How to solve this using Double Set Matrix? DS

[DileepW137]

Hello there,

I am struggling to do this question using your overlapping sets matrix. How do you do this using the technique as described in the Word Problems book?


At the beginning of each month, a certain book club offers its members the opportunity to purchase one or more books at a discount. At the beginning of January, the number of book club members who elected to purchase both The Brothers Karamazov and Pride and Prejudice was equal to the number of members who elected to purchase neither of those books. How many members were in the book club at the beginning of January?

1) 128 book club members purchased The Brothers Karamazov and 212 book club members purchased Pride and Prejudice.
2) Exactly half of the book club members who purchased The Brothers Karamazov also purchased Pride and Prejudice.



I have the following so far..


B------not B-------Total
P --------- x-------212
not P---------------x
Total----128---------------------?


Using Venn Diagram method, I am getting A as the answer. (since the x's cancel out below)
(128 − x) + x + (212 − x) + x = 128 + 212 = 340


Thank you,

[Sage Pearce-Higgins]

First of all, please could you tell me the source of this problem?

For this problem, I would set up a double-set matrix initially just as a rephrase of the question:
----------- B----------------- not B--------------Total
P----------x
not P ----------------------- x
Total-----------------------------------------------??

Then introduce the information from statement 1:
----------- B----------------- not B--------------Total
P------------x---------------- 212
not P------------------------- x
Total----------128------------------------------??

Then deduce the following:
------------- B----------------- not B--------------Total
P-------------x--------------212 - x------------- 212
not P--------128 - x--------------x
Total -------128 -----------------------------------??

From this you can fill in more numbers:
----------- B----------------- not B--------------Total
P ---------x------------------212 - x-------------212
not P----128 - x---------- x----------------------128
Total-----128-------------- 212-------------- ??

This shows how statement 1 is sufficient. If you fill in the information from statement 2 in a similar way, you will see that there are no actual numbers and no way to reach an answer to the question. I agree with you that the answer is A.

However, using a double-set matrix for a DS problem like this can be cumbersome and time-consuming. It seems that your formula approach (Total = Both + Neither + A not B + B not A) takes you to an answer more quickly by showing how the x cancels out, and is probably a better method for a problem like this one.