 riders × $2.50 = $112.50 in potential fare revenue, but 10 of those riders pay no fare, so actual fare revenue is 35 × $2.50 = $87.50. Operating cost is $120 per hour, so even before other costs the route runs a deficit. Thus the conclusion fails on both formal and numerical grounds.
2. What additional information would most directly help the auditor determine whether Route 12 is actually profitable?
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The auditor needs (a) whether the 40-rider threshold is intended as a necessary condition, a sufficient condition, or a proxy for revenue, and (b) whether there are non-fare revenues (subsidies, advertising) or allocated costs not captured by the $120/hour figure. If the threshold is merely a necessary condition, meeting it says nothing about profitability; the auditor should compare total revenue per hour (fare plus other sources) with total cost per hour. The given fare and cost numbers already suggest a deficit, but confirming other revenue streams would settle the question.
3. Suppose the authority clarifies that the 40-rider threshold is necessary but not sufficient, and that a route must also generate at least $130 per hour in fare revenue. Using the given numbers, evaluate whether Route 12 meets the revenue condition.
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No. The 45 riders include 10 free-transfer riders, so paying riders = 35. Fare revenue = 35 × $2.50 = $87.50 per hour, below the $130 requirement. Even if all 45 riders paid, revenue would be 45 × $2.50 = $112.50, still below $130. Therefore Route 12 fails the revenue condition regardless of how the free transfers are treated.