Yesterday Bookstore B sold twice as many softcover books as hardcover

Question

Yesterday Bookstore B sold twice as many softcover books as hardcover books. Was Bookstore B's revenue from the sale of softcover books yesterday greater than its revenue from the sale of hardcover books yesterday?

(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.
(2) The average price of the softcover and hardcover books sold at the store yesterday was greater than $14.

ID: 700294
DS90820.02

nick1816 · Accepted Answer

N_S = 2*N_H

Statement 1-

Assume the avg price Of softcover books = x
 the avg price Of hardcover books = y = x+10

Revenue of softcover books =  x*N_S  =  x*2N_H   = xN_H + xN_H

Revenue of hardcover books =  (x+10)*N_H = xN_H + 10*N_H

We have no idea whether x is greater than 10

Insufficient

Statement 2 -   \frac{xN_S+yN_H}{N_S +N_H }> 14

\frac{x*2N_H + y*N_H }{ 3N_H }> 14

2x+y>42

We can't compare revenues

Insufficient

Combining both statements

2x+y>42

2x+x+10> 42

x> \frac{32}{3}

x> 10.xyz > 10

Sufficient

NiftyNiffler · Answer

Yesterday Bookstore B sold twice as many softcover books as hardcover books. Was Bookstore B's revenue from the sale of softcover books yesterday greater than its revenue from the sale of hardcover books yesterday?

(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.
(2) The average price of the softcover and hardcover books sold at the store yesterday was greater than $14.

Let's assign some variables

S = no. of soft copies sold
H = no. of hard copies sold

x = avg. price of each soft copy
y = avg. price of each hard copy

Given: S = 2H

Statement 1: 
y = x + 10

We have to find if R S  > R H  
=> Sx > Hy 
=> 2Hx > H(x +10)
=> Hx > 10H 
=> x > 10 (Since we are not given the value of x INSUFFICIENT)

Statement 2:
Makes certain that x > 14 & y > 14, but not absolute values INSUFFICIENT

St1 and St2 taken together, SUFFICIENT

So, answer is  C .

aritrar4 · Answer

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.

GMATGuruNY · Answer

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.

C

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.

chetan2u · Answer

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

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=x6lQeAPoGuQ

rahulbanka · Answer

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)?

KarishmaB · Answer

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.

woohoo921 · Answer

@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.

egmat · Answer

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

KarishmaB · Answer

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. Screenshot 2024-04-05 at 10.56.50 PM.png 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

Aslamkhanshovon · Answer

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

Bunuel · Answer

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.

GMATinsight · Answer

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

Gemmie · Answer

s: number of softcover books
h: number of hardcover books
a: avg price of a softcover book
b: avg price of a hardcover book

s = 2h

Question: a * s > b * h ?
=> a * 2h > b * h ?
=> 2a > b ?

Statement 1:   b = a + 10

=> Insufficient

Statement 2:    \frac{as + bh}{s+h} > 14

=>  \frac{2ah + bh}{2h + h} > 14

=>  \frac{2a + b}{3} > 14

=> 2a + b > 42

=> Insufficient

Combined:  
b = a + 10
2a + b > 42

=> 3a + 10 > 42

=>  a > \frac{32}{3} \approx{10.67}

=> 2a > a + 10.67 > a + 10 = b

=> 2a > b

=> Sufficient

Answer is C

SimantaSourav · Answer

Why the second statement can't be interpreted as this : (Avg Price of S + Avg Price of H)/2 > 14?

Bunuel · Answer

Because statement (2) refers to the average price of all the books sold, not the average of the average prices of the two types of books.

Let S be the average price of a softcover book and H be the average price of a hardcover book.

If n hardcover books and 2n softcover books were sold, then the average price of all the books is

(2nS + nH)/3n = (2S + H)/3.

So statement (2) means

(2S + H)/3 > 14,

not (S + H)/2 > 14.

Yesterday Bookstore B sold twice as many softcover books as hardcover

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Yesterday Bookstore B sold twice as many softcover books as hardcover

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