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Percentages

by cypherskull » Sun Aug 19, 2012 10:52 am
I found the language of the problem statement somewhat ambiguous.

A new sales clerk in a department store has been assigned to mark sale items with red tags, and she has marked 30% of the store items for sale. However, 20% of the items that are supposed to be marked with their regular prices are now marked for sale, and 55% of the items that are supposed to be marked for sale are marked with regular prices. What percent of the items that are marked for sale are supposed to be marked with regular prices?
1.30%
2.35%
3.40%
4.45%
5.50%
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by GMATGuruNY » Sun Aug 19, 2012 12:03 pm
cypherskull wrote:I found the language of the problem statement somewhat ambiguous.

A new sales clerk in a department store has been assigned to mark sale items with red tags, and she has marked 30% of the store items for sale. However, 20% of the items that are supposed to be marked with their regular prices are now marked for sale, and 55% of the items that are supposed to be marked for sale are marked with regular prices. What percent of the items that are marked for sale are supposed to be marked with regular prices?
1.30%
2.35%
3.40%
4.45%
5.50%
This is a mixture problem.
Ingredient 1: Items that should be marked REGULAR.
Of these items, the percentage marked for sale = 20%.
Ingredient 2: Items that should be marked FOR SALE.
Of these items, the percentage marked for sale = 45%.
When the two types of items are MIXED, the percentage marked for sale = 30%.

Use alligation.

Step 1: Plot the 3 percentages on a number line, with the two starting percentages (20% and 45%) on the ends and the goal percentage (30%) in the middle.
should be regular (20%)----------30%------------------(45%) should be for sale

Step 2: Calculate the distances between the percentages.
should be regular (20%)-----10-----30%----------15--------(45%) should be for sale

Step 3: Determine the ratio in the mixture.
(should be regular) : (should be for sale) is equal to the RECIPROCAL of the distances in red.
Thus:
(should be regular) : (should be for sale) = 15:10 = 3:2.

Since the sum of the parts of the ratio = 3+2 = 5:
Of every 5 items, 3 should be regular and 2 should be for sale, implying that the fraction that should be regular = 3/5.

Let the total number of items = 100.
Should be regular = (3/5)100 = 60.
Since 20% of these items are marked for sale, should be regular but marked for sale = .2(60) = 12.
Since 30% of all the items are marked for sale, the total marked for sale = .3(100) = 30.
Thus:
(should be regular but marked for sale)/(total marked for sale) = 12/30 = 2/5 = 40%.

The correct answer is C.
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by neelgandham » Sun Aug 19, 2012 12:20 pm
Let the total number of items be 100.
Let the total number of items that should be marked for sale be S.
Then the total number of items that should be at the regular price is 100-S.

Total number of items that are actually marked for sale by the clerk = (30/100)*100=30.

20% of the items that are supposed to be marked with their regular prices are now marked for sale. i.e. Items that are supposed to be marked with their regular prices but now marked for sale =
(20/100)*(100-S)

55% of the items that are supposed to be marked for sale are marked with regular prices. i.e. Items that are supposed to be marked for sale are marked with sale price = [(100-55)/100]*S

Total number of items that are actually marked for sale by the clerk = 30 = Items that are supposed to be marked for sale are marked with sale price + Items that are supposed to be marked with their regular prices but now marked for sale = [(100-55)/100]*S + (20/100)*(100-S)
[(100-55)/100]*S + (20/100)*(100-S) = 30
0.45*S + 20 - 0.2*S = 30
0.25*S = 10
S = 40.

Total number of the items that are supposed to be marked with regular prices = (20/100)*(100-S) = (1/5)*(100-40) = 12(From the question: 20% of the items that are supposed to be marked with their regular prices are now marked for sale)

What percent of the items that are marked for sale are supposed to be marked with regular prices?
?
= Total number of the items that are supposed to be marked with regular prices/Total number of items that are actually marked for sale by the clerk = 12/30 = 4/10 = 40/100 = 40%

IMO C
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