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Each new car produced in a certain factory includes at least
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17 Aug 2019, 05:50

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Question Stats:

Each new car produced in a certain factory includes at least one of the following three options: Bluetooth, navigation system, and sunroof. Of the cars that came off the production line yesterday, 25 had Bluetooth, 13 had a navigation system, and 30 had a sunroof. Yesterday, 50 cars were completed. Which of the following could be true about the cars that came off the production line yesterday?

Indicate all possible answers.

A) There were 2 cars that had exactly two of the options and there were 8 cars that had all three options.

B) There were 3 cars that had exactly two of the options and there were 7 cars that had all three options.

C) There were 4 cars that had exactly two of the options and there were 7 cars that had all three options.

D) There were 6 cars that had exactly two of the options and there were 6 cars that had all three options.

E) There were 7 cars that had exactly two of the options and there were 5 cars that had all three options.

F) There were 8 cars that had exactly two of the options and there were 4 cars that had all three options.

Indicate all possible answers.

A) There were 2 cars that had exactly two of the options and there were 8 cars that had all three options.

B) There were 3 cars that had exactly two of the options and there were 7 cars that had all three options.

C) There were 4 cars that had exactly two of the options and there were 7 cars that had all three options.

D) There were 6 cars that had exactly two of the options and there were 6 cars that had all three options.

E) There were 7 cars that had exactly two of the options and there were 5 cars that had all three options.

F) There were 8 cars that had exactly two of the options and there were 4 cars that had all three options.

Retired Moderator

Joined: **10 Apr 2015 **

Posts: **6218**

Given Kudos: **136 **

Re: Each new car produced in a certain factory includes at least
[#permalink]
18 Aug 2019, 04:55

3

GreenlightTestPrep wrote:

Each new car produced in a certain factory includes at least one of the following three options: Bluetooth, navigation system, and sunroof. Of the cars that came off the production line yesterday, 25 had Bluetooth, 13 had a navigation system, and 30 had a sunroof. Yesterday, 50 cars were completed. Which of the following could be true about the cars that came off the production line yesterday?

Indicate all possible answers.

A) There were 2 cars that had exactly two of the options and there were 8 cars that had all three options.

B) There were 3 cars that had exactly two of the options and there were 7 cars that had all three options.

C) There were 4 cars that had exactly two of the options and there were 7 cars that had all three options.

D) There were 6 cars that had exactly two of the options and there were 6 cars that had all three options.

E) There were 7 cars that had exactly two of the options and there were 5 cars that had all three options.

F) There were 8 cars that had exactly two of the options and there were 4 cars that had all three options.

Indicate all possible answers.

A) There were 2 cars that had exactly two of the options and there were 8 cars that had all three options.

B) There were 3 cars that had exactly two of the options and there were 7 cars that had all three options.

C) There were 4 cars that had exactly two of the options and there were 7 cars that had all three options.

D) There were 6 cars that had exactly two of the options and there were 6 cars that had all three options.

E) There were 7 cars that had exactly two of the options and there were 5 cars that had all three options.

F) There were 8 cars that had exactly two of the options and there were 4 cars that had all three options.

This is a 3-criteria overlapping sets question.

So, we can sketch the given information as follows:

Let's start with answer choice A.

If 2 cars had exactly 2 of the options, and 8 cars had all three options, then we can add this information as follows:

ASIDE: It makes no difference which two overlapping options we use for the "2 cars had exactly 2 of the options" part.

In the above diagram, I've placed those two cars in the navigation/sunroof overlap, but I could have also placed them in the navigation/Bluetooth overlap or in the sunroof/Bluetooth and still answer the question correctly.

Since there are no other cars that have overlapping features, we can complete the rest of the diagram as follows:

Now that we've completed the diagram, we must ask, "HAVE WE ACCOUNTED FOR ALL 50 CARS?"

To find out, we must add all of the values in the diagram.

We get: 17 + 3 + 8 + 2 + 20 = 50

Perfect!

So, answer choice A is possible

Now follow the same steps for answer choice B to get:

HAVE WE ACCOUNTED FOR ALL 50 CARS?

Add all of the values in the diagram to get: 18 + 3 + 7 + 4 + 20 = 51

No good!

Answer choice B is NOT possible

For answer choice C, we get:

Add values: 18 + 2 + 7 + 4 + 19 = 50

Perfect!

So, answer choice C is possible

For answer choice D, we get:

Add values: 19 + 1 + 6 + 6 + 18 = 50

Perfect!

So, answer choice D is possible

For answer choice E, we get:

Add all of the values in the diagram to get: 21 + 1 + 5 + 7 + 18 = 53

No good!

Answer choice E is NOT possible

For answer choice F, we get:

Add all of the values in the diagram to get: 21 + 1 + 4 + 8 + 18 = 52

No good!

Answer choice F is NOT possible

Answer: A, C, D

Cheers,

Brent

Re: Each new car produced in a certain factory includes at least
[#permalink]
03 Oct 2021, 19:48

3

Here is a quicker solution.

We know that for three overlapping sets A, B, C:

Total = A + B + C - (sum of all 2s) - 2(sum of all threes)

Putting in the values from the prompt/question, we get;

50 = 25 + 13 + 30 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - (sum of all 2s) - 2(sum of all threes)

We have to plug values from each option in the aforementioned equation to find the correct answers.

Option A: 2 cars with two options and 8 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - (2) - 2(8)

50 = 68 - 18

50 = 50

Correct

Option B: 3 cars with two options and 7 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 3 - 2(7)

50 = 68 - 17

50 = 51

Incorrect

Option C: 4 cars with two options and 7 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 4 - 2(7)

50 = 68 - 18

50 = 50

Correct

Option D: 6 cars with two options and 6 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 6 - 2(6)

50 = 68 - 18

50 = 50

Correct

Option E: 7 cars with two options and 5 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 -7 - 2(5)

50 = 68 - 17

50 = 51

Incorrect

Option F: 8 cars with two options and 4 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 8 - 2(4)

50 = 68 - 16

50 = 52

Incorrect

We know that for three overlapping sets A, B, C:

Total = A + B + C - (sum of all 2s) - 2(sum of all threes)

Putting in the values from the prompt/question, we get;

50 = 25 + 13 + 30 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - (sum of all 2s) - 2(sum of all threes)

We have to plug values from each option in the aforementioned equation to find the correct answers.

Option A: 2 cars with two options and 8 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - (2) - 2(8)

50 = 68 - 18

50 = 50

Correct

Option B: 3 cars with two options and 7 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 3 - 2(7)

50 = 68 - 17

50 = 51

Incorrect

Option C: 4 cars with two options and 7 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 4 - 2(7)

50 = 68 - 18

50 = 50

Correct

Option D: 6 cars with two options and 6 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 6 - 2(6)

50 = 68 - 18

50 = 50

Correct

Option E: 7 cars with two options and 5 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 -7 - 2(5)

50 = 68 - 17

50 = 51

Incorrect

Option F: 8 cars with two options and 4 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 8 - 2(4)

50 = 68 - 16

50 = 52

Incorrect

Re: Each new car produced in a certain factory includes at least
[#permalink]
20 Jul 2022, 13:46

computerbot, when we should subtract all three three times, not 2 times? IN the explanation, it says that items in all three categories are triple counted.

Re: Each new car produced in a certain factory includes at least
[#permalink]
20 Jul 2022, 17:47

computerbot wrote:

Here is a quicker solution.

We know that for three overlapping sets A, B, C:

Total = A + B + C - (sum of all 2s) - 2(sum of all threes)

Putting in the values from the prompt/question, we get;

50 = 25 + 13 + 30 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - (sum of all 2s) - 2(sum of all threes)

We have to plug values from each option in the aforementioned equation to find the correct answers.

Option A: 2 cars with two options and 8 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - (2) - 2(8)

50 = 68 - 18

50 = 50

Correct

Option B: 3 cars with two options and 7 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 3 - 2(7)

50 = 68 - 17

50 = 51

Incorrect

Option C: 4 cars with two options and 7 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 4 - 2(7)

50 = 68 - 18

50 = 50

Correct

Option D: 6 cars with two options and 6 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 6 - 2(6)

50 = 68 - 18

50 = 50

Correct

Option E: 7 cars with two options and 5 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 -7 - 2(5)

50 = 68 - 17

50 = 51

Incorrect

Option F: 8 cars with two options and 4 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 8 - 2(4)

50 = 68 - 16

50 = 52

Incorrect

We know that for three overlapping sets A, B, C:

Total = A + B + C - (sum of all 2s) - 2(sum of all threes)

Putting in the values from the prompt/question, we get;

50 = 25 + 13 + 30 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - (sum of all 2s) - 2(sum of all threes)

We have to plug values from each option in the aforementioned equation to find the correct answers.

Option A: 2 cars with two options and 8 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - (2) - 2(8)

50 = 68 - 18

50 = 50

Correct

Option B: 3 cars with two options and 7 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 3 - 2(7)

50 = 68 - 17

50 = 51

Incorrect

Option C: 4 cars with two options and 7 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 4 - 2(7)

50 = 68 - 18

50 = 50

Correct

Option D: 6 cars with two options and 6 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 6 - 2(6)

50 = 68 - 18

50 = 50

Correct

Option E: 7 cars with two options and 5 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 -7 - 2(5)

50 = 68 - 17

50 = 51

Incorrect

Option F: 8 cars with two options and 4 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - 8 - 2(4)

50 = 68 - 16

50 = 52

Incorrect

why do we substract it with 2(sum of all threes)? do we require at least 1 that has all the 3?

Posted from my mobile device

Re: Each new car produced in a certain factory includes at least
[#permalink]
21 Jul 2022, 03:39

1

When we counted the sum of all 3, it was counted thrice while considering the sum of all 2. Hence, we need to subtract twice from thrice to get one.

ademeutia wrote:

why do we substract it with 2(sum of all threes)? do we require at least 1 that has all the 3?

Posted from my mobile device

Posted from my mobile device

Re: Each new car produced in a certain factory includes at least
[#permalink]
12 Nov 2023, 02:06

1

How I solved:

50=25+13+30-Both-(2*All)+Neither

Since we are told that neither is 0 in the stem (at least one of the modifications..), we can exclude that.

50=68-Both-(2*All)

18=B+2A

9-B/2=A

Since A must be an integer (there cannot be half of a completed car), we can eliminate options B and E, since B would be odd and thus give A as a non integer.

A is possible, if B=2, 9-1=8=A

C is possible, if B=4, 9-2=7=A

D is possible, if B=6, 9-3=6

E is not possible, as if B=8, 9-4=5=A. Note that if neither was not zero, then E could be possible - we would just need neither to be equal to 1 in that case.

Therefore A, C, and D are good.

50=25+13+30-Both-(2*All)+Neither

Since we are told that neither is 0 in the stem (at least one of the modifications..), we can exclude that.

50=68-Both-(2*All)

18=B+2A

9-B/2=A

Since A must be an integer (there cannot be half of a completed car), we can eliminate options B and E, since B would be odd and thus give A as a non integer.

A is possible, if B=2, 9-1=8=A

C is possible, if B=4, 9-2=7=A

D is possible, if B=6, 9-3=6

E is not possible, as if B=8, 9-4=5=A. Note that if neither was not zero, then E could be possible - we would just need neither to be equal to 1 in that case.

Therefore A, C, and D are good.

gmatclubot

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