During a 10-week summer vacation, was the average (arithmetic mean)

Yes, you can do this way as well.

my approach to prove that A, B, and C are not sufficient.

1. suppose average for C is 4, and so is for J.
2x4=8
2x4 - 5 = 3
so A for C > than A for J.

but it can be that A read 5/week and J read 4/week
10>3 true again, but values differ.

A and D are out

2. clearly insufficient.
B is out.

1+2
40 total books
C in first 5 weeks read 40/2 -3 = 17 books, and 23 in last 5 weeks
J on the contrary read 20 books in the first 5 weeks, and 20 in the last 5 weeks.
satisfies both 1 and 2, but averages are equal

now.
suppose C=5books/week = 50 books total
J = 4books/week = 40 total
C during first 5 weeks read 46 books, and 4 in the last 5 weeks
J read 39 in the first 5 weeks, and 1 in the last 5 weeks.
again, satisfies both conditions from 1 and 2..
2 outcomes - C is out

Asked: During a 10-week summer vacation, was the average (arithmetic mean) number of books that Carolyn read per week greater than the average number of books that Jacob read per week?

Let the average (arithmetic mean) number of books that Carolyn read per week be c
Let the average number of books that Jacob read per week be j

(1) Twice the average number of books that Carolyn read per week was greater than 5 less than twice the average number of books that Jacob read per week.
2c > 2j-5 => c>j may or may not be true
NOT SUFFICIENT

(2) During the last 5 weeks of the vacation, Carolyn read a total of 3 books more than Jacob.
No data is provided for first 5 weeks
NOT SUFFICIENT

IMO E

let C = average number of books that Carolyn read per week
let J = average number of books that Jacob read per week

(1) \(2C > 2J -5\)
\(2C - 2J = -5\)
\(C - J = -\frac{5}{2}\)

We can't say for certain whether Carolyn's average was higher or lower.

(2) Clearly insufficient.

(1&2) Combined, we still can't determine whether Carolyn's average is higher or lower. INSUFFICIENT.

Answer is E.

what does "greater than 5 less than" even mean lol?

Let \(c\) be the average # of books that Carolyn read per week;
Let \(j\) be the average # of books that Jacob read per week;

Then "Twice the average number of books that Carolyn read per week was greater than 5 less than twice the average number of books that Jacob read per week" would mean \(2c>2j-5\).

Bunuel, Sajjad1994 kindly update the question tag to GMAT prep (it is part of Gmat Official practice question 1 )

Here is the official solution

Let C and J represent the average numbers of books Carolyn and Jacob read per week during their 10-week summer vacation. Determine if C > J, which is the same as determining if C –J is positive.

(1) It is given that 2C > 2J – 5, from which it follows that 2C – 2J > –5 and C –J > –​five halves​. Thus C –J could be positive, but it could also be negative; NOT sufficient.

(2) If C 2 and J 2 represents the numbers of books Carolyn and Jacob read during the last 5 weeks of the summer, then C 2 = J 2 + 3, but this is not enough information to determine if C > J, because the numbers of books Carolyn and Jacob read during the first 5 weeks of the summer are unknown. If C 2 = 20, from which it follows that J 2 = 17, and Carolyn read 20 books during the first 5 weeks of the summer, then C = ​the fraction with numerator 20 plus 20 and denominator 10​ = 4. If Jacob read 13 books during the first 5 weeks, then J = ​the fraction with numerator 17 plus 13 and denominator 10​ = 3 and C > J, but if Jacob read 43 books during the first 5 weeks, then J = ​the fraction with numerator 17 plus 43 and denominator 10​ = 6 and C < J; NOT sufficient.

Since the values for C –J obtained in the examples used for (2), namely 4 – 3 = 1 and 4 – 6 = –2, also satisfy (1) because each is greater than –​five halves​, it follows that (1) and (2) together are NOT sufficient.

The correct answer is E; both statements together are still not sufficient.