Counting on Math Tutoring Services

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Today's problem comes from the 2018 Raytheon Mathcounts. An eighth grader solved this within a few seconds. It took me much longer.

There is a set of seven numbers. The average of the highest four is 10. The average of the lowest four is 5. What is the lowest possible sum of all seven numbers?

To tackle this problem, let's convert each number into a variable- a,b,c,d,e,f,g - ranked from highest to lowest.

Then the average of the first four is 10. Therefore a+b+c+d= 40.

The average of the last four is 5. Therefore d+e+f+g= 20.

If we add both numbers we get 60=a+b+c+2d+e+f+g.

Now we need to find a value of d and subtract it from 60. We are looking for the highest value for d to get the smallest sum of all numbers. What do we know about the median term? We know that when averaged with the three lowest terms, the average is five. So the smallest possible value for d is five (that is d=5, e=5, f=5,g=5). We also know that the average of the four highest terms is ten. Therefore, the largest value that d can have is 10 (that is, a=10, b=10, c=10, d=10). Thus d is greater or equal to 5 and less than or equal to 10.

Let's take d=10.

a+b+c+2d+e+f+g-d=60-10=50.

The lowest sum for the seven numbers is 50.

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