Question on weighted averages

[tomslawsky]

This question is based from my head, but inspired by a weighted average problem in the OG quant.

An adult bookstore generates all of it's revenue from only 2 movies, both of which are holiday themed.

*Chris Kringle's Tail (C)
*Peter Cotton's Tail (P)

During the course of the year, the total revenue from the sale of these movies increased by 1.0% from 1969 to 1970. If the sales of C decreased by 13% from 1969-1970 and the sales of P increased by 8% from 1969 to 1970, what is the ratio of sales of movie C to movie P in 1969?

Following the template that I use to do these problems, I calculated the ratio as follows:

C= -13%
P= +8%
Overall (O)= +1%

Therefore, ratio = (C-O)/(P-O) = (13-1)/(7-1) = 14/7 = 2/1

However, paralleling the answer for the official problem, my answer should be the reciprocal, or 1/2 and nor 2/1.

I'm confused- where did I go wrong here?

Thank you.

[rajkapoor]

Consider this : The weight of the decrease in C(13%) will tilt the scale towards negative side , so you need more weight on P(8%) to be able to balance it closer to it (+1%)
................................+1
-|-----------------|--|-------|-----------
-13......................0.............8



Where did you go wrong - is more of a CR question of the type Resolving the Paradox

=====
You follow certain template to solve these type of problems.
using that template gives wrong answer.
OG answers are correct(fact).

Each explains this paradox EXCEPT -

choice A - Template being used is wrong
Choice B - You copied the template wrong or put in the wrong values.
Choice C - You are looking at the answer through car mirror and hence getting
reciprocal result
Choice D - The writer of this response is clueless
choice E - Values of C and P have been reversed.

Hint : How did you get the ratio to be (C-O)/(P-O) ? shouldn't it be reversed ?

[RonPurewal]

Therefore, ratio = (C-O)/(P-O) = (13-1)/(7-1) = 14/7 = 2/1

actually, if you follow this technique, you will ALWAYS get the reciprocal of the ratio that you're supposed to get.

in other words, (number/amount of C) : (number/amount of P)
will actually equal
| P - O | / | C - O |
(not the other way around).

--

here's the way to think about it: the FARTHER you are away from the weighted average, the LESS influence you have on the average. this means that, the greater the distance |x - o|, the FEWER/LESS of "x" there must be.

[thoppae.saravanan]

Following a template formula may not work always. Best thing to do is understand the problem and then work accordingly.

In this problem, let C and P be the sales in the year 1970 and C1 and P1 be the sales in the year 1969.

From problem statement,

C = 0.87C1
P = 1.08P1

Also C+P = 1.01 (C1+P1). Solving these two equations for C/P we get 7/14 = 1/2.

[mschwrtz]

thoppae.saravanan, nice algebraic approach, and I don't disagree with your sentiment, but I'd temper it with a few observations.

1) "Understanding" the problem requires either insight or algorithm. Insight can be hard to come by when you're fatigued or stressed, so we'd like our students to have the option of using well-understood and well-oiled mechanical means.

2) The OP was a very small step from reliably applying one such means.

3) Ron's answer sought to reveal the logic beneath the algorithm.