## Arithmetic - Unlikely Uses of Weighted Averages

We know the usual questions in which we use weighted averages - two groups coming together, two solutions getting mixed etc but sometimes weighted averages come in very handy in unlikely Algebra questions.

If you see an expression that looks something like this: (ax + by)/(x+y), think weighted averages. So the value of (2x + 3y)/(x+y) will lie between 2 and 3 for all positive values of x and y. I came across a question today which made me write this post. This is what the question looked like:

Question: If x and y are positive numbers such that x + y = 3, then which of the following could be the value of 300x + 400y?

I. 864
II. 1020
III. 1198

Now note here that x and y are positive but not necessarily integers. x could be 0.1 and y will be 2.9 in that case. x could be 2.1 and y will be 0.9 in that case etc. I hope you see that infite such combinations are possible so you cannot plug in values.

But we do notice this:

Weighted Avg (W) = (300x + 400y)/(x + y) = (300x + 400y)/3

This represents the weighted average of 300 and 400 such that the weights are x and y. The weighted average of 300 and 400 will certainly lie between 300 and 400.

So the value of (300x + 400y) will be 3 * W

So the value of this will lie between 3 * 300 and 3 * 400 i.e. between 900 and 1200.

Hence values II and III are possible but not I.

Alternativley, think about it like this: We need the value of 300x + 400y

Now since x and y are positive number, the smallest value of x will be infinitesimally greater than 0. In that case y will be almost 3. So the value of the expression will be slightly less than 1200.

Similarly, the smallest value of y will be infinitesimally greater than 0. In that case x will be almost 3. So the value of the expression will be slightly more than 900.

The expression will take all values between 900 and 1200.

Now, let's bring it all together - the very definition of weighted averages is that it is the weight given to each quantity. When you give all the weight to 300 and none to 400, the average becomes 300. When you give all the weight to 400 and none to 300, the average becomes 400. So the two trains of thought are alctually the same!