Four Friends Go to Macy’s for Shopping and Buy a Top each GMAT Data Sufficiency

Question: Four friends go to Macy’s for shopping and buy a top each. Three of them buy a pillow case each too. The prices of the seven items were all different integers, and every top cost more than every pillow case. What was the price, in dollars, of the most expensive pillow case if the total price of the seven items was $89?

1. The most expensive top cost $16
2. The least expensive pillow case cost $9

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Solution and Explanation:

Approach Solution (1):

S1: The most expensive top cost $16
Top prices = 16, 15, 14, 13
Pillow case prices = 12, 10, 9
No other higher pillow prices are possible as all numbers still have to sum to 89
Sufficient

S2: The least expensive pillow case cost $9
Here, pillow prices can range based on an unfixed top price
For example:
Top prices could be 17, 16, 15, 14
Pillow prices could be 12, 11, 4 or 13, 10, 4
Insufficient

Correct option: A

Approach Solution (2):

Given that:
4T + 3P = 89
(1) From S1, we know that the most expensive top = $16
So let’s take other tops as = $13, $14, $15
So total money spend on tops = 13 + 14 + 15 + 16 = $58
Money left for pillow covers = 89 – 58 = $31
Now let’s divide 31 among 3 pillows keeping in mind that the maximum value of a pillow cover can be 12 only
So a combination can be = 9, 10 and 12
We can’t have any other combination other than this one whose sum will be = 31
And if we lower price of one of the tops say from 13 to 12, the total cost of tops will be = 12 + 14 + 15 + 16 = $57
So we will have $32 for pillow covers
Now it is impossible to distribute 32 among 3 when max value can be 11, as max value for this combination = 11 + 10 + 9 = 30 only
So price of most expensive pillow cover = $12
Hence S1 is sufficient

(2) From S2, we know that:
Least expensive pillow cover = $9
So we can take pillow cover as 9 + 10 + 11 =$30
So we will have to distribute remaining $59 among 4 tops
We can have 1 combination = 12 + 14 + 16 + 17 = 59
Now let’s take pillow covers as = 9 + 10 + 12 = $31
So we will have to distribute remaining $58 among 4 tops
This can be achieved as = 13 + 14 + 15 + 16 = 58
So from 2, we are getting two answers for price of most expensive pillow cover
Thus not sufficient

Correct option: A

Approach Solution (3):

Let x1 + x2 + x3 + x4 be the top costs and x5 + x6 + x7 be the pillow case costs
Essentially, x1 > x2 > x3 > x4 > x5 > x6 > x7 = 89
Let x1 be the most expensive top and x5 be the most expensive pillow case

S1: The most expensive top cost $16
Therefore, x1 + x2 + x3 + x4 = 16 + 15 + 14 + 13 (since all numbers are different integers) then x5 = 12, x6 and x7. To arrive at the most expensive pillowcase (and different integer values), we need to minimize the differences between each of the integer values starting with $16 (upper limit)
Sufficient

S2: The least expensive pillow case cost $9.
Since we have a lower limit, the most expensive pillowcase can hold multiple values. Therefore, insufficient.

Correct option: A

“Four friends go to Macy’s to shop and buy a top each. Three of them buy a pillow case each too. The prices of the seven items were all different integers, and every top cost more than every pillow case. What was the price, in dollars, of the most expensive pillow case if the total price of the seven items was $89?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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