Four friends go to Macys for shopping and buy a top each. Three of th

(1) 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

(2) 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

Answer : A

x1, x2, x3 - pillows, y1, y2, y3, y4 - tops, lets just say that x1 < x2 < x3 < y1 < y2 < y3 < y4.
x1 + x2 + x3 + y1 + y2 + y3 + y4 = 89.

#1 y4 = 16, which means that y3 <= 15, y2 <= 14, y1 <= 13, if we summ all these inequalities we'll get y1 + y2 + y3 + y4 <= 58, which means that
x1 + x2 + x3 >= 89 - 58 = 31 (1)
We also have to keep in mind that x3 <= 12, x2 <= 11 and x1 <= 10
Now lets look at our question, wich asks us the value of x3. To check if we can answer the question explicitly we should check if we can change x3 and still make our inequalities work: lets do it
1: x3 = 12 (max possible), in this case y4 = 16, y3 = 15, y2 = 14, y1 = 13, x2 = 11, x1 = 9 - that works and gives us perfect 89.
2: x3 = 11 (second max), we can't really add to values of y so we have to compensate with x: in this case x2 will be 10 at max, x1 will be 9 at max and the summ will be 11+10+9 = 30 which doesn't satisfy (1) - fail. Going lower aggravates the situation even further so 12 is the only possible value for x3 thus the answer to question is SUFFICIENT

#2 x1 = 9
that means x2 >= 10, x3 >=11, y1>=12, y2>=13, y3 >=14, y4 >=15
and so x1+x2+x3+y1+y2+y3+y4 >= 84. We also have to keep in mind that x1+x2+x3+y1+y2+y3+y4 <= 89
The situation like this implies that our variables aren't exactly fixated coz the gap between 84 and 89 is rather big, lets see if we can answer our question. To do that lets check these 2 equalities: for the first one lets take minimum possible values of x1-x3 and y1-y4:
1)x1 + x2 + x3 + y1+ y2 +y3+y4 = 84 (9+10+11+12+13+14+15)
lets increase x3 by 1
2)x1+x2+(x3+1)+(y1+1)+(y2+1)+(y3+1)+(y4+1) = x1+x2+x3+y1+y2+y3+y4 +5 = 89.
The fact that we got 89 and not above it means that even if we increase our value of x3 we can still get respective values of y1-y4 that would let us get the 89 total price (9 10 12 13 14 15 16)
So we got 2 different possible values for x3 thus #2 is INSUFFICIENT

Answer is A.

Should be A

Given that,

4T + 3P = 89

1) From 1st statement we know that most expensive top= $16

so lets 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 lets divide 31 among 3 pillows keeping in mid that the max value of a pillow cover can be 12 only

so a combination can be = 9,10 &12

we cant have any other combination other than this one whose sum will be =31

and if we lower the 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 exp pillow cover = $12

hence statement 1 is sufficient

2) From 2nd statement we know that ,

Least exp Pillow cover = $9

so we can take pillow covers as = 9 +10 + 11 = $30

So we will have to distribute remaining $59 among 4 tops.

we can have 1 combination as = 12+14+16+17 =59

Now lets 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

Hence answer should be A

St (1) The most expensive top cost $16.
To maximize the vale of 1 PC we need to min the value of Tops, Max vale of top = 16; so since all 4 tops are of different prices and we need to min their value lets take the lowest distinct value of the remaining tops = 16+15+14+13+12+x+y = 89 where x and y could be anything that adds up to give 19 and are not of a concern. Max value of PC =12. Thus Suff
(2) The least expensive pillow case cost $9.
max value of PC could be anything NS
Answer A

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. (x6 and x7 could be any combinations totalling to 19). 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.

St2: The least expensive pillow case cost $9.
Since we have a lower limit, the most expensive pillowcase can hold multiple values. Therefore insufficient.

Ans: A

VERITAS PREP OFFICIAL SOLUTION

The first problem here is figuring out the starting point. There must be many ways in which you can price the seven items such that the total cost is $89. So we need to establish a base case (which conforms to all the conditions given in the question stem) first and then we will tweak it around according to the additional information obtained from our statements.

‘Seven items for $89’ means the average price for each item is approximately $12. But 12 is not the exact average. 12*7 = 84 which means another $5 were spent.

A sequence with an average of 12 and different integers is $9, $10, $11, $12, $13, $14, $15.

But actually another $5 were spent so the prices could be any one of the following variations (and many others):

$9, $10, $11, $12, $13, $14, $20 (Add $5 to the highest price)

$9, $10, $11, $12, $13, $16, $18 (Split $5 into two and add to the two highest prices)

$9, $10, $12, $13, $14, $15, $16 (Split $5 into five parts of $1 each and add to the top 5 prices)

$7, $9, $13, $14, $15, $16, $17 (Take away some dollars from the lower prices and add them to the higher prices along with the $5)

etc

Let’s focus on another piece of information given in the question stem: “every top cost more than every pillow case.”

This means that when we arrange all the prices in the increasing order (as done above), the last four are the prices of the four tops and the first three are the prices of the three pillow cases. The most expensive pillow case is the third one.

Now that we have accounted for all the information given in the question stem, let’s focus on the statements.

Statement 1: The most expensive top cost $16.

We have already seen a case above where the maximum price was $16. Is this the only case possible? Let’s look at our base case again:

$9, $10, $11, $12, $13, $14, $15

(a further $5 needs to be added to bring the total price up to $89)

Since the prices need to be all unique, if we add 1 to any one price, we also need to add at least $1 to each subsequent price. E.g. if we increase the price of the least expensive pillow case by $1 and make it $10, we will need to increase the price of every subsequent item by $1 too. But we have only $5 more to give.

If the maximum price is $16, it means the rightmost price can increase by only $1. So all prices before it can also only increase by $1 only and except the first two prices, they must increase by $1 to adjust the extra $5.

Hence the only possible case is $9, $10, $12, $13, $14, $15, $16.

So the cost of the most expensive pillow case must have been $12.

Statement 1 is sufficient alone.

Statement 2: The least expensive pillow case cost $9.

A restriction on the lowest price is much less restrictive. Starting from our base case

$9, $10, $11, $12, $13, $14, $15,

we can distribute the extra $5 in various ways. We can do what we did above in statement 1 i.e. give $1 to each of the 5 highest prices: $9, $10, $12, $13, $14, $15, $16

We can also give the entire $5 to the highest price: $9, $10, $11, $12, $13, $14, $20

So the price of the most expensive pillow case could take various values. Hence, statement 2 alone is not sufficient.

Answer (A)

Note that the answer is a little unexpected, isn’t it? If we were to read the question and guess within 20 secs, we would probably guess that the answer is (C), (D) or (E). The two statements give similar but complementary information. It would be hard to guess that one will be sufficient alone while other will not be. This is what makes this question interesting and hard too.

Our strategy here was to establish a base case and tweak it according to the information given in the statements. This strategy is often useful in DS – not just in max-min questions but others too.

In ascending order, let the 7 distinct integers be as follows:
a, b, c, d, e, f, g
Since a, b and c represent the three pillow cases, the question stem can be rephrased as follows:
What is the value of c?

Statement 1: g=16
Resulting values:
a, b, c, d, e, f, 16 --> a+b+c+d+e+f = 89-16 = 73
Here:
a, b, c, d, e, and f must be 6 distinct integers that are less than 16 and sum to 73.
If c=13, then c, d, e and f cannot be distinct integers less than 16.
If c=11, the greatest possible sum for a, b, c, d, e and f = 9+10+11+13+14+15 = 72
Too small.
Thus:
c=12
SUFFICIENT.

Statement 2: a=9
Resulting values:
9, b, c, d, e, f, g --> b+c+d+e+f+g = 89-9 = 80
Here:
b, c, d, e, f and g must be 6 distinct integers that are greater than 9 and sum to 80.
Case 1: 10, 11, 12, 13, 14, 20
Case 2: 10, 12, 13, 14, 15, 16
Since c=11 in Case 1 but c=12 in Case 2, INSUFFICIENT.

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.

Bunuel KarishmaB
Not sure why A is sufficient for definite answer. I would appreciate if you can provide your comments on my thinking.

A is sufficient if we assume that all top cost is in sequence but question do not have such clarification.

Statement 1 : not definite yes
case 1:
four top : $13 + $14 + $15 + $16 = $58 so need to allocate $31 (89-58) between three pillow cases and only way to do that is $9, $10, $12 so we get $12 as an answer.

case 2
four top: $12 + $13 + $15 + $16 = $56 so need to allocate $33 (89-56) between three pillow cases and we can not do that so no answer

Statement 2:
Minimum $9 is not sufficient itself as agreed by everyone

Both statement:
still we can have 2 cases mentioned in statement 1 so not a single answer.

if we can assume that all top cost is in sequence then answer is A otherwise both is not sufficient.

Please let me know if i am missing anything.

Thank you
Mayur A.

You took two cases. Case 1 gave you an answer of 12. Case 2 you found was not possible. Then answer must be 12.
You got a unique definite value.
Why do you think statement 1 is not sufficient alone?

I would say let me place them in decreasing order of cost from most expensive to leats expensive: __ __ __ __ __ __ __
I need the value of the highlighted blank.

(1) The most expensive top cost $16.

So let me write the possible cases:
16, 15, 14, 13, 12, 11, 10
But this sums up to 91 ( = 13 * 7)
So I need to reduce the sum by 2. I can do it in only two ways - reduce the last price by 2 or reduce last two prices by 1 each:
16, 15, 14, 13, 12, 11, 8
or
16, 15, 14, 13, 12, 10, 9
If I try to reduce any other price by 1, I will need to reduce all the following prices by 1 too to maintain the distinct pricing. But that will reduce the sum by more than 2. So only these two are possible and in each, the price of the third last item is 12.

Sufficient alone.

(2) The least expensive pillow case cost $9.

So let me write the possible cases: 9, 10, 11, 12, 13, 14, 15 - Total sum is 12*7 = 84. I need to add 5 to the sum. I can do it in many ways.

9, 10, 11, 12, 13, 14, 20
or
9, 10, 11, 12, 13, 15, 19
or
9, 10, 12, 13, 14, 15, 16
etc.

The price of the third item may be 11 or 12 here.
Not suffiicent

Answer (A)

Tip: Start with the base case and then adjust.
Reminds me of a max-min question: https://anaprep.com/algebra-game-plan-with-max-min/

Hard question...
Here is my solution on why 1) is sufficient.
Prices of the seven items:
16 > 16 - (p1) > 16 - (p1 + p2) > ... > 16 - (p1+p2+...+p6)
where p_k is integer > 0. (in other words, p_k >= 1)
we want to know the value of the third cheapest item, which the price is P = 16-(p1+p2+p3+p4)
Since p1,p2,p3,p4 >= 1 -> P <= 16-(1+1+1+1)=12

if we sum all the prices, we will get 7*16 - (6p1+5p2+...+p6) = 89. We can rearange the terms so we get P in this equation:
P = (25+(3p1+2p2+p3+2p5+p6))/3 >= (25+(3+2+1+2+1))/3 = 34/3 = 11,3.... -> P >= 11,3... . Given that P is integer, P>=12.

12<=P<=12 -> P=12

Here is my solution on why 2) is insufficient.
Prices of the seven items:
9 < 9 + q1 < ... < 9+(q1+q2+...+q6)
where q_k is integer > 0. (in other words, q_k >= 1)
we want to know the value of the third cheapest item, which the price is Q = 9+(q1+q2)
Since q1,q2 >= 1 -> Q >= 9-(1+1)=11

if we sum all the prices, we will get 7*9 + (1q6+2q5+3q4+4q3+5q2+6q1) = 89. We can rearange the terms so we get Q in this equation:
Q = 9 + (26-(1q6+2q5+3q4+4q3+q1))/5 <= 9+(26-11)/5 -> Q <= 12.

11<=Q<=12 -> Q can be 12 or 11

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