A new sales clerk in a department store has been assigned to

Question

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%

A. 30% 
B. 35% 
C. 40% 
D. 45% 
E. 50%

Bunuel · Accepted Answer

# of items 100 (assume);
# of items that should be marked for sale is x;
# of items that should be at the regular price is 100-x;
# of items that are actually marked for sale 0.3*100=30.

Let's see how many items are marked for sale:
20% of 100-x (20% of the items that are supposed to be marked with their regular prices (100-x)  are now marked for sale ); 
100-55=45% of x (since 55% of the items that are supposed to be marked for sale (x) are marked with regular prices, then the remaining  45% of x are marked for sale );

So, 0.2(100-x)+0.45x=30 --> x=40 (# of items that should be marked for sale).

So, # of the items that are supposed to be marked with their regular prices is 0.2*(100-40)=12, which is 12/30*100=40% of # of items that are actually marked for sale.

Answer: C.

EvaJager · Answer

Here is a pure algebraic approach:

If in the store there are  R  items that should sell at regular price, and  S  items that should sell at reduced price,
then the total number of items is  R + S . 
 30%  of them, or  0.3(R + S)  items are now marked for sale and this is comprised of  0.2R  and  0.45S , 
as wrongly  20%  of the regular items, and only  45%  of the sale items were marked for sale ( 55%  of the sale items were marked regular).
So,  0.3(R + S) = 0.2R + 0.45S , from which we can deduce that  0.1R = 0.15S , or  2R = 3S.

We have to evaluate the ratio  \frac{0.2R}{0.3(R+S)}  - out of those marked for sale, what fraction/percentage should be marked regular.

\frac{0.2R}{0.3(R+S)}=\frac{2R}{3R+3S}=\frac{2R}{3R+2R}=\frac{2R}{5R}=\frac{2}{5}=40% .

Hence, answer C.

KarishmaB · Answer

Say there are 100 items in the store. Some of them are sale items (those that should be marked for sale) and the rest are regular items (should have regular prices)

20% of regular items are marked for sale. 45% of sale items are marked for sale (since 55% of sale items have regular prices). Total 30% of the items are marked for sale. So 30 items are marked for sale.
Does it remind you of something? Weighted Average!

regular items/sale items = w1/w2 = (45 - 30)/(30 - 20) = 3/2

Total 60% (=3/5) of the items are regular items. 20% of them are marked for sale so number of regular items marked for sale = 20% of 60 = 12
Out of the 30 items marked for sale, 12 are actually regular items which is 12/30 *100 = 40%

shreya717 · Answer

Thanks Bunnel. That was very helpful.

I arrived at C as well, however I used plugging in. (I happened to plug in 100 as total and 40 as number of items that should be on sale, hence arrived at C as well!)

kuttingchai · Answer

Book marking for future reference : Vote for C

Total items : 100
Marked for Sale: 30%(100) = 30

Total Retail P items = x
Total Sales P items = (100-x)

20% of the items that are supposed to be marked with their regular prices are now marked for sale
20%(x) - marked for sale by mistake

55% of the items that are supposed to be marked for sale are marked with regular prices. 
therefore, 45% of items marked correctly with sales price

So total we have

20%(x) + 45% (100-x) = 30

(20x + 4500 - 45x) / 100 = 30

4500 - 3000 = 25x

x = 60 (total items for retail price)

therefore total items for sales are = 40

20%(x) = 20%(60) = 12 items tagged wrongly for sale

therefore,

12 = z% (30)
z = 40%

farukqmul · Answer

Hi,
Can you explain more about the weighted average ?Thanks

KarishmaB · Answer

Check out posts on this concept on the blog link given in my signature below.

Dienekes · Answer

Can this problem be solved using the double-set matrix technique?

jorgearredondoc · Answer

I think this could be better understood if using this 2x2 Matrix attached.

PareshGmat · Answer

Regular ................... Sale .................... Total

100-x ............................. x .................... 100 (Assume)

20% of regular is marked sale  = \frac{20(100-x)}{100}

55% of sale is marked regular, which also means 45% of sale is "actually" for sale  = \frac{45x}{100}

Total sale = 30

\frac{20(100-x)}{100} + \frac{45x}{100} = 30

x = 40

Answer = C

GMATinsight · Answer

Answer: Option C

Check solution as attached

JeffTargetTestPrep · Answer

Since this a percent problem, we can assign a “good” number as the total number of items in the store. So, let’s say the total number of items in the store is 100. Since the sales clerk has marked 30% of the store items for sale, she has marked 30 items as sale items, and therefore 70 items are regular-price items.

We assume the total number of items is 100, and let’s assume that x items were supposed to be marked as sales items. Thus, 100 - x items were supposed to be marked as regular-price items.

Looking back at the given information, we know that 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. Thus:

0.2(100 - x) = number of items that are marked as sale items but should be marked as regular-priced items, and thus:

0.8(100 - x) = number of items that are marked (correctly) as regular-price items.

0.55x = number of items that are marked as regular-price items but should be marked as sale items, and thus:

0.45x = number of items that are marked (correctly) as sale items.

Recall that 30 items are marked as sale items and 70 items as regular-price items. Therefore, we have:

0.45x + 0.2(100 - x) = 30

and

0.55x + 0.8(100 - x) = 70

Let’s solve the first equation:

0.45x + 0.2(100 - x) = 30

45x + 20(100 - x) = 3000

45x + 2000 - 20x = 3000

25x = 1000

x = 40

The problem asks: “What percent of the items that are marked for sale are supposed to be marked with regular prices?”

Since we have that 0.2(100 - x) is the number of items marked as sale items when they should be marked as regular-price items, and we have that x = 40, there are:

0.2(100 - 40) = .2(60) = 12 such items.

We also have that a total of 30 items are marked for sale, so the percentage of the marked sale items that are supposed to be marked with regular prices is 12/30 = 4/10 = 0.4 = 40%.

Answer: C

sinhap07 · Answer

This is very vague Karishma. Can you draw the weighted average diagram to make it clearer?

KarishmaB · Answer

We have used the weighted average formula here which is much faster than drawing the diagram. Here is a discussion on both: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... -averages/

archana89 · Answer

The Picking Numbers strategy would be ideal for this problem, since there are percentages without given values. Let's use 100 as the total number of items in the store. This means that 30 of the items are marked for sale and 70 are marked at regular price. We are also told that 20% of the regular-priced items are marked for sale, and 55% of the sale items are marked at regular price. This means that 45% of the sale items are marked for sale. We can set up an equation with x as the number of regular-priced items and y as the number of sale items: 0.2x + 0.45y = 30.

Since 20% of the regular-priced items are marked incorrectly and 45% of the sale items are marked correctly, 80% of the regular-priced items are marked at regular price and 55% of the sale items are marked at regular price. This information gives us the next equation: 0.8x + 0.55y = 70.

Combining these 2 equations, we have a system of equations that we can solve. Multiply both sides of the equation 0.2x + 0.45y = 30 by –4 so that we can subtract it from 0.8x + 0.55y = 70 and isolate the y variable.

Now we know that the number of sale items is 40, so the number of regular-priced items must be 100 – 40 = 60. We can now calculate the number of regular-priced items marked for sale: Since 20% of 60 is 12 and the total number of items marked for sale is 30, the percent would be .

So, 40% of the items marked for sale are supposed to be marked at regular price.

Answer Choice (C) is correct.
 
 Confirm Your Answer:
 
Plug the numbers we figured back into the problem. Out of the 30 items marked for sale, 12 are marked incorrectly, so 30 – 12 = 18 of the sale items are actually on sale. Eighteen is indeed 45% of 40, so the correct number of sale items is 40. Our answer is confirmed.

viphaneev · Answer

One of the things that I have been struggling with is that how do you know that you need two variables R and S instead of assuming just one, say N = total number of items. 
I understand that with two variables, in this case, it is simpler to solve the problem. 
But my struggle is how do you come up with that intuitively in the first place ? 
I have been having this problem where I don't know what variables should I assume.

In addition,  EvaJager , if you are out there and see this message, I see your tag as an applied mathematician. 
How did you find a way to make mathematics come intuitive for you? In GMAT and non-GMAT. 
Plus, did you come up with the two variables in this question intuitively or by trial and error?

Thanks in advance.

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