Question

If the average of 5 positive integers is 70, what is the largest possible value of their median?

Answer 1

Answer: C) 116

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Explanation:

We want the median to be as large as possible, so let's go for the largest answer choice and see if it works or not. I'll show that it doesn't work.

So we have 5 values, in which the median is 346. This must mean there are 2 values larger than 346 (since this value is in slot 3, aka the middle). But if one value is 346, then the other 4 values must be 1 so that we end up with a mean of 70, as shown below

add up the five values: 1+1+1+1+346 = 350

divide by five: 350/5 = 70

In the set {1, 1, 1, 1, 346} the median is 1,and not 346. So this is where things fall apart. If we wanted 346 in the middle, then we'd need to have two other values larger than 346, or equal to it. However that would make the sum get over 350. We want the sum to be 350 so dividing it over 5 leads to the proper mean of 70.

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The median can be 116 as this example shows

{1, 1, 116, 116, 116}

sum = 1+1+116+116+116 = 350

divide that sum over five: 350/5 = 70

we get the proper mean we want