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Palindromic Milometer

Age 11 to 14 Short Challenge Level:

If Alan travelled with a large average speed, then his journey was longer, so his final milometer reading would have been higher.

The fastest that Alan could have travelled, or the upper bound on his average speed, is the speed limit, 75 miles per hour. If Alan had travelled for 3 hours at 75 miles per hour, he would have travelled 3$\times$75 = 225 miles, so his journey can't have been longer than 225 miles.

29792 + 225 = 30017, so his final milometer reading can't have been more than 30017. So it must have been the largest palindromic number less than 30017. Numbers beginning with 3 are larger than numbers beginning with 2, so the first digit of the final reading should be a 3.

The second and third digits must both be 0, as if either of them weren't a 0, then the number would be greater than 30017. So the final milometer reading must have been 30003.



Using total distance/total time
Alan's journey was 30003 $-$ 29792 = 211 miles long. So, since it took him 3 hours, his average speed was 211$\div$3 = 70$\frac{1}{3}$ miles per hour.



Finding the difference in average speed from the difference in distance
Alan travelled 30017 $-$ 30003 = 14 miles less than he would have travelled if he had travelled at 75 miles per hour. So, since his journey took 3 hours, he travelled 14$\div$3 = 4$\frac{2}{3}$ miles per hour slower than 75 miles per hour. So his average speed was 70$\frac{1}{3}$ miles per hour.

You can find more short problems, arranged by curriculum topic, in our short problems collection.