Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of x km till then, where x is a whole number and is palindromic, i.e., $x$ remains unchanged when its digits are reversed. At the end of the trip, the car had traveled a total of $26862 \mathrm{~km}$ till then, this number again being palindromic. If Brishti never drove at more than $110 \mathrm{~km} / \mathrm{hr}$ , then the greatest possible average speed at which she drove during the trip, in km/hr, was

$80$

$90$

$110$

$100$

✅ Correct Option: 4

Trip duration: 8 hours

Starting odometer reading: x km (palindromic whole number)

Ending odometer reading: 26862 km (palindromic)

Speed limit: Never more than 110 km/hr

We need to find the greatest possible average speed.

Distance traveled during trip = 26862 - x km

Average speed = $\frac{26862 - x}{8}$ km/hr

Our goal: Maximize the average speed while keeping it ≤ 110 km/hr

Since Brishti never drove more than 110 km/hr, her average speed also cannot exceed 110 km/hr.

$\frac{26862 - x}{8} \leq 110$

26862 - x ≤ 880

x ≥ 26862 - 880

x ≥ 25982

To maximize the average speed, we need to minimize x (the starting odometer reading).

We need the smallest palindromic number that is ≥ 25982.

Checking palindromes near 25982:

25952 → Too small (< 25982)

26062 → This works! (≥ 25982)

With x = 26062:

Distance during trip = 26862 - 26062 = 800 km

Average speed = $\frac{800}{8} = 100$ km/hr

Check: 100 km/hr ≤ 110 km/hr

Any smaller palindromic starting reading would give us an average speed > 110 km/hr, which violates the constraint.

Any larger palindromic starting reading would give us a smaller average speed.

Therefore, the greatest possible average speed is 100 km/hr.

When maximizing a rate like average speed, we often need to work backwards from constraints to find the optimal conditions. Here, the speed limit constraint determined the minimum starting odometer reading, which then gave us the maximum average speed.

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