Is the sum of the prices of the 3 books that Shana bought less than $4

Target question: Is the sum of the prices of the 3 books less than $48 ?

Let A = price of the LEAST expensive book (in dollars)
Let B = price of the mid-priced expensive book (in dollars)
Let C = price of the MOST expensive book (in dollars)

So, A < B < C

Statement 1: The price of the most expensive of the 3 books that Shana bought is less than $17.
So, C < 17
Let's TEST some values.
There are several cases that satisfy statement 1. Here are two:
Case a: A = $1, B = $2 and C = $3, in which case A + B + C = 1 + 2 + 3 = 6. In this case, the answer to the target question is YES, the sum of the prices IS less than $48
Case b: A = $16.25, B = $16.50 and C = $16.75, in which case A + B + C = 16.25 + 16.50 + 16.75 = 49.50. In this case, the answer to the target question is NO, the sum of the prices is NOT less than $48
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The price of the least expensive of the 3 books that Shana bought is exactly $3 less than the price of the second most expensive book.
In other words, A = B - 3
Let's TEST some values again.
There are several cases that satisfy statement 1. Here are two:
Case a: A = $1, B = $4 and C = $10, in which case A + B + C = 1 + 4 + 10 = 15. In this case, the answer to the target question is YES, the sum of the prices IS less than $48
Case b: A = $1, B = $4 and C = $100, in which case A + B + C = 1 + 4 + 100 = 105. In this case, the answer to the target question is NO, the sum of the prices is NOT less than $48
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that C < 17
Statement 2 tells us that A = B - 3
Let's MAXIMIZE ALL of the values
If C < $17, then the greatest possible value of C = $16.99
Since B must be less than C, the greatest possible value of B = $16.98
Since A is $3 less than B, greatest possible value of A = $12.98
So, when we MAXIMIZE all 3 values, A + B + C = $16.99 + $16.98 + $12.98 = $46.95, which is less than $48
If if the GREATEST values of A, B and C have a sum that's less than $48, we can conclude that is must be the case that A + B + C < 48
In other words, the answer to the target question is YES, the sum of the prices IS less than $48
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

Let \(x_1, x_2, x_3\) be the prices in increasing order and let \(S\) be the sum of the prices. The original question: Is \(S<$48\) ?

1) Using that \(x_3<17\), we can determine an upper bound for \(S\).

\(S\leq 3x_3<51\)
\(S<$51\)

We can't get a definite answer to the original question. \(\implies\) Insufficient

2) We know that \(x_1=x_2-3\), but no information is given about \(x_3\). Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

1&2) Using all statement information, we can determine an upper bound for \(S\).

\(S\leq x_3-3+x_3+x_3\)
\(S\leq 3x_3-3<48\)
\(S<$48\)

Thus, the answer to the original question is a definite Yes. \(\implies\) Sufficient

Answer: C

1) The most expensive is B<17 so the total could 48 or less than 48; Insufficient.

2) The most expensive could be any value so insufficient.

Both:
If the most expensive can be maximum 16, the second-highest price can be 15 then the third on will 12; The maximum price will be 43. Even in case of extreme values such as 16.9, 15.9 and 12.9 the less than \(48\). Sufficient.

Ans C.

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Solution:

Question Stem Analysis:

We need to determine whether the total amount spent on the 3 books is less than $48.

Statement One Alone:

Since the most expensive of the 3 books Shana bought is less than $17, the total she has spent is less than 17 x 3 = $51. Since it’s less than $51, it could be less than $48 (e.g., $47), or it could be more than $48 (e.g., $50). Statement one alone is not sufficient.

Statement Two Alone:

Since we don’t know the actual amount spent on any of the 3 books, statement two alone does not allow us to determine whether the total amount spent is less than $48 or not.

Statements One and Two Alone:

With the two statements, let’s assume that the most expensive book is $17 (even though we know it’s less than $17). Let’s also assume the second most expensive book is also $17 and hence the least expensive book is $14. If this is the case, the total amount spent on the 3 books would be exactly 17 + 17 + 14 = $48. However, since we know the prices of the 3 books must be actually less than $17, $17, and $14, respectively, then the total spent is indeed less than $48.

Answer: C

Answer: Option C

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Posted from my mobile device

How do you know B<C? we just know most expensive book costs a certain amount. WHy can't second most expensive book be same price as most expensive? ScottTargetTestPrep BrentGMATPrepNow KarishmaB

If the prices of books A, B and C were something like $4, $7, and $7 respectively, I suppose we could still say book C is the most expensive book.
Fortunately, the issue doesn't really come into play for this question.
That is, even if we allow for A ≤ B ≤ C, the correct answer is still C.

I know it will not affect but I am trying to understand what would it be if it were testing us on this concept let's say. Why do we assume that they are necessarily lesser than each other? How does gmat treat the case to be?

I checked the official solution, and the test makers begin by writing: \(B_1 ≤ B_2 ≤ B_3\).
Later in their solution, when combining both statements, they write the following:


So, it looks like they're allowing for the possibility that \(B_2 = B_3\)

Awesome thanks for sharing.
Also how would the “second most expensive” in the statement 2 affect the whole situation? I mean it could be that as per statemtn1 they can all be equated but not as per statement 2 since B2 is “second most expensive” which means it has to be lesser expensive than b3. Pls clarify BrentGMATPrepNow



Posted from my mobile device

Hi,
This solution really helped! Esp the 'maximising' logic in the end.
Btw, I think there is a typo here-
'Since A is $3 less than B, greatest possible value of A = $12.98'
Should be $13.98, thus $47.95
The inferences remain the same though.