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Of the 20 people who each purchased 2 tickets to a concert,

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Of the 20 people who each purchased 2 tickets to a concert,

by sachin_yadav » Thu Jan 15, 2015 9:40 am
Of the 20 people who each purchased 2 tickets to a concert, some used both tickets, some used only 1 ticket, and some used neither ticket. What percent of the tickets that were purchased by the 20 people were used by those people?
(1) Of the 20 people, 10 used only 1 ticket.
(2) Of the 20 people, 4 used neither ticket.

OA is C

It seems to be overlapping sets question. How to create set-matrix table for this question ?

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by Brent@GMATPrepNow » Thu Jan 15, 2015 4:19 pm
sachin_yadav wrote:Of the 20 people who each purchased 2 tickets to a concert, some used both tickets, some used only 1 ticket, and some used neither ticket. What percent of the tickets that were purchased by the 20 people were used by those people?

(1) Of the 20 people, 10 used only 1 ticket.
(2) Of the 20 people, 4 used neither ticket.
Here's how I'd solve it....

Target question: What percent of the tickets that were purchased by the 20 people were used by those people?

Given: 20 PAIRS of tickets were bought by 20 PEOPLE. Some people used both tickets, some used only 1 ticket, and some used 0 tickets

So, we have the following table to represent what happened with each of the 20 PEOPLE
Image
To answer the target question, we need to know the number of people in each of the two HIGHLIGHTED boxes.

Statement 1: Of the 20 people, 10 used only 1 ticket.
So, we know that there are 10 people in the middle box, but we don't know about the other two boxes. This leaves several possible cases. Here are two:
Image
In the top case, we see that 10 of the 40 tickets were used by the purchaser. So, 25% of the tickets were used by the purchaser.
In the bottom case, we see that 30 of the 40 tickets were used by the purchaser. So, 75% of the tickets were used by the purchaser.

Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Of the 20 people, 4 used neither ticket.
So, we know that there are 4 people in the right-hand box, but we don't know about the other two boxes. This leaves several possible cases. Here are two:
Image
In the top case, we see that 32 of the 40 tickets were used by the purchaser. So, 80% of the tickets were used by the purchaser.
In the bottom case, we see that 16 of the 40 tickets were used by the purchaser. So, 40% of the tickets were used by the purchaser.

Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
When COMBINED the statements give us one and only one possible case:
Image
In this case, we see that 22 of the 40 tickets were used by the purchaser. So, 55% of the tickets were used by the purchaser.

Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

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by Brent@GMATPrepNow » Thu Jan 15, 2015 4:25 pm
sachin_yadav wrote:Of the 20 people who each purchased 2 tickets to a concert, some used both tickets, some used only 1 ticket, and some used neither ticket. What percent of the tickets that were purchased by the 20 people were used by those people?
(1) Of the 20 people, 10 used only 1 ticket.
(2) Of the 20 people, 4 used neither ticket.

OA is C

It seems to be overlapping sets question. How to create set-matrix table for this question ?

Regards
Sachin
We COULD solve the question using the Double Matrix method, but it's a little confusing.
We can examine the status of each individual ticket (1st ticket and 2nd ticket)
We'll examine all 20 PAIRS of tickets.
For the 2 rows, we can have 1st ticket used and 1st ticket not used
For the 2 columns, we can have 2nd ticket used and 2nd ticket not used

I'll let you try it this way.

HINT: One of the boxes must automatically contain a zero.

Cheers,
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by GMATGuruNY » Fri Jan 16, 2015 5:28 am
sachin_yadav wrote:Of the 20 people who each purchased 2 tickets to a concert, some used both tickets, some used only 1 ticket, and some used neither ticket. What percent of the tickets that were purchased by the 20 people were used by those people?
(1) Of the 20 people, 10 used only 1 ticket.
(2) Of the 20 people, 4 used neither ticket.
This is an EITHER/OR group problem.
Each of the 20 people EITHER used ticket 1 OR did not.
Each of the 20 people EITHER used ticket 2 OR did not.

For an EITHER/OR group problem, use a GROUP GRID (also known as a double-matrix) to organize the data.
Let 1 = used ticket 1, N1 = did not use ticket 1, 2 = used ticket 2, N2 = did not use ticket 2.
Here is the grid:
Image
In a group grid, the entries in any given row or column must add up to the TOTAL of that row or column.

Statement 1: Of the 20 people, 10 used only 1 ticket.
Let x = the number of people who used only ticket 2, implying that the number of people who used only ticket 1 = 10-x.
The following grid is yielded:
Image
No way to determine how many tickets were used.
INSUFFICIENT.

Statement 2: Of the 20 people, 4 used neither ticket..
The following grid is yielded:
Image
No way to determine how many tickets were used.
INSUFFICIENT.

Statements combined:
The following grid is yielded:
Image

Since the entries in any given row or column must sum to the total of that row or column, we get the following grid:
Image
Total tickets used = (total ticket 1) + (total ticket 2) = (16-x) + (x+6) = 22.
Since each of the 20 people purchased 2 tickets -- for a total of 40 tickets -- and 22 of the 40 tickets were used, we get:
Percentage used = (22/40) * 100 = 55%.
SUFFICIENT.

The correct answer is C.
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by Brent@GMATPrepNow » Fri Jan 16, 2015 8:48 am
I thought I'd point out that Mitch's "group grid" approach is also known as the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it.
Here, we have a population of people, and the two characteristics are:
- used ticket 1 OR did not use ticket 1
- used ticket 2 OR did not use ticket 2

This question type is VERY COMMON on the GMAT, so be sure to master the technique.

To learn more about the Double Matrix Method, watch our free video: https://www.gmatprepnow.com/module/gmat- ... ems?id=919

Once you're familiar with this technique, you can attempt these additional practice questions:

Easy Problem Solving questions
- https://www.beatthegmat.com/the-aam-aadm ... 72242.html
- https://www.beatthegmat.com/finance-majo ... 67425.html

Medium Problem Solving questions
- https://www.beatthegmat.com/probability- ... 73360.html
- https://www.beatthegmat.com/posted-speed ... 72374.html
- https://www.beatthegmat.com/motel-t271938.html
- https://www.beatthegmat.com/of-the-appli ... 70255.html
- https://www.beatthegmat.com/opening-nigh ... 64869.html
- https://www.beatthegmat.com/ds-french-ja ... 22297.html
- https://www.beatthegmat.com/prblem-solving-t279424.html

Difficult Problem Solving questions
- https://www.beatthegmat.com/ratio-problem-t268339.html
- https://www.beatthegmat.com/overlapping- ... 65223.html
- https://www.beatthegmat.com/fractions-t264254.html
- https://www.beatthegmat.com/overlapping- ... 64092.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-2

Easy Data Sufficiency questions
- https://www.beatthegmat.com/for-what-per ... 70596.html
- https://www.beatthegmat.com/ds-quest-t187706.html

Medium Data Sufficiency questions
- https://www.beatthegmat.com/sets-matrix-ds-t271914.html
- https://www.beatthegmat.com/each-of-peop ... 71375.html
- https://www.beatthegmat.com/a-manufacturer-t270331.html
- https://www.beatthegmat.com/in-costume-f ... 69355.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-1

Difficult Data Sufficiency questions
- https://www.beatthegmat.com/double-set-m ... 71423.html
- https://www.beatthegmat.com/sets-t269449.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-3

Cheers,
Brent
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by sachin_yadav » Sat Jan 17, 2015 9:33 am
Mitch and Brent,

Thank you so much for your explanations. They are extremely good. The practice questions on this type will definitely help.

Regards
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by subhakimi » Mon Jan 19, 2015 10:25 am
sachin_yadav wrote:Of the 20 people who each purchased 2 tickets to a concert, some used both tickets, some used only 1 ticket, and some used neither ticket. What percent of the tickets that were purchased by the 20 people were used by those people?
(1) Of the 20 people, 10 used only 1 ticket.
(2) Of the 20 people, 4 used neither ticket.

OA is C

It seems to be overlapping sets question. How to create set-matrix table for this question ?

Regards
Sachin
Your approach is redundant. It's basically an equation question.

We have three variables of which one is known and since each statement gives the value of one variable any of then can answer the question.
Each statement alone is sufficient
Answer is C.
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by Matt@VeritasPrep » Sun Jan 25, 2015 8:57 pm
subhakimi wrote:We have three variables of which one is known and since each statement gives the value of one variable any of then can answer the question.
Each statement alone is sufficient
Not quite: neither statement alone is sufficient, but the two statements TOGETHER are sufficient. You're essentially treating the question as

x + y + z = 20. What is (2x + y)/40?

S1:: y = 10
S2:: z = 4

for which we need both statements to solve.
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by Scott@TargetTestPrep » Tue Dec 12, 2017 9:57 am
sachin_yadav wrote:Of the 20 people who each purchased 2 tickets to a concert, some used both tickets, some used only 1 ticket, and some used neither ticket. What percent of the tickets that were purchased by the 20 people were used by those people?
(1) Of the 20 people, 10 used only 1 ticket.
(2) Of the 20 people, 4 used neither ticket.
We can create the following formula:

Total people = # who only used only 1 ticket + # who used both tickets + # who used neither ticket

20 = # who only used only 1 ticket + # who used both tickets + # who used neither ticket

Since each of the 20 people bought 2 tickets, a total of 40 tickets were bought. Of course, not all tickets were used since some used only 1 ticket and some used neither ticket. We need to determine what percentage of the tickets that were purchased by the 20 people were used by those people.

Statement One Alone:

Of the 20 people, 10 used only 1 ticket.

We see that the # who only used only 1 ticket is 10. However, since we know nothing about the number of people who used neither ticket, we still cannot answer the question. Statement one alone is not sufficient.

Statement Two Alone:

Of the 20 people, 4 used neither ticket.

We see that the # who used neither ticket is 4. However, since we don't know the number of people who used only 1 ticket, we still cannot answer the question. Statement two alone is not sufficient.

Statements One and Two Together:

Using the information from statements one and two, we see that:

20 = 10 + # who used both tickets +4

Thus, 20 - 14 = 6 people used both tickets.

Since a total of 40 tickets were purchased and only (6 x 2) + (10 x 1) = 22 were used, the percentage of tickets purchased by the 20 people that were used is 22/40 x 100 = 11/20 x 100 = 55 percent.

Answer: C

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