weighted averages

[shawndx]

if you know any 2 of the following 3, then you can find the third one:
1. the ratio of 'weights' of the different quantities
2. the values of the quantities
3. the weighted average

I understand everything except the scenario when you are given the ratio of weights and the weighted average. How does one go about finding the values of the quantities? I am able to find the ratio of the quantities, but cannot find the exact quantities? I solved some questions backwards and tried omitting the values and seeing if they were solvable, but I ended up with many possibilities. However, the ratio of the values of quantities could be calculated... Could you confirm that the actual values of the quantities cannot be calculated or if they can, can you please explain how?

[shawndx]

bump... anyone care to help?

[Stuart@KaplanGMAT]

Here's a very useful rule to remember for data sufficiency:

In order to find the actual value of a quantity, you need at least one actual quantity somewhere in the information.

For example, if all you're given is a ratio, there's no way to determine the actual quantities involved; similarly, if all you're given is the proportions of a shape, there's no way to determine any of the actual dimensions (e.g. area, length, perimeter) of that shape.

Following this rule, you're 100% correct - if all you know is the ratio of the weights and the weighted average, there's no way to figure out the values of the quantities.

For example, let's say that males make up 60% of a class and that the weighted average of male and female grades on a test is 75%. Based on this information alone, we can't even tell whether the males scored lower or higher than the females! Each of the following scenarios is consistent with that information:

1) Males averaged 71, females averaged 81;
2) Males averaged 67, females averaged 87;
3) Males averaged 79, females averaged 69; and
4) Males averaged 83, females averaged 63.

All we can determine is the following formula:

weighted average = (avg group 1)(weight group 1) + (avg group 2)(weight group 2)
75 = .6(male average) + .4(female average)

Since we have 1 equation and 2 unknowns, there are infinite possible solutions.

[hey_thr67]

Question_1: A company consists of managers or directors. What percentage of the group are directors ?

1) The average of managers is $5000 less than that of company.

2) The average of directors is $15000 more than that of company.


I have tweaked the question little bit. Please see,

Question 2: A company consists of managers or directors. What percentage of the group are directors ?

1) The average of managers is $20000 less than that of directors.

2) The average of directors is $15000 more than that of company.

[lunarpower]

if you know any 2 of the following 3, then you can find the third one:
1. the ratio of 'weights' of the different quantities
2. the values of the quantities
3. the weighted average

I understand everything except the scenario when you are given the ratio of weights and the weighted average. How does one go about finding the values of the quantities? I am able to find the ratio of the quantities, but cannot find the exact quantities? I solved some questions backwards and tried omitting the values and seeing if they were solvable, but I ended up with many possibilities. However, the ratio of the values of quantities could be calculated... Could you confirm that the actual values of the quantities cannot be calculated or if they can, can you please explain how?

yeah, so that's what happens when i come up with things spontaneously. haha

if you have #1 and #2 then you can get #3, and if you have #2 and #3 then you can get #1.
however, if you have #1 and #3, then, as this poster correctly points out, you can't get #2. but, what you can find is the ratio of the distances between the quantities and the weighted average.

for instance, let's say that two quantities are "weighted" in the ratio 2:1 -- like, there are twice as many men as women in a room. then, let's say you have the weighted average of something; for instance, let's say that the average height of everyone in the room is 170 centimeters.
then, you can't find the actual average heights of the men and the women, but you can find the ratio of how far away they are from 170.
specifically, since there are only half as many women, the women's average has to be twice as far away from 170 to make the weighted average work.
(more generally, if the weights are in the ratio a:b, then the distances from the weighted average are in the ratio b:a.)

so, if there are twice as many men and the average height is 170 cm, then the distance between the men's height and 170 must be half the distance between the women's height and 170.
so, the men's and women's heights (on average) could be ...
... 171 and 168 cm
... 172 and 166 cm
... 175 and 160 cm
... 166 and 178 cm (a demographic anomaly)
... more generally, (170 + x) cm and (170 - 2x) cm, respectively, or (170 - x) cm and (170 + 2x) cm

hope that helps.