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LostInParadise's avatar

Can you do the algebraic manipulation to solve this problem?

Asked by LostInParadise (24584points) September 16th, 2010

A while back I gave an intuition problem regarding weighted averages. I said that if as the weights increased the associated values also increased, what conclusion could be drawn? A number of people correctly answered that the weighted average would be greater than the simple average.

Intuition is one thing but an actual proof is something else. I wanted to see if there was a relatively easy way to confirm the intuition. Arranging the weights so that 0 <W1 < W2 < .... Wn we are given that the associated values satisfy 0 <V1 < V2 < ... Vn. What we want to show is that
(W1V1 + ... + WnVn)/(W1+...Wn) > (V1+...Vn)/n.

Multiplying by the terms in the denominator and bringing all the terms together, this is the same as showing that
n (W1V1+...+WnVn) – (V1+...Vn)(W1+...Wn) > 0

I was hoping that the formula could be reworked so as to make the inequality obvious. It turns out that it can and that the reworked formula for the difference can be expressed rather compactly using appropriate algebraic notation. If you try some simple cases, the formula should be fairly clear.

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5 Answers

the100thmonkey's avatar

Do you know the answer to this question?

iamthemob's avatar

Is this a test? Am I in my underwear?

weeveeship's avatar

The answer is always 42.

LostInParadise's avatar

@the100thmonkey , I know the answer. I made up the problem.

LostInParadise's avatar

I am using Rot13 to encode the answer, so you can work on the problem without being distracted by the answer.

Hint:
Hfr a=2 naq snpgbe gur erfhygvat rkcerffvba. Qb lbh frr ubj guvf pna or trarenyvmrq? Gel vg jvgu a=3.

Answer:
Sbe a=2, lbh fubhyq trg 2(J1I1+J2I2)=(J2-J1)(I2-I1)

Guvf trarenyvmrf gb a (J1I1+...+JaIa) – (I1+...Ia)(J1+...Ja) = gur fhz bs grezf
(Jv – Jw)(Iv – Iw) jurer v > w. Fvapr nyy gur grezf ner cbfvgvir, vg sbyybjf gung guvf vf terngre guna 0.

Gb frr jul gur rdhngvba vf evtug, svefg abgvpr gung nyy gur grezf ba gur yrsg ner bs gur sbez JvIw. Gurer ner gjb glcrf bs grezf, ubzbtrarbhf jurer v=w naq urgrebtrarbhf jurer v <> w.

Rnpu ubzbtrarbhf grez JvIv nccrnef (a-1) gvzrf, a gvzrf sebz gur svefg grez naq bapr sebz gur frpbaq grez. Ba gur evtug fvqr, n ubzbtrarbhf grez sbe, fnl J1 jvyy or cebqhprq va rnpu grez bs gur fhz pbagnvavat J1. Gurer ner a-1 bgure ahzore gung pna nccrne jvgu 1, fb gurer ner a-1 ubzbtrarbhf grezf ba gur evtug.

Rnpu cnve bs urgrebtrarbhf grezf JvIw naq JwIv jvyy or trarengrq bapr, sebz (Jv-Jw)(Iv-Iw).

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