At a certain library, 1600 books were checked out by patrons yesterday

Question

At a certain library, 1600 books were checked out by patrons yesterday. If each patron checked out at least 1 book and at most 10 books yesterday, was the number of patrons who checked out books yesterday greater than 300?

(1) 80 patrons checked out 1 or 2 books each.
(2) 150 patrons checked out 3 or 4 books each.

(1) 80 patrons checked out 1 or 2 books each.
(2) 150 patrons checked out 3 or 4 books each.

MartyMurray · Accepted Answer

At a certain library, 1600 books were checked out by patrons yesterday. If each patron checked out at least 1 book and at most 10 books yesterday, was the number of patrons who checked out books yesterday greater than 300?

(1) 80 patrons checked out 1 or 2 books each.
(2) 150 patrons checked out 3 or 4 books each.

This is a max/min question. To evaluate whether the statements provide sufficient information, we'll seek to determine whether we can use the maximum or minimum possible number of patrons indicated by the statements to determine whether the number of patrons who checked out books yesterday was greater than 300.

This kind of question isn't super hard, but we'll be making multiple calculations for each statement. So, we have to be careful to execute well so that we don't arrive at an incorrect answer.

(1) 80 patrons checked out 1 or 2 books each.

Max:

To find the maximum number of patrons indicated by this statement, we'll minimize the number of books checked out by each patron.

80 patrons × 1 book = 80 books

1520 books left --> 1 book per patron --> 1520 more patrons

No more work necessary since the max is over 1520 and thus must be over 300.

Min:

To find the minimum number of patrons indicated by this statement, we'll maximize the number of books checked out by each patron.

80 patrons × 2 books = 160 books

1440 books left --> 10 books per patron --> 144 more patrons

Min = 80 + 144 = 224 patrons

The maximum indicated by this choice is over 1520, and the mininum is 224. So, given what this choice says, the number could be greater or less than 300.

Insufficient.

(2) 150 patrons checked out 3 or 4 books each.

Max:

To find the maximum number of patrons indicated by this statement, we'll minimize the number of books checked out by each patron.

150 patrons × 3 books = 450 books

1150 books left --> 1 book per patron --> 1150 more patrons

No more work necessary since the max is over 1150 and thus must be over 300.

Min:

To find the minimum number of patrons indicated by this statement, we'll maximize the number of books checked out by each patron.

150 patrons × 4 books = 600 books

1000 books left --> 10 books per patron --> 100 more patrons

Min = 150 + 100 = 250 patrons

The maximum indicated by this choice is over 1150, and the mininum is 250. So, given what this choice says, the number could be greater or less than 300.

Insufficient.

Statements (1) and (2) combined

Max:

To find the maximum number of patrons indicated by the statements combined, we'll minimize the number of books checked out by each patron.

Notice that, since we'll have 80 patrons who checked out 1 book and 150 who checked out 3, this number will be the same as the max for Statement (2), which we arrived at by having 150 patrons check out 3 books and the rest check out 1.

Thus, the maximum for the statements combined is over 1150.

Min:

To find the minimum number of patrons indicated by the statements combined, we'll maximize the number of books checked out by each patron.

80 patrons × 2 books = 160 books

150 patrons × 4 books = 600 books

840 books left --> 10 books per patron --> 84 more patrons

Min = 80 + 150 + 84 = 314 patrons

The maximum indicated by this choice is over 1150, and the mininum is 314. So, given what this choice says, the number must be greater than 300.

Sufficient.

Correct answer:  C

GMATGuruNY · Answer

An alternate approach is to test the threshold: to test whether it's possible for exactly 300 patrons to check out 1600 books.

Statement 1:
80 patrons, 1 or 2 books each --> 80 to 160 books
220 patrons, 3 to 10 books each --> 660 to 2200 books
Combining the two ranges, we get:
Total checked out --> 1460 to 2360 books ---> 300 patrons can check out 1600 books
Clearly, the number of patrons could just as easily be 301.
Since the answer is NO in the first case but YES in the second, INSUFFICIENT.

Statement 2:
150 patrons, 3 or 4 books each --> 450 to 600 books
150 patrons, a minimum of 1 and a maximum of 10 books each --> 150 to 1500 books
Combining the two ranges, we get:
Total checked out --> 600 to 2100 books ---> 300 patrons can check out 1600 books
Clearly, the number of patrons could just as easily be 301.
Since the answer is NO in the first case but YES in the second, INSUFFICIENT.

Statements combined:
80 patrons, 1 or 2 books each --> 80 to 160 books
150 patrons, 3 or 4 books each --> 450 to 600 books
70 patrons, 5 to 10 books each --> 350 to 700 books
Combining the three ranges, we get:
Total checked out --> 880 to 1460 books ---> 300 patrons are too few to check out 1600 books
Implication:
The number of patrons must exceed 300, with the result that the answer to the question stem is YES.
SUFFICIENT.

C

mmdfl · Answer

(1) books checked out by 80 patrons => *80= books (minimum and maximum) Books left = 1600 - = books (minimum and maximum) left to patrons that checked out between 3 and 10 books each. Minimum #patrons =80 + min( / ) = 80 + 144 < 300 Since "Minimum #patrons < 300", we need calculate the maximum value of #patrons to evaluate if #patrons < 300 Maximum #patrons = 80 + max( / ) = 80 + 1520/3 > 300 Insufficient information. (2) books checked out by 150 patrons => *150 = books (minimum and maximum) Books left = 1600 - = books (minimum and maximum) left to patrons that checked out between 1 and 2 books each, and patrons that checked out between 5 and 10 books each. Minimum #patrons =150 + min( / ) = 150 + 100 < 300 Since "Minimum #patrons < 300", we need calculate the maximum value of #patrons to evaluate if #patrons < 300 Maximum #patrons = 150 + max( / ) = 150 + 1150/1 > 300 Insufficient information. (1 and 2) Books left = 1600 - - = books (minimum and maximum) left to patrons that checked out between 5 and 10 books each. Minimum #patrons =80 + 150 + min( / ) = 230 + 84 > 300 Sufficient information => (C)

CorporateAsset · Answer

In such questions, we can define min & max ranges for clarity. Below is the solution considering the ranges. Note: the logic we need to test is that if given people took max books they can for eg in statement 1 it was 2 people, and then the remaining also took the max they can (cap is 10 books) that will give us the minimum number of people taking maximum number of books.

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KarishmaB · Answer

Application of "use the information you have". Let's see if 1600 books could be checked out by 300 patrons. If not, then the number of patrons would need to be more than 300. I will review statement 2 first here because it gives bigger numbers and hence makes it more likely that my constraints are satisfied. (2) 150 patrons checked out 3 or 4 books each. It means 450 to 600 books are accounted for by 150 people. Now we have 150 people left who need to account for leftover books (at most 1150 are left) Each of the 150 people could have checked out 10 books and hence they can account for up to 1500 books. So number of people can certainly be less than 300. Not sufficient alone. If (2) is not sufficient alone, (1) alone cannot be since 220 people can account for 2200 books, much more than 1600. Using both together: From stmnt 1, 80 patrons can account for at most 160 books. (Each checks out 2 books) From stmnt 2, 150 patrons can account for at most 600 books. (Each checks out 4 books) Now I have 300 - 150 - 80 = 70 patrons left. They can account for at most 700 books. (Each checks out 10 books) We can account for only 160+600 + 700 = 1460 books. But we need to account for 1600 books. Hence there must have been more than 300 patrons. Answer (C) Another question in which you should start with what you have been given: https://youtu.be/Nlb6hg-njvQ

ShivamH · Answer

In the (1), when it says 80 patrons checked out 1 or 2 books , does this mean only 80 could have checked out 1 or 2 and beyond these 80 will choose a number anything except 1 or 2. If that is the case, then the max calculation of 80+1520 patron may not hold good. Please correct me in my understanding if I am wrong. Thank you.

Bunuel · Answer

“80 patrons checked out 1 or 2 books each” means exactly 80 patrons did so, so those 80 account for between 80 and 160 books.

soma7 · Answer

Hi team,

Can someone tell whether my approach is correct?

let 'n' be the number of patron and x be the no.of books/patron
Therefore, 1<=x<=10

Statement 1: 80 patrons 1 or 2 books.

If 80 patrons 1 book --> 80 books ---> remaining 1600-80=1520.
if all the 1520 books are checked as 10 books by each patron ---> patrons = 152
total patrons = 152+80 = 232 < 300 ------> "1"

if 80 patrons 2 books ---> 160 books ---> remaining 1600-160 = 1440.
if all the 1440 books are checked as 10 books by each patron ---> patrons = 144.
total patrons = 144+80 = 224 < 300

if 1520 books are checked as 4 books be each patron in "1", patrons = 1520/4 = 380
total patrons = 380 + 80 = 460 > 300

Hence, Insufficient. Similarly, we can prove statement 2.

But combining both the statements, we can get n > 300 in both the cases.

At a certain library, 1600 books were checked out by patrons yesterday

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