Mental Math Shortcuts

Today my MathCounts team practiced their first Countdown Round.  This is the spelling bee style competition in which students compete head-to-head on stage to answer the question before their opponent.  Identifying elegant shortcuts is critical to getting the answer asap.  For example:


(Source: 2016 MathCounts Chapter Countdown Round)

The question asks for the difference between the average time and 10 minutes.  While one could find the average of the 3 times and then subtract from 10 minutes, a more clever approach (using smaller numbers) is to average the 3 differences from the start.  Since 10:13 > 10:00t_1 = -13.  The other time differences are 9 and 22.  The sum is -13+9+22=18 and the average is 6.  Smaller numbers means quicker, more accurate calculations.

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