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Each of the 30 boxes in a certain shipment weighs either 10

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Each of the 30 boxes in a certain shipment weighs either 10

by rakeshd347 » Fri Oct 11, 2013 7:16 pm
Each of the 30 boxes in a certain shipment weighs either 10 pounds or 20 pounds, and average (arithmetic mean) weight of the boxes in the shipment is 18 pounds. If the average weight of the boxes in the shipment is to be reduced to 14 pounds by removing some of the 20-pound boxes, how many 20-pound boxes must be removed?

A. 4
B. 6
C. 10
D. 20
E. 24

OA is .D
Last edited by rakeshd347 on Sat Oct 12, 2013 2:12 am, edited 1 time in total.
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by GMATGuruNY » Fri Oct 11, 2013 8:26 pm
Each of the 30 boxes in a certain shipment weighs either 10 pounds or 20 pounds, and average (arithmetic mean) weight of the boxes in the shipment is 18 pounds. If the average weight of the boxes in the shipment is to be reduced to 14 pounds by removing some of the 20-pound boxes, how many 20-pound boxes must be removed?

A. 4
B. 6
C. 10
D. 20
E. 24
We can plug in the answers, which represent the number of 20-pound boxes that must

Current total weight = 30*18=540.

Answer choice C: 10
After 10 20-pound boxes are removed, the new total weight = 540 - 10*20 = 340.
New number of boxes = 30-10 = 20.
New average = 340/20 = 17.
The resulting average needs to be smaller, so more 20-pound boxes must be removed.
Eliminate A, B, and C.

Answer choice D: 20
After 10 20-pound boxes are removed, the new total weight = 540 - 20*20 = 140.
New number of boxes = 30-20 = 10.
New average = 140/10 = 14.
Success!

The correct answer is D.
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by [email protected] » Sat Oct 12, 2013 12:44 am
Hi rakeshd347,

Mitch provides an explanation that TESTS THE ANSWERS, which is a great way to solve this problem. There's a Number Property rule that can actually make that solution a bit faster.

We're told that the average weight is 18 pounds, which makes the total weight 540 pounds (18 pounds x 30 boxes = 540 pounds)

Since the box weights are 10 or 20 pounds each, you can figure out the number of each.

If there were 30 20-pound boxes, the total weight would be 600 pounds.
Since the total weight is 540 pounds, that means that 6 of the boxes weight 10 pounds instead of 20.

So, there are:
6 10-pound boxes
24 20-pound boxes

Since we want to remove some number of the 20-pound boxes and get an average weight of 14 pounds, we know that the number of 10-pound boxes MUST BE greater than the number of 20-pound boxes (this is a weighted average Number Property). Since we have 6 10-pound boxes, we need to remove at least 19 20-pound boxes.

We can certainly TEST answer D first (which would eliminate the initial TEST that Mitch did and find the correct answer in 1 try) OR we can note that answer E (24) would eliminate ALL of the 20-pound boxes, which would make the average of the remaining boxes 10 pounds, which is too low (and would prove that answer D was correct).

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by mevicks » Sat Oct 12, 2013 2:03 am
Alt. algebraic approach (prone to errors & slower):

Let the no of 10 lbs boxes be x & no of 20 lbs boxes be y.
Given: x + y = 30

Original total weight: 10x + 20y
Avg weight = 18
Thus, (total weight)/30 = 18
10x + 20y = 18*30 ... (i)

Let a be the no of removed 20 lbs boxes.
New total weight: 10x + 20(y - a)
New avg weight = 14
Thus, (new total weight)/(30 - a) = 14
10x + 20(y - a) = 14 (30 - a) ... (ii)

Substitute (i) in (ii)
10x + 20y - 20a = 14*30 - 14a
18*30 - 14*30 = 6a
4*30 = 6a
[spoiler]a = 20 is the answer (D)[/spoiler]

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by GMATGuruNY » Sat Oct 12, 2013 2:50 am
rakeshd347 wrote:Each of the 30 boxes in a certain shipment weighs either 10 pounds or 20 pounds, and average (arithmetic mean) weight of the boxes in the shipment is 18 pounds. If the average weight of the boxes in the shipment is to be reduced to 14 pounds by removing some of the 20-pound boxes, how many 20-pound boxes must be removed?

A. 4
B. 6
C. 10
D. 20
E. 24

OA is .D
At its core, this is a WEIGHTED AVERAGE/MIXTURE problem.
Ingredient 1: 10-pound boxes
Ingredient 2: 20-pound boxes

Case 1: Average weight of the mixture = 18 pounds.
Case 2: Average weight of the mixture = 14 pounds.

An alternate -- and very fast -- approach is use ALLIGATION to determine the required ratio of 10-pounds boxes to 20-pound boxes in each case.

Case 1: Average weight = 18 pounds
Step 1: Plot the 3 weights on a number line, with the weights of the boxes on the ends and the average weight of mixture in the middle.
10-pound-----------18-----------20-pound

Step 2: Calculate the distances between the weights.
10-pound-----8-----18-----2-----20-pound

Step 3: Determine the ratio in the mixture.
The required ratio of 10-pound boxes to 20-pound boxes is equal to the RECIPROCAL of the distances in red.
10-pound : 20-pound = 2:8 = 6:24.
Thus, in Case 1, there are 6 10-pound boxes and 24 20-pound boxes, for a total of 30 boxes.

Case 2: Average weight = 14 pounds
Using the same approach, we get:
10-pound----4-----14----6-----20-pound
10-pound : 20 pound = 6:4.
Thus, in Case 2, there remain 6 10-pound boxes, while the number of 20-pound boxes decreases from 24 to 4 -- a decrease of 20 boxes.

The correct answer is D.

For more practice with alligation, check here:

https://www.beatthegmat.com/posting.php? ... t&p=678346
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