Softcover books -- tricky problem in the OG for Quant 2021

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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.

C
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GMATGuruNY wrote:
Wed Oct 14, 2020 3:17 pm
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.

C
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.
(total revenue)/(total sold) > 14
(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.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

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I unlock the best way for YOU to solve problems.

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