Question

Brishti went on a 8-hour trip in a car, before the trip the car had traveled a total of x kms till then, where x is a whole number and is palindromic, At the end of his trip the car had had traveled a total of 26862 km. If Bristi never drove at more than 110 km/h, then the greatest possible average speed at which see dove is?

• A. 90
• B. 80
• C. 110
• D. 100

Answer 1

The Correct Option is D

Approach: Rather than testing each option, find the smallest palindromic starting odometer the trip allows — the smaller the start, the longer the distance, hence the faster the average speed.

Step 1: Let the average speed be $s$ km/h. The reading just before the trip is $26862-8s$, and this must itself be a palindrome.

Step 2: Since $s\le 110$, the trip is at most $8\times110=880$ km, so the start lies in $[26862-880,\ 26862]=[25982,\ 26862]$.

Step 3: A 5-digit palindrome in this range has the form $\overline{2\,b\,c\,b\,2}$; to be at least $25982$ we need $b=6$, giving $26062,26162,\dots,26862$.

Step 4: Greatest speed needs the greatest distance, i.e. the smallest valid start $=26062$. Then $8s=26862-26062=800\Rightarrow s=100$.

Answer: The greatest possible average speed is $100$ km/h.