Yesterday Bookstore B sold twice as many softcover books as hardcover

From stem we know that no. of soft-cover books = twice no. of hard-cover books
S = 2H

Let no. of soft cover books = x
Let no. of hard cover books = y
We need to prove => Sx > Hy
=> 2Hx > Hy (since S=2H)


Statement 1:
y = x + 10

is 2H(x) > H(x + 10) ?

if x= 5, then 2*5*H > H*15 is not true
if x = 15, then 2*15H > H*25 is true

so statement 1 is not sufficient

Statement 2:
x,y > 14
We don't know anything else, so isn't sufficient

Combining both together:
So, if x = 14,then 2*14H > H(14 + 10) => 28H > 24H is true
if x = 15, then 2*15H > H(15 + 10) => 30H > 25H is true

Since the difference increases with increasing value of x, we can be sure that for all further values, revenues from soft cover will be greater than that from hard cover. Therefore C is sufficient.

Let the hardcover books sold be h, so softcover books sold become 2h.
For revenue we require the average prices too, so let the average price of softcover and hardcover be x and y.
We are looking for : Is 2hx>hy? Or Is 2x>y?

(1) The average (arithmetic mean) price of the hardcover books sold at the store yesterday was $10 more than the average price of the softcover books sold at the store yesterday.
y=x+10.
So the question becomes: Is 2x>x+10? OR Is x>10?
We don’t know whether x is greater than 10 or not.
Insufficient

(2) The average price of the softcover and hardcover books sold at the store yesterday was greater than $14.
Total price = h*y+2h*x
Total books = h+2h = 3h
Average = \(\frac{hy+2hx}{3h}=\frac{y+2x}{3}>14………y+2x>42\)
We cannot say anything about relation of x and y.
Insufficient


Combined
y+2x>42 and y=x+10
\(x+10+2x>42……..3x>32……..x>10.67>10\)
Sufficient


C

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Yesterday Bookstore B sold twice as many softcover books as hardcover books.
Let h = the number of hardcover books sold and 2h = the number of softcover books sold, implying that 3h = the total number of books sold.

Statement 1: The average (arithmetic mean) price of the hardcover books sold at the store yesterday was $10 more than the average price of the softcover books sold at the store yesterday
Case 1:
softcover price = $1 --> softcover revenue = 1*2h = 2h
hardcover price = $11 --> hardcover revenue = 11*h = 11h
In this case, softcover revenue is LESS THAN hardcover revenue, so the answer to the question stem is NO.

Case 2:
softcover price = $10 --> softcover revenue = 10*2h = 20h
hardcover price = $20 --> hardcover revenue = 20*h = 20h
In this case, softcover revenue is EQUAL TO hardcover revenue, so the answer to the question stem is NO.

Case 3:
softcover price = $20 --> softcover revenue = 20*2h = 40h
hardcover price = $30 --> hardcover revenue = 30*h = 30h
In this case, softcover revenue is GREATER THAN hardcover revenue, so the answer to the question stem is YES.

Since the answer is NO in Cases 1 and 2 but YES in Case 3, INSUFFICIENT.
The 3 cases indicate an inflection point of $10, as follows:
If the softcover price is LESS THAN OR EQUAL TO $10, the answer to the question stem is NO.
If the softcover price is GREATER THAN $10, the answer to the question stem is YES.

Statement 2: The average price of the softcover and hardcover books sold at the store yesterday was greater than $14.
\(\frac{total-revenue}{3h} > 14\)
total revenue > 42h
No way to determine whether softcover revenue is greater than hardcover revenue.
INSUFFICIENT.

Statements combined:
In Case 1, total revenue = (softcover revenue) + (hardcover revenue) = 2h + 11h = 13h
In Case 2, total revenue = (softcover revenue) + (hardcover revenue) = 20h + 20h = 40h
Neither case is viable, since Statement 2 requires that total revenue > 42h.
Implication:
To satisfy Statement 2, the softcover price must be GREATER THAN $10, as in Case 3.
Thus, the answer to the question stem is YES.
SUFFICIENT.



The inflection point discussed under Statement 1 can also be determined algebraically:
Let p = the softcover price and p+10 = the hardcover price.
Softcover revenue = (number sold)(unit price) = (2h)(p) = 2hp
Hardcover revenue = (number sold)(unit price) = (h)(p+10) = hp + 10h
For softcover revenue to exceed hardcover revenue, we get:
2hp > hp + 10h
hp > 10h
p > 10
For softcover revenue to exceed hardcover revenue, the softcover price must be GREATER THAN $10.

Hello woohoo921,

You have made a minor mistake in translating one of the statements (statement 2) and thus, you could not reject the incorrect set of data that you are considering. Let me correct that for you and clear your confusion.

You have taken the following values for the variables here:

• Average Price per softcover book (PS) = $10
• Average Price per hardcover book (PH) = $20
• Number of softcover books sold (NS) = 12
• Number of hardcover books sold (NH) = 6

Let’s check this data, one by one, with the question stem, statement 1, and finally with statement 2.

Question stem: “Bookstore B sold twice as many softcover books as hardcover books.”

• So, NS must be equal to 2 × NH.
• This is justified by the data we consider: 12 = 2 × 6.

Statement 1: “The average (arithmetic mean) price of the hardcover books sold at the store yesterday was $10 more than the average price of the softcover books sold at the store yesterday.”

• Our data satisfies this as well since PH = 20 and PS = 10, and 20 is indeed 10 more than 10.

Statement 2: “The average price of the softcover and hardcover books sold at the store yesterday was greater than $14.”

This is where you went wrong with your translation.

You considered the average of the average price per softcover book and average price per hardcover book. This gave you (20 + 10)/2 = 15.
BUT the statement talked about the average price of ALL (softcover and hardcover) the books. So, you need to consider the total number of books when calculating the average – not just 2 that signifies the two types of books.
You would get the correct average price by calculating the sum of the prices of all books divided by the total number of books.

• Thus, Average price = \(\frac{(PS × NS) +(PH × NH)}{(NS + NH)}\)
  
  • Using your values, we get average price = (10 × 12 + 20 × 6)/(12 + 6)
  • = 240/18 = 13.33
  • This is less than $14, and NOT greater than 14 as statement 2 wanted.

Hence, the set of data you took does NOT satisfy Statement 2. And thus, we cannot consider it.

I hope this clarifies the confusion you had.

Best Regards,
Ashish Arora,
Quant Expert, e-GMAT

The highlighted is not correct.
It solves to R(s) = 2R(h) - 20x

Use as few variables as you can.

Avg price of softcover books is P. Revenue from softcover books = P*2x
Avg price of hardcover books is (P + 10). Revenue from hardcover books = (P + 10)*x

The question is:
Is 2Px greater than (Px + 10x)?
i.e. Is (Px + Px) greater than (Px + 10x)?
i.e. Is Px greater than 10x?
That depends on what P is. If P is less than 10, then 10x is greater. If P is greater than 10, then Px is greater.

We know the ratio of Softcover to Hardcover book ratio, i.e. s:h = 2:1

Question: Is revenue from S > Revenue from B ?
For the calculation of Revenue we need to know the selling price of books and number of books (or the ratio of number of book and Ratio fo revenue per book)

Statement 1: Price of H = 10+Price of S
NOT SUFFICIENT

Statement 2: Avg of S and H > 14
This again takes us to the sum of price but comparison of prices is still unknown
NOT SUFFICIENT

Combining the statements

Let hardcover book sold = x
then softcover sold = 2x

Total price = x*(10+S)+2x*S >14*3x
i.e. 10+3S > 42
i.e. S > 10.66, Revenue = 10.66*2x
i.e. H > 20.66, Revenue = 20.66*x

Revenue of S > Revenue of H

SUFFICIENT

Answer: Option C

­

Hi Bunuel VeritasKarishma, chetan2u

I used the weighted avg scale method described in Karishma's blog here on gmatclub, just wanted to understand if I did it the right way or just got luck with the answer :

number of hardcover books : h, soft 2h,

Statement 1 : Avg of hardcover books is greater : we don't know if the total revenue was greater or the average is higher only becasue the denominator is smaller [ h <2h] [If it was the other way round and we were given that softcover books had higher avg then we would be sure that softcover books had higher revnue beacuse the denominator is smaller there and the avg still happens to be greater]

Statement 2 : insuff

both together :

Scale : Softcover h [x avg] ------14------ [x+10 avg] Harcover 2h

num of softcover books = 2h = x + 10 - 14/ 10

so x + 10 = 20h -14 , and x = 20h - 24,

Now for revnue comparison we multiply x * h

and x + 10 * 2h,

clearly x+10 * 2h is bigger, so (C)

Certain flaws
Average is greater than 14, so you should have taken it as 14+y
Next you seem to have taken 2h for hardcover in the final step but it is for softcover.

May be it is better to follow algebraic approach here.

Hello Bunuel, KarishmaB

To me, it seems the answer should be A based on the workings below. Appreciate your help to clarify. Thanks in advance.

Let (i) quantity of hardcovers = x, (ii) quantity of softcovers = 2x, (iii) revenue from hardcovers be R(h), and (iv) revenue from softcovers be R(s).

Based on Statement 1
R(h)/x - R(s)/2x = 10
This solves to R(s) = 2R(h) + 20x
Since R(h) and x must be positive, it follows that R(s) > R(h)?

@avitgutman
Thank you for this helpful video and for your insights as always.

To clarify, I was confused as to why you cannot have the price for H be 20 and price for S be 10.
E.g.,
Let the volume for H=6
Let the price for S=10

H: 6 hardcover books * $20= $120
S: 12 softcover books * $10= $120

($20+$10)/2=$15 > $14

I am probably making a silly mistake, and I realize that you mentioned $10 vs. $20, but I am not fully grasping something here.

Thanks again.

 

­You can visualize what is given if you understand the scale method of weighted averages very well.

Question: Was Bookstore B's revenue from the sale of softcover books yesterday greater than its revenue from the sale of hardcover books yesterday?

Revenue = Price * Quantity
We know that quantity of softcovers were twice that of hardcovers. If price of softcovers is half that of hardcovers, then the revenue from both would become equal. So if price of softcover is less than half of price of hardcover, then revenue of softcover would be less. And if price of softcover is more than half of price of hardcover, then revenue of softcover would be greater.
 
Each individual statement is obviously not sufficient. Are both statements together sufficient? Let's see.

Weight of softcover books is 2 times weight of hardcover books. Ratio of weight is 2:1.
Price of softcover is x, then hardcover price is x+10. This 10 will be divided in the ratio 1:2 on the scale i.e. 3.33 and 6.66.

Let's assume that the average is about 14 (very slightly more than 14). This is what it will look like then.



Since average is more than 14, then x is more than 10 for sure. Let's say cost of softcover books is 10.67 (the lowest possible). Then cost of hardcover books is 20.67. Cost of softcover books is more than half the cost of hardcover books. Hence revenue from softcover books will be more. Since average is always more than 14, cost of softcover books is always more than half the cost of hardcover books. Hence revenue from softcover books will always be more.

Answer (C)
 ­
Check out how to use weighted averages here:
https://www.youtube.com/watch?v=_GOAU7moZ2Q
 

What if
Number of hardcover books =5
Number of soft cover books =10

They said: average price more than 14. That means together they are priced more than 28. We can assume together they can be 30 (average 15). If their combined prices are 30, according to statement they are priced software 10 and hard cover 20.

That means 5 hardcover book prices are 100 and and 10 softcover book prices are also 100.

I think the answer should be E. Please tell me where i am wrong.

Posted from my mobile device

The point is that if there are 5 hardcover books priced at $20 each and 10 softcover books priced at $10 each, then the overall average price would be (10 * $10 + 5 * $20)/15 = $13.3..., which is less than $14. This illustrates that the overall average price of all books is not simply the average of the average prices of hardcover and softcover books. If the numbers of hardcover and softcover books were equal, then averaging their prices would be accurate. However, since there are more softcover books, the overall average is skewed closer to the average price of softcover books rather than that of hardcover books.

Hope it's clear.