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A convenience store stocks soda in 12-ounce and 24-ounce bottles. If t

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A convenience store stocks soda in 12-ounce and 24-ounce bottles. If t  [#permalink]

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New post 15 Oct 2018, 09:09
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A convenience store stocks soda in 12-ounce and 24-ounce bottles. If the average capacity of all the bottles in the store is 22 ounces, then what fraction of the bottles in the store are 12 ounces?

A) 1/3
B) 1/6
C) 2/3
D) 5/6
E) 2/5

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Re: A convenience store stocks soda in 12-ounce and 24-ounce bottles. If t  [#permalink]

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New post 15 Oct 2018, 09:25
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Top Contributor
dimmak wrote:
A convenience store stocks soda in 12-ounce and 24-ounce bottles. If the average capacity of all the bottles in the store is 22 ounces, then what fraction of the bottles in the store are 12 ounces?

A) 1/3
B) 1/6
C) 2/3
D) 5/6
E) 2/5


We can apply the weighted average formula, which says:

Weighted average of groups combined = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...

So, we get: group average = (proportion of 12-ounce sodas in stock)(average weight of 12-ounce sodas) + (proportion of 24-ounce sodas in stock)(average weight of 24-ounce sodas)

Let F = the fraction (proportion) of 12-ounce sodas in stock
This means 1 - F = the fraction (proportion) of 24-ounce sodas in stock
We also know the weighted average of all sodas is 22 ounces.

So, we can write: 22 = (F)(12) + (1 - F)(24)
Expand: 22 = 12F + 24 - 24F
So: 22 = 24 - 12F
So: -2 = -12F
So: F = 2/12 = 1/6

Answer: B

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A convenience store stocks soda in 12-ounce and 24-ounce bottles. If t  [#permalink]

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New post Updated on: 18 Oct 2018, 01:43
1
dimmak wrote:
A convenience store stocks soda in 12-ounce and 24-ounce bottles. If the average capacity of all the bottles in the store is 22 ounces, then what fraction of the bottles in the store are 12 ounces?

A) 1/3
B) 1/6
C) 2/3
D) 5/6
E) 2/5


Let
The number of 12 ounce bottles = X and
The number of 24 ounce bottles = Y
Then, it is given in question stem that average of all the bottles are 22 ounces
so, (12*X+24*Y)/(X+Y)= 22
Solving this we get-> 12X+24Y=22X+22Y
=> 2Y=10X
=>Y/X=5

Now, we have to find what fraction of the bottles are 12 ounces=> X/(X+Y)=>1/(1+Y/X)
=> 1/(1+5)=>1/6. Hence correct option is B.

Originally posted by AnupamKT on 15 Oct 2018, 09:38.
Last edited by AnupamKT on 18 Oct 2018, 01:43, edited 1 time in total.
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Re: A convenience store stocks soda in 12-ounce and 24-ounce bottles. If t  [#permalink]

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New post 15 Oct 2018, 18:35
AnupamKT wrote:
dimmak wrote:
A convenience store stocks soda in 12-ounce and 24-ounce bottles. If the average capacity of all the bottles in the store is 22 ounces, then what fraction of the bottles in the store are 12 ounces?

A) 1/3
B) 1/6
C) 2/3
D) 5/6
E) 2/5


Let
The number of 12 ounce bottles = X and
The number of 24 ounce bottles = Y
Then, it is given in question stem that average of all the bottles are 22 ounces
so, (12*X+24*Y)/(X+Y)= 22
Solving this we get-> 12X+24Y=22X+22Y
=> 2Y=10X
=>Y/X=5

Now, we have to find what fraction of the bottles are 12 ounces=> X/(X+Y)=>1/(1-Y/X)
=> 1/(1-5)=>1/6. Hence correct option is B.


Hi AnupamKT,
Please check X/(X+Y)=1/(1+Y/X)
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Re: A convenience store stocks soda in 12-ounce and 24-ounce bottles. If t  [#permalink]

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New post 16 Oct 2018, 05:41
PKN wrote:
AnupamKT wrote:
dimmak wrote:
A convenience store stocks soda in 12-ounce and 24-ounce bottles. If the average capacity of all the bottles in the store is 22 ounces, then what fraction of the bottles in the store are 12 ounces?

A) 1/3
B) 1/6
C) 2/3
D) 5/6
E) 2/5


Let
The number of 12 ounce bottles = X and
The number of 24 ounce bottles = Y
Then, it is given in question stem that average of all the bottles are 22 ounces
so, (12*X+24*Y)/(X+Y)= 22
Solving this we get-> 12X+24Y=22X+22Y
=> 2Y=10X
=>Y/X=5

Now, we have to find what fraction of the bottles are 12 ounces=> X/(X+Y)=>1/(1-Y/X)
=> 1/(1-5)=>1/6. Hence correct option is B.


Hi AnupamKT,
Please check X/(X+Y)=1/(1+Y/X)


Thanks. Noted. Typo error.
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Re: A convenience store stocks soda in 12-ounce and 24-ounce bottles. If t  [#permalink]

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New post 17 Oct 2018, 18:06
dimmak wrote:
A convenience store stocks soda in 12-ounce and 24-ounce bottles. If the average capacity of all the bottles in the store is 22 ounces, then what fraction of the bottles in the store are 12 ounces?

A) 1/3
B) 1/6
C) 2/3
D) 5/6
E) 2/5


We can let a = the number of 12-ounce bottles and b = the number of 24-ounce bottles. We can create the equation:

12a + 24b = 22(a + b)

12a + 24b = 22a + 22b

2b = 10a

b = 5a

Thus, a/(a + b) = a/(a + 5a) = a/6a = 1/6 of the bottles are 12 ounces.

Answer: B
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Re: A convenience store stocks soda in 12-ounce and 24-ounce bottles. If t  [#permalink]

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New post 23 Oct 2018, 14:10
dimmak wrote:
A convenience store stocks soda in 12-ounce and 24-ounce bottles. If the average capacity of all the bottles in the store is 22 ounces, then what fraction of the bottles in the store are 12 ounces?

A) 1/3
B) 1/6
C) 2/3
D) 5/6
E) 2/5


This question can be solved by using allegation method :
Attachments

Untitledbottles.png
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Re: A convenience store stocks soda in 12-ounce and 24-ounce bottles. If t  [#permalink]

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New post 27 Oct 2018, 00:35
"Teeter Totter" method for solving in 1 minute.

12----------22--24

12 is further away from 22 than 24 is (the respective differences are: 10 and 2).
Take the differences and write them above the teeter totter.
Take the total difference (24-12) and write that below the teeter totter.

10 2
12----------22--24
12

Since the total average (22) is much closer to 24, it means there must be a greater number of 24 ounce bottles than 12 ounce bottles.
To find proportion, take the individual differences and divide by total difference.

Hence, proportion of 24 ounce bottles = 10/12 = 5/6
and proportion of 12 ounce bottles = 2/12 = 1/6

Answer B.

P.S the Manhattan guides discuss this method in greater detail.
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Re: A convenience store stocks soda in 12-ounce and 24-ounce bottles. If t &nbs [#permalink] 27 Oct 2018, 00:35
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