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

Understanding the Question

We need to determine whether Carolyn read more books per week on average than Jacob during their 10-week summer vacation. This is a yes/no question - we need to be able to answer definitively YES or definitively NO for the information to be sufficient.

Let's define:
- Carolyn's average books per week = C
- Jacob's average books per week = J

Our question: Is \(\mathrm{C} > \mathrm{J}\)?

For information to be sufficient, it must constrain the possibilities enough that we can only get one answer - either always YES or always NO.

Analyzing Statement 1

Statement 1 says: "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."

In simpler terms: Carolyn's average can't be too far below Jacob's average. But does "not too far below" guarantee she's actually above? Let's test with concrete examples:

Test Case 1: What if they both averaged 5 books per week?
- Carolyn: 5 books/week, Jacob: 5 books/week
- Check: Is twice Carolyn's (10) > twice Jacob's minus 5 (5)?
- 10 > 5? YES! ✓ Statement satisfied
- But \(\mathrm{C} = \mathrm{J}\), so our answer is NO (Carolyn is not greater)

Test Case 2: What if Carolyn averaged 6 and Jacob averaged 5?
- Check: Is 12 > 5? YES! ✓ Statement satisfied
- And \(\mathrm{C} > \mathrm{J}\), so our answer is YES

Since we can get both YES and NO answers while satisfying Statement 1, it is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now let's completely forget Statement 1 and analyze Statement 2 independently.

Statement 2 says: "During the last 5 weeks of the vacation, Carolyn read a total of 3 books more than Jacob."

This tells us about only HALF the vacation. The first 5 weeks remain a complete mystery. Let's explore the possibilities:

Scenario 1: What if they read the same amount in the first 5 weeks?
- First 5 weeks: Carolyn = 10 books, Jacob = 10 books
- Last 5 weeks: Carolyn = 8 books, Jacob = 5 books (satisfies the +3 condition)
- 10-week totals: Carolyn = 18 books, Jacob = 15 books
- Averages: \(\mathrm{C} = 1.8\) books/week, \(\mathrm{J} = 1.5\) books/week
- Answer: YES, \(\mathrm{C} > \mathrm{J}\)

Scenario 2: What if Jacob dominated the first 5 weeks?
- First 5 weeks: Carolyn = 0 books, Jacob = 10 books
- Last 5 weeks: Carolyn = 8 books, Jacob = 5 books (still satisfies the +3 condition)
- 10-week totals: Carolyn = 8 books, Jacob = 15 books
- Averages: \(\mathrm{C} = 0.8\) books/week, \(\mathrm{J} = 1.5\) books/week
- Answer: NO, \(\mathrm{C} < \mathrm{J}\)

Since we can get both YES and NO answers, Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices B and D.

Combining Both Statements

Now we use BOTH pieces of information together:
1. Carolyn's average is "somewhat close" to Jacob's (she can't be too far behind)
2. Carolyn read 3 more books in the last 5 weeks

The critical question: Is being "close" to Jacob's average PLUS being ahead by 3 books in the last half enough to guarantee \(\mathrm{C} > \mathrm{J}\)?

Let's think about this strategically. The constraint from Statement 1 is fairly weak - it still allows Carolyn to be behind Jacob overall. Meanwhile, her 3-book advantage covers only the last 5 weeks. What about the first 5 weeks?

Can we construct a scenario where Jacob still wins?
- Imagine Jacob had a substantial lead in the first 5 weeks (say, 8-10 books ahead)
- As long as their averages still satisfy the "closeness" constraint from Statement 1, this is allowed
- Despite losing by 3 books in the second half, Jacob could maintain a higher overall average

Can we construct a scenario where Carolyn wins?
- If the first 5 weeks were more balanced (or if Carolyn was ahead)
- Combined with her guaranteed 3-book advantage in the last 5 weeks
- Carolyn could easily have the higher average

Since both outcomes remain possible even with both statements combined, they are NOT sufficient together.

[STOP - Not Sufficient!] This eliminates choices A, B, C, and D.

The Answer: E

Even with both statements combined, we cannot definitively determine whether Carolyn's average was greater than Jacob's average. The constraints are simply too weak - they allow for multiple scenarios with different answers.

Answer Choice E: "The statements together are not sufficient."