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Mind-blowingly fun math programs and resources for grades 6-8!
Complete a quick request below to receive your FREE digital copy of the 2021-2022 MATHCOUNTS School Handbook, with 250 creative, awesome math problems (scroll down to see an example problem). It's one of the best middle school math resources out there...and we're only a little bit biased.
The MATHCOUNTS School Handbook is the primary preparation resource for our Competition Series program. After submitting this form, you will be redirected to a page to download this year's handbook. From there you can also learn more about the Competition Series.
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Without using a calculator, how fast can you solve this?
The digits 3, 4 and 8 can be used to make six different three-digit numbers in which each digit is used exactly once. What is the average of these six numbers?
Click the buttons to the left to reveal two possible solutions.
The six three-digit numbers that can be made are 348, 384, 438, 483, 834 and 843. The sum of these six numbers is 3,330. The average is 3,330 ÷ 6 = 555.
This solution is the approach most take first. It is correct, but adding all six of those three-digit numbers without a calculator can take a bit of time. Can you think of a faster solution?
Since all six of the three-digit numbers use the digits 3, 4 and 8, we know each digit will appear in each place value the same number of times. Thus, the average of the digits in each place value is just the average of the digits we are working with. That would be (3 + 4 + 8) ÷ 3 = 15 ÷ 3 = 5, so the average of all six numbers is 555.
Although the text is longer, this approach can save time by simplifying the mental math required. At MATHCOUNTS, we encourage coming up with multiple solutions and always trying to find the "shortcut" in the problem... it's not doing less math, it's creative problem solving!