Properties of Sets

[Valentino99]

Hello,

could somebody please help with an easier solution for this problem:

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?
a) 30%
b) 35%
c) 40%
d) 45%
e) 50%

ans: [spoiler]c)[/spoiler]

[GMATGuruNY]

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.

[Valentino99]

Hi Mitch,

Thank you for the detailed explanation.

Could you please let know what was your thought process to identify this problem as a mixture problem?

When I attempted this problem, I interpreted the meaning of the sentence - "she has marked 30% of the store items for sale" as : 30% of the total no. of items were supposed to be marked for sale. I found out soon that this is wrong as the following percentage (55%)in the problem when applied to 70 items (100 - 30) gives no. of items as non-integer. I just wanted to share what I had thought so that you may correct the way I approached/thought about the problem.

Thanks!
Saurabh

[getherdonedave]

Hi Mitch,

Thank you for the explanation. Could you please explain how you arrived at 45%? Thanks again.


*********Nevermind, just re-read the question. got it!

[teejaycrown]

45% was deduced as 1-55% can you please explain why?