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Convenience store - Average question

by sreak1089 » Mon Jun 14, 2010 8:46 pm
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 must be sold for the average volume per bottle to be reduced to 25 ounces if no 20 ounce bottles are sold.

A. 10
B. 20
C. 30
D. 32
E. 34
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by Rich@VeritasPrep » Mon Jun 14, 2010 9:24 pm
Hey there,

There are a couple ways you can do this. One really sneaky thing you can do is notice that the weighted average of 35 is three-fourths of the way from 20 to 40. That simply means there will be 3 times as many 40-oz bottles as 20-oz bottles. Since the sum of the number bottles is 48, that means there are 12 20-oz bottles and 36 40-oz bottles.

You want to go to an average of 25, and the number of 20-oz bottles will stay the same. 25 is one-fourth of the way from 20 to 40, which means you'll now have 3 times as many 20 oz bottles as 40 oz bottles. That makes for 12 20-oz bottles and 4 40-oz bottles.

So you go from 36 to 4 40-oz bottles for a decrease of [spoiler]
32[/spoiler] and a final answer of D

Or, you can do the straight algebra...

Here's a detailed outline:

x = # of bottles of 20-oz water
y = # of bottles of 40-oz water

x+y=48 total # of bottles

20x+40y= 48*35 total volume

x+2y= 24*7/2 = 168/2 divide both sides by 20

x+2y = 84

Now create system of two equations:

x+2y = 84
x+y=48

From this, you get:

x=12
y=36

So we now know there are 12 bottles of 20-ounce water and 36 bottle of 40-oz water.

We want to reduce the average volume to 25 oz.

The number of 20-ounce bottles will stay at 12 for a total of 240 oz.

New average will be:

(240+40k) / (12 + k) = 25 where k is the new number of 40-oz bottles

240+40k = 300 + 25k

15k = 60

k = 4

So the number of 40-oz bottles will go from 36 to 4.

We must sell 32 bottles, and the answer is D

Hope this makes sense! Let me know if you have any further questions.
Last edited by Rich@VeritasPrep on Mon Jun 14, 2010 9:38 pm, edited 1 time in total.
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by GMATGuruNY » Mon Jun 14, 2010 9:30 pm
The problem tells us that the average volume per bottle is 35 ounces and that there are 48 bottles.

When a problem gives you the average, figure out the TOTAL.

The total volume of all 48 bottles is 48 * 35 = 1680 ounces.

What are we looking for? The problem asks for the number of 40-ounce bottles that must be sold in order for the average to decrease to 25 ounces per bottle.

Whenever a PS question asks for a specific amount, consider trying out the answers in order to determine which is correct.

When you're trying out the answer choices, start with the one right in the middle (normally answer choice C).

In this problem, the answer choices represent how many 40-ounce bottles need to be sold. So let's start with answer choice C:

If we sell 30 40-ounce bottles, we'll lose 30 * 40 = 1200 total ounces.
Since we started with 1680 total ounces, we'll now have 1680 - 1200 = 480 total ounces left.
Since we started with 48 bottles and just sold 30, we'll have 48-30 = 18 bottles left.
The new average volume per bottle will be 480/18 = 26.666.

Too high. We want the new average to be 25 ounces. So we need to sell more 40-ounce bottles.

Let's try answer choice D:

If we sell 32 40-ounce bottles, we'll lose 32 * 40 = 1280 total ounces.
Since we started with 1680 total ounces, we'll now have 1680 - 1280 = 400 total ounces left.
Since we started with 48 bottles and just sold 32, we'll have 48-32 = 16 bottles left.
The new average volume per bottle will be 400/16 = 25.

Success!

The correct answer is D.
Last edited by GMATGuruNY on Mon Jun 14, 2010 10:04 pm, edited 9 times in total.
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by Testluv » Mon Jun 14, 2010 9:31 pm
The current average is 35 ounces. This means there are three times as many forty ounce bottles as there are twenty ounce bottles.* Thus, there are (3/4)*48 = 36 forty ounce bottles, and 48 - 36 = 12 twenty ounce bottles.

The desired average is 25. Since we have 20 and 40 ounce bottles, we need three times as many twenty ounce bottles as forty ouncers.*.....we have 12 twenty ounce bottles right now....12 is three times what number? Clearly, 12 is three times 4. So, to get to the desired average of 25, we need 4 forty ounce bottles. We have 36 right now, so we need to sell off 32.

Choose D.

*If x is x units from the grand average, and y is y units from the grand average, then the ratio x/y is equivalent to the proportion Y/X. Suppose the average grade of males in a class is 60, the average grade of females is 75, and the class' overall average is 70. Well, the males average of 60 is ten units from the grand average of 70. But the females average of 75 is only five units from the grand average. Because the grand average is closer to the females' average, there are clearly more females. In fact, the females to males ratio is exactly 10:5 or 2:1.
Last edited by Testluv on Mon Jun 14, 2010 9:38 pm, edited 1 time in total.
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by Testluv » Mon Jun 14, 2010 9:33 pm
Wow. Three teachers posted virtually simultaneously.
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by sreak1089 » Tue Jun 15, 2010 7:44 pm
Wonderful !! That's called a Student's delight :)
I like different approaches to solving this problem.

Weighted average concept clearly got drilled down.. Thanks a lot.

Testluv wrote:Wow. Three teachers posted virtually simultaneously.
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by amising6 » Tue Jun 15, 2010 9:37 pm
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 must be sold for the average volume per bottle to be reduced to 25 ounces if no 20 ounce bottles are sold.

A. 10
B. 20
C. 30
D. 32
E. 34



my take:
let there be x no of bottles of 20 ounces
so no of 40 ounces of bottle will be (48-x) (since total number of bottle of 48)
average volume is 35
so we can say
48*35 is total volume
48*35=20*x +40(48-x)( total volume is equal to no of bottles 8 average volume which will be equal to volume of 20 and 40 ounce bottle)
solving we will get x as 12
so there are 12 number of 20 ounce bottle
that means there will be 48-12=36 bottle of 40 ounces

now second condition
let the no of 40 ounces bottle sold be Y

so 25 (48-y)=20*12+40(36-y)
solving we will get Y as 32
so 32 bottles of 40 ounces should be sold
thanks
AKS
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by Raghuram R Menon » Mon Mar 07, 2016 9:53 pm
@gmat guru ny brilliant explanation.got the basic idea right.Thank you sir.
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