# What do you think of this math problem?

I thought of this one while stuck in a traffic jam. There was a story on the radio that said that teachers were striking because their pay was below the national average. That got me to wondering if the national average included their salaries. Would it be possible for their salary to be below average when their salaries are not included but above average when included? I hope that your intuition says that this is not possible. Can you come up with an argument to prove this assertion?

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## 10 Answers

If they get an increase the new average would be higher, you calculate a new average with their new income.

Three states 1) $59,000 2)$50,000 and 3) $41,000 equals $50,000 now state 3 gets an increase to $45,000 and the average is now $51,333.

**The average includes state 3; at all times.**

Not following your logic. If you calculate the average including the state, it lowers the average calculated without them. Why can’t the new overall average be lower than the salaries for the state?

All the states are in the average **ALL**

Both before and after the increase.

Oh I see what you are saying, but the problem I posed still remains as a hypothetical.

The problem can be easily solved without algebra. All you need to use is that adding a new value below/above the average of the others will decrease/increase the overall average.

It’s not possible. Including teachers pay in the average can only bring the average a little closer to teachers’ pay it cannot make it equal or more than.

@flutherother , You got it right. Maybe it is more obvious to you than to me. One way of reasoning is to say that including the new teachers will lower the average. Removing them will increase the average to what it was, but that means that the new teachers’ salaries must be lower than the new average.

I did not want to bring politics into this, but I strongly support the North Carolina teachers and all the other teachers who have gotten a pretty raw deal.

This is much simpler than I originally thought. The new average is a weighted average of the previous average and the salaries for the state. As such, the new average will fall between the previous average and the state value.

One final note. If you compute the average including the state and then raise the salaries to the average, the state will still be below average.

For example, consider the numbers 8, 16 and 24. The average is 16. Raising 8 to 16 gives 16, 16 and 24, with an average of 18⅔, which is greater than 16.

If the 8 is first removed, the average of the rest is the average of 16 and 24, which is 20. Giving a value of 20 gives us 16, 20 and 24, with an average of 20. This will hold in general, since adding something to a collection with the same value as the average for the collection leaves the average unchanged.

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