CAT - weighted average

[ElizabethS105]

A convenience store currently stocks 48 bottles of mineral water. The bottles have two sizes of either 20 or 40 ounces each. The average volume per bottle the store currently has in stock is 35 ounces. How many 40 ounce bottles are in stock?
20
24
28
32
36

CAT Answer Explanation
20x + (40)(48) – 40x = (35)(48)
20x = (40)(48) – (35)(48)
20x = (48)(40 – 35)
20x = (48)5
20x = 240
x = 12

The answer explanation makes sense, but I am wondering why my approach is wrong. Here is how I thought about it..

If the average is 35, that is 15 above the 20 and 5 below the 40; 5 = 1/3 of 15... there has to be 1/3 of the 40oz bottles and 2/3rds of the 20oz bottles to even out. If that is the case, there should be 16 of the 40 oz bottles - but that is not an answer choice. Can you please explain why my reasoning is flawed?

Thanks!

[RonPurewal]

yes, there is a shortcut like that. but you're not doing it right.

before discussing the correct shortcut, it's worthwhile to note how you could disprove your version—a nice example of the type of critical thinking that you should bring to these things.

A COUNTEREXAMPLE TO YOUR VERSION of this shortcut:
• let's say male french bulldogs weigh 26 pounds each, and females weigh 23 pounds each.
• say the average weight of some group of french bulldogs is 25 pounds.
with these numbers, your procedure would become '1 is half of 2. therefore, the bulldogs must be half male and half female.' but clearly they aren't, because in that case the average would be 24.5 pounds (halfway between the male and female weights).

ANOTHER COUNTEREXAMPLE:
• same bulldog weights
• say the average weight of the group is 26 pounds.
with these numbers, your fraction becomes 2/1—in other words, you would conclude that '200 percent of the bulldogs are male'. clearly nonsense.
(in this case i'm guessing that you would just start using the reverse fraction—i.e., 1 over 2, rather than 2 over 1. but, if that happens, it's a sign of trouble; a formula shouldn't just WHOA stop working.)

i'll discuss the correct version in the next post. but the most important point is this: whenever you come up with some sort of shortcut, TEST IT WITH EASY EXAMPLES.

[RonPurewal]

actually, PERHAPS THE EASIEST COUNTEREXAMPLE would be to look at the simplest situation possible—i.e., equal weights (a 'normal' average).

say you have a group of teenagers, all either 16 or 18 years old, and that there are equal numbers of 16- and 18-year-olds.
then, clearly, the average age should be 17 years.
...but if you apply the method above, you'll get the fraction 1/1, and you'll conclude that the entire group consists of 18-year-olds.
huh? that's definitely the wrong result. so that shortcut goes in 'file 13'.

('file 13' is what one of my favorite high-school teachers called her wastebasket.)

[RonPurewal]

HOW IT ACTUALLY WORKS:

• take the same two 'distances' between the extremes and the weighted average. (you are already figuring these correctly.)

• take the RATIO of those distances.

• the weights will have the REVERSE ratio.

there's no need to memorize the reversal, since it's just simple common sense. (if you have more X's than Y's, the average should be closer to the 'X' value. so, GREATER weight <--> SHORTER distance from weighted average.)

--

in your original example:

• the 'distances' for the twenty- and forty-ounce bottles are, respectively, 5 and 15. thus the RATIO is 1 to 3.

• so the relative weights (= ratio of how many you have of each) should be 3 to 1.

• in terms of fractions, the big bottles should account for 3/(3 + 1) = 3/4 of the total, and the small bottles should be the other 1/(3 + 1) = 1/4 of the total.

[RonPurewal]

now, you should...

...apply the CORRECT shortcut to my additional examples above (french bulldogs and teenagers), and make sure you get correct results;

...then make up some additional examples of your own.

this latter step is key.
when you construct your own examples, you'll have to think much more completely about how the situation works. specifically, you have to understand HOW to choose the numbers, as well as WHY.
that step is a non-issue when it comes to your normal 'problem solving', because you're presented with numbers that someone else already made up. so, making the numbers up yourself—especially making up numbers that will do something specific—will present an additional challenge, and will also leave you with a more complete understanding.

[ElizabethS105]

For the bulldog example:
let's say male french bulldogs weigh 26 pounds each, and females weigh 23 pounds each.
• say the average weight of some group of french bulldogs is 25 pounds.
* if there were 15 bulldogs together, then there would be 10 male bulldogs and 5 female
* Weighted ratio = 2:1 (since mean is closer to the male weight, there will be more males)

If the average weight was 26 though, there would be 0 female bulldogs in the group.

Correct?

[tim]

Exactly!

[NL]

Nicely posted, Ron!

[RonPurewal]

If the average weight was 26 though, there would be 0 female bulldogs in the group.

Correct?

yes, this is correct—and it's also an excellent thing for you to be doing.

namely, you are TESTING a method/hypothesis/other such thing by using EASY / OBVIOUS CASES.

here 26 is the average weight of the male bulldogs.
the females weigh less.
so, it should be obvious that there's only one way to get an overall average of 26... by not letting any girls into the treehouse.

the point, of course, is that you already know what the answer SHOULD be for this case. so, if your hypothesis/rule/method gives ANYTHING ELSE then you can immediately toss it.