Two Fare Rules and the Repeated Receipt
A city bus operator in Mysuru is testing two fare rules for a new airport shuttle. Distance travelled is x km, with x ≥ 0. Rule P charges ₹40 plus ₹18 for each complete kilometre, so P(x) = 40 + 18 floor(x). Rule Q charges ₹40 plus ₹18 per km and then rounds the total down to the nearest ₹5, so Q(x) = 5 floor((40 + 18x)/5). An accountant notes that trips of 2.5 km and 2.6 km both produced an ₹85 receipt under Rule Q and concludes that Rule Q is not a function. A passenger group complains that under Rule P a trip of 1.99 km costs ₹58 while a trip of 2.00 km costs ₹76, a jump of ₹18. The operator wants a fare rule that is a function, never charges less than ₹40, and never increases by more than ₹10 when distance increases by 0.1 km. Trial receipts show Q(2.5) = Q(2.6) = 85 and P(1.99) = 58, P(2.00) = 76.
1. Evaluate P and Q at 2.5 km and 2.6 km. Is the accountant correct that Q is not a function? Justify using the definition of a function.
Read the worked answer
P(2.5) = 40 + 18 floor(2.5) = 40 + 36 = ₹76. P(2.6) = 40 + 18 floor(2.6) = 40 + 36 = ₹76.
Q(2.5) = 5 floor((40 + 45)/5) = 5 floor(17) = ₹85. Q(2.6) = 5 floor((40 + 46.8)/5) = 5 floor(17.36) = ₹85.
The accountant is wrong. A relation is a function when each input has exactly one output. Different inputs may share the same output; that does not break the definition. For each distance x, the expression 5 floor((40 + 18x)/5) gives one definite fare, so Q is a function. What fails is the inverse relation: fare ₹85 corresponds to more than one distance, so the inverse is not a function.
2. Test the condition 'no increase of more than ₹10 when distance increases by 0.1 km' for P near 1.99 km and for Q near 2.49 km. Which rule should be adopted?
Read the worked answer
For P: P(1.99) = 40 + 18(1) = ₹58, while P(2.00) = 40 + 18(2) = ₹76. The increase is ₹18, so P fails the condition.
For Q: Q(2.49) = 5 floor((40 + 44.82)/5) = 5 floor(16.964) = ₹80. Q(2.50) = 5 floor(85/5) = ₹85. The increase is ₹5, so this step passes.
The underlying amount 40 + 18x rises by ₹1.80 when x rises by 0.1 km. Rounding down to the nearest ₹5 can add at most one ₹5 step when a multiple of 5 is crossed, so Q never jumps by more than ₹5. Rule Q is therefore the rule that satisfies the operator's smoothness condition, while Rule P does not.
3. A passenger pays ₹100 under Q. Find the interval of distances that could produce this fare. What does the interval imply about the inverse relation of Q?
Read the worked answer
Q(x) = 100 means 5 floor((40 + 18x)/5) = 100, so floor((40 + 18x)/5) = 20. Hence
100 ≤ 40 + 18x < 105 60 ≤ 18x < 65 10/3 ≤ x < 65/18.
So every distance from about 3.33 km up to but not including about 3.61 km gives the same ₹100 fare. The inverse relation assigns fare ₹100 to many distances, so it is one-to-many and therefore not a function. This is consistent with Q itself being a function: the failure is in reversing the rule, not in the original rule.