A store sold 60 percent of the hats from a shipment of hats at a

Question

A store sold 60 percent of the hats from a shipment of hats at a selling price that was 50 percent greater than the store's cost for each hat. Then the store reduced the selling price by 66 2/3 percent and sold 70 percent of the remaining hats at the reduced selling price. If the store did not sell any other hats form the shipment, then the store's gross profit from the sale of the hats from the shipment was what percent of the store's cost for the hats from the shipment?

A) 1%
B) 4%
C) 9%
D) 13%
E) 25%

A) 1%
B) 4%
C) 9%
D) 13%
E) 25%

gmatophobia · Accepted Answer

Choosing the right numbers is very crucial to solve this question.

Let's assume that the number of hats in the shipment = 100

A store sold 60 percent of the hats from a shipment of hats at a selling price that was 50 percent greater than the store's cost for each hat

Number of hats sold = 60

Also given - The initial selling price of the hats =  \frac{3}{2}  * times the store's cost for each hat.

Let's assume the stores cost for each hat = $2
Cost price of 100 hats = $2 * 100 = $200

The initial selling price of the hats =  \frac{3}{2} * 2 = $3

Hats remaining =  100 - 60 = 40

Then the store reduced the selling price by 66 2/3 percent and sold 70 percent of the remaining hats at the reduced selling price.

66 \frac{2}{3} %  = \frac{2}{3}

New selling price =  \frac{1}{3} rd * old selling price

New selling price =  \frac{1}{3} rd * 3 = $1

Number of hats sold =  \frac{70}{100} * 40 = 28

Total Revenue = (60 * 3) + (28 * 1)= 208

Profit = $(208-200) = $8

Profit % =  \frac{8}{200} * 100 = 4 %

Option B

JeffTargetTestPrep · Answer

If the store bought q hats at p dollars each, we have:

Total cost = q × p = qp

The total revenue from the sale of 60 percent of the hats:

Total revenue₁ = 0.6q × 1.5p = 0.9qp

(Note: “50 percent greater” corresponds to a factor of 1.5.)

The total revenue from the sale of 70 percent of the remaining 40 percent of the hats:

Total revenue₂ = (0.7)(0.4)q × (1/3)(1.5)p = 0.14qp

(Note: “66 2/3 percent decrease” corresponds to a factor of 0.333, or 1/3.)

Total profit = Total revenue₁ + Total revenue₂ – Total cost

Total profit = 0.9qp + 0.14qp – qp = 0.04qp

Therefore, the percentage in question is:

Percentage = Total profit/Total cost = 0.04qp/qp = 0.04 = 4%

Answer: B

Kinshook · Answer

Given: A store sold 60 percent of the hats from a shipment of hats at a selling price that was 50 percent greater than the store's cost for each hat. Then the store reduced the selling price by 66 2/3 percent and sold 70 percent of the remaining hats at the reduced selling price.

Asked: If the store did not sell any other hats form the shipment, then the store's gross profit from the sale of the hats from the shipment was what percent of the store's cost for the hats from the shipment?

Let the cost of the hats from the shipment be x.

Let the total hats be 100.
A store sold 60 percent of the hats from a shipment of hats at a selling price that was 50 percent greater than the store's cost for each hat.
60 hats sold at selling price of 1.5x.

Then the store reduced the selling price by 66 2/3 percent and sold 70 percent of the remaining hats at the reduced selling price. 
40*70% = 28% of hats were sold at 1.5x(100% - 662/3)% = .5x

Total selling price = 1.5x *60 + .5x *28 = 90x + 14x = 104x
Total cost price = 100x 
Gross profit = 104x - 100x = 4x

Gross Profit as percent of the store's cost for the hats from the shipment = 4x/100x *100% = 4%

IMO B

Aishyk97 · Answer

Hi  Bunuel   JeffTargetTestPrep   Kinshook   gmatophobia ,
Can you help me understand where I am going wrong in my logic.

I asssumed there are 100 hats in the shipment. The CP = 100 for each hat.

Now 60 percent are sold at 1.5 times the CP. SP = 150. So profit on 60 hats = 60*50 = 3000
The SP is then reduced by 2/3rd. New SP = 150*1/3 = 50
Loss per hat = 150 - 100 = 50
So loss for 28 hats (70 percent of remaining) = 28*50 = (1400)

So I got total Profit as 3000 - 1400 = 1600

So  \frac{1600}{100*100}*100

But I'm really far from the required answer.

Any help is appreciated.

chetan2u · Answer

You have not included the loss incurred due to non sale of 12 hats (30% of remaining).
This will incur a loss of 12*100 or 1200.

Total profit now is 1600-1200 or 400.
% =100 * 400/(100*100) = 4

chetan2u · Answer

As the question deals with %, take the amounts as 100. Let the CP and also the number of hats purchased be 100.

1).  Total money spent  = 100*100 = 10,000

2)  Money received  
(I) 60 hats at 50% profit = 60*150 = 9000
(II) 70% of remaining 40 hats, that is 28 hats, at 66 2/3 % discount. 
33 1/3, 66 2/3 and 100 are equivalent to three parts 1/3, 2/3 and 3/3 respectively. Thus a discount of 2/3 the amount means the new SP is 1/3 of initial SP = 150*1/3 = 50. Amount received = 28*50 = 1400
 Total  = 9000+1400 = 10400

3)  Profit  = 10400-10000 = 400
Profit % =  \frac{400}{10000}*100=4

B

GMATCoachBen · Answer

In the video below, I breakdown 2 key habits for ones like this:

1) Organize the info into a P*Q = R table, just like many people do on Rate problems (R*T = D)

2) For the Cost, pick the number 100 for the Price and Quantity

https://www.youtube.com/watch?v=38hPR2amVeQ

einstein801 · Answer

Hi there,

Im trying to figure out where I went wrong and would appreciate if you could give me some help.
I have the same steps and calculations as you but start to go wrong when I calculate gross profit immediately for each selling price, 3c/2 and c/2.

Gross profit for 60% of hats = (3c/2-c)*0.6x = 0.3xc
Gross profit for 70% of remainder of hats = (c/2-c)*0.28x = -0.14xc

Total gross profit = 0.3xc -0.14xc = 0.16xc
Gross profit as % of total cost = 0.16xc / xc = 16%.

Bunuel · Answer

If you calculate the profit/loss on individual batches of hats, then you should also take into account the third batch of 0.12x hats, which were not sold and thus generated a loss of -0.12cx. Thus, the gross profit equals 0.3xc - 0.14xc - 0.12xc = 0.04xc.

KarishmaB · Answer

Method 1: Assuming Values Say there were 100 hats and each was bought for $6 (LCM of 2 and 3 so that we can easily increase it by 50% and reduce by 2/3rd). Total cost price = $600 60 hats were sold for selling price that was 50% more i.e. $9 each Revenue = 60*9 = 540 70% of the remaining 40 hats i.e. 28 hats were sold for selling price that for 2/3rd less than 9 i.e. at $3 Revenue = 28*3 = $84 Total Profit = 540 + 84 - 600 = $24 Percentage of profit = 24/600 * 100 = 4% Answer (B) Method 2: Use weighted averages 60 hats were sold at 50% profit. 70% of 40 hats i.e. 28 hats were sold at "increase of 50% and then reduction of 66.67%". We know how to handle successive percentage changes using multiplication factors. It will be (3/2) * (1/3) = 1/2. Hence this is 50% loss. Rest of the 12 hats are sold at 100% loss Avg profit = \frac{(60*50 + 28*(-50)+12*(-100))}{100} = 4 Concept Videos of weighted averages: https://www.youtube.com/watch?v=_GOAU7moZ2Q Post on successive percentage changes: https://anaprep.com/arithmetic-successi ... e-changes/

SOFIA19 · Answer

Hi. How did you come to the conclusion that 66 2/3 percent is equal to 2/3? Where did the 66 go? When I was doing it I thought it was 66 * 2/3 %, which in my mind would equal 44%. Thank you.

Krunaal · Answer

66 2/3 percent translates to 66 + 2/3 percent => 66 + 0.66667 percent => 66.66667 percent => 66.66667 * 1/100 => 0.6666667 => 2/3

Usually you will find multiplication denoted by a * or brackets. For eg. 2*3 = 6 or 2(3) = 6. Hope this helps.

GMATGuruNY · Answer

Let the shipment = 100 hats
Since the price for each hat is initially 1/2 greater than the cost and later reduced by 2/3, let the cost for each hat = the product of the two denominators = 2*3 = 6
Total cost for the 100 hats = (100 hats)($6 per hat) = 600
 
A store sold 60 percent of the hats at a selling price that was 50 percent greater than the store's cost for each hat .
At a selling price of $9 per hat -- 50% greater than the $6 cost -- the revenue for the first 60 hats = (60 hats)($9 per hat) = 540

The store reduced the selling price by 66 2/3 percent and sold 70 percent of the remaining hats at the reduced  selling price.
70% of the 40 remaining hats = 28 hats
At a selling price of $3 per hat -- 2/3 less than the initial $9 price -- the revenue for the next 28 hats = (28 hats)($3 per hat) = 84

Profit = total revenue - total cost = (540+84) - 600 = 24

The store's gross profit from the sale of the hats was what percent of the store's cost for the hats from the shipment? 
 \frac{Profit}{Cost} = \frac{24}{600} = \frac{4}{100} =  4%

B

hattie · Answer

I also think it's 14%. I followed  gmatophobia  method to calculate, assuming there are 100 hats in total, with a cost of 2 $ per hat, so the selling price for the first case is 3 $.

So, for the first case, the revenue is 3 * 60 = 180
The cost is 2 * 60 = 120
So, the profit for the first case is 180 - 120 = 60

For the second case, the revenue is 1 * 28 = 28
The cost is 2 * 28 = 56

So, the profit for the second case is 56 - 28 = -28
So, the total gross profit is 60 - 28 = 32

Therefore, the percentage of the total cost is 32 / 200 = 0.16 = 16%

Why isn't it 4%? Where did I go wrong in my calculation?

In the reply, I saw that if we calculate the gross profit for each batch, we need to include the third batch's loss of -24.
But why, according to your method, is there no need to calculate the gross profit for the third batch? In my method, we would need to account for the third batch's gross profit.

I’ve looked at it many times and couldn’t find any errors. I’d like to ask the experts to point out where I went wrong! Thank you!

Krunaal · Answer

The key misstep lies in this batch-wise profit calculation approach, as it does not directly account for the overall gross profit across the shipment.

Gross profit is not the sum of the profits from individual batches; instead, it is the difference between total revenue and total cost across the entire shipment.

In your calculation:

You found the profit for the first and second sales correctly but missed incorporating the overall structure of gross profit as a single calculation. 
 Unsold hats were excluded, but their costs were already accounted for in the total cost. This led to a misalignment between total revenue and total cost.

This led to an overestimation.

Gross profit is the total revenue from all sales minus the total cost of all hats. It must always be calculated holistically

Gross Profit =   Total Revenue - Total Cost

Now, first adding revenue you calculated for two batches = 180 + 28 = 208
Total cost of the shipment = 200
 Gross Profit = 208 - 200 = 8

8/200 = 4%

Hope it helps.

siddhantvarma · Answer

Hi experts.  JeffTargetTestPrep   Bunuel   KarishmaB 
Going through all the replies in this post, I still have a small doubt:

Why is gross profit taken as total revenue - total cost across all shipments and not as just revenue - cost (across shipments sold)? Is this because of how "gross profit" defined? If yes, please expand on what "gross profit" actually means. I googled the exact definition of "gross profit" and found this:
 
It says cost of  goods sold , so why are we taking the cost of goods that actually weren't sold? 
In my head, I assumed gross profit as profit, which we define as "revenue from goods sold" - "cost of the goods sold". By this definition, the cost should only consider the "goods that are sold".

Krunaal · Answer

Let me try to answer your doubt too,

The definition you found is correct:
 
COGS typically refers to the total cost of producing or procuring all the goods available for sale, not just the goods that are sold. This is because the cost of goods in inventory (whether sold or unsold) is considered part of the store's operational cost for the shipment.
In this specific question, we are dealing with  total shipment costs   rather than a dynamic inventory accounting where unsold goods can be carried forward.  Since the problem states that no other hats from the shipment were sold, the cost of the hats in the shipment is fully accounted for, regardless of how many were sold.
In short, COGS refers to the total cost of acquiring or producing the goods that were available for sale, regardless of how many were sold.

Let me know if this helps put things in perspective, or if you need more clarification.

KarishmaB · Answer

You are given in the question: ...  If the store did not sell any other hats form the shipment,...

It means the remaining hats were unsold and hence a loss was taken on them. That needs to be taken into consideration in the calculation of gross profit. 
When something is to be sold later, we do not consider its cost price in our current calculations then. But if something remains unsold, that is a loss and its cost price needs to be adjusted in gross profit.

siddhantvarma · Answer

Thanks  Krunaal  and  KarishmaB 
It makes sense now. One last follow up - whoever wants to pitch in on this, say we were not to take the cost of unsold goods into account, how would the question be phrased differently in that case? My goal is to learn how I can differentiate between the two cases. 
Had the last line directly been " the store's gross profit from the sale of the hats from the shipment was what percent of the store's cost for the hats from the shipment?"
instead of "  If the store did not sell any other hats form the shipment,  then the store's gross profit from the sale of the hats from the shipment was what percent of the store's cost for the hats from the shipment?"
Would we then just take the cost of the goods sold into account and not the unsold goods?

A store sold 60 percent of the hats from a shipment of hats at a

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A store sold 60 percent of the hats from a shipment of hats at a

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