# Calling all Logic Experts!

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OK, this is a shameless request for help with DS's 5th grade homework. This assignment, and too many more like them, are why we are homeschooling again starting in the fall. Actually, I like that the teacher is doing logic, but with everything else she is giving right now, we don't have the time to work this one out, and it's due Wednesday.

Normally I am not the kind of mom that would ask for help like this, but with 15 other logic problems in tonight's homework, this is the one that I'm too tired to wrap my head around.

Here's the question:

The book fair is here, and many wonderful books are available! Several genres are represented, including fantasy, non-fiction, and historical fiction. One class was surveyed to determine the most popular genre for that class. From the clues, determine how many people were in the class and which genre was most popular.

*Everyone in the class bought at least one book.

*14 people bought only one book

*Only 3 people bought all three genres

*7 people bought fantasy books but not non-fiction or historical fiction

*4 people bought fantasy and non-fiction but not historical fiction

*14 people did not buy historical fiction

*13 people did not buy non-fiction

*8 people did not buy fantasy books

How many people were in the class?

What was the most popular genre?

At first blush, my answer would be 21 in the class, with fantasy as the most popular. What are your thoughts?

Thank you so much for any insight or help anyone can provide!

Suzanne

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information given. I'm not saying this very well. Place the options along the margins and then fill in and deduce answers from the information given.

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Are you sure you typed out the problem correctly? Even with a logic box I can't get it to come out right. I get 14 people for Fantasy. 10 for Non-fiction. 9 for Historical Fiction. BUT - that doesn't come out to 8 people who do not buy fantasy books (I get 9). I get 23 people in all.

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This is a math problem where you solve systems of simultaneous equations with seven variables. Every student bought at least one book, and there are three genres. This gives us seven possible combinations of genres. Each student will fall into one, and only one of these:

(F) Fantasy only

(nF) Non-fiction only

(hF) Historical fiction only

(FnF) Fantasy & non-fiction

(FhF) Fantasy & historical fiction

(nFhF) Non-fiction & historical fiction

(FnFhF) Fantasy, non-fiction & historical fiction

Now to set up the equations:

*14 people bought only one book

(F) + (nF) + (hF) = 14

*Only 3 people bought all three genres

(FnFhF) = 3

*7 people bought fantasy books but not non-fiction or historical fiction

(F) = 7

*4 people bought fantasy and non-fiction but not historical fiction

(FnF) = 4

*14 people did not buy historical fiction

(F) + (nF) + (FnF) = 14

But we already know the values for (F) and (FnF), so we can substitute

(7) + (nF) + (4) = 14

(nF) = 3

Additionally, now that we know the value for (nF), we can solve for (hF) using the first equation.

(F) + (nF) + (hF) = 14

(7) + (3) + (hF) = 14

(hF) = 4

*13 people did not buy non-fiction

(F) + (hF) + (FhF) = 13

(7) + (4) + (FhF) = 13

(FhF) = 2

*8 people did not buy fantasy books

(nF) + (hF) + (nFhF) = 8

(3) + (4) + (nFhF) = 8

(nFhF) = 1

Now that all the variables are solved, summing the categories will give us the number of students:

(F) = 7

(nF) = 3

(hF) = 4

(FnF) = 4

(FhF) = 2

(nFhF) = 1

(FnFhF) = 3

(F) + (nF) + (hF) + (FnF) + (FhF) + (nFhF) + (FnFhF) = 24

There are 24 students in the class.

16 students bought at least one work of fantasy. 11 students bought at least one work of non-fiction. 10 students bought at least one work of historical fiction. Fantasy is the most popular genre.

Edited by Jaxon

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I agree with Jaxon's answers but we find the problem much easier to figure out by using a Venn or set diagram. Draw the 3 circles intersecting with one another (I label the circles F, NF, HF).

From the clues:

14 bought only 1 - set this aside for now

3 bought all 3 - put the 3 where the 3 circles intersect

7 bought F not NF or HF, put 7 in the F circle

4 bought F and NF but not HF, place the 4 in the intersection of F and NF

14 bought no HF - The only places for no HF are F, NF and the intersection of F and NF, we already know F has 7 and the intersection has 4, so we can figure out that NF is 14-11 or 3, so put that in the

NF circle

Go back to the 14 only bought 1 kind - now that we know that F=7 and NF=3 we can figure out that HF = 14-10 or 4 so put that in the HF circle

13 bought no NF - The only places with no NF are F, HF and the intersection of F&HF - we know 2 out of the 3 (F=7 and HF=4) so we can figure that the intersection is 13-7-4 or 2, put 2 in the intersection between HF and F

8 bought no F - No F is the sum of NF, HF and the intersection of NF and HF; Again we know 2 of the 3 so we can calculate the intersection = 8-3-4 or 1; Put 1 in the intersection of NF and HF.

Add all of the numbers on your Venn diagram to get 24 students in the class. Add each genre and find that F = 16, NF = 11, HF = 10

Hope that helps!!

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I knew that the hive mind, minds immeasurably wiser than mine, could help me! Thank you so so so much! While I am not the sharpest tack when it comes to logic, if any of you are in need of a good recipe for something, I am your go-to girl!

Suzanne

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