For each order, a mail order bookseller charges a fixed processing fee and an additional shipping fee for each book...

Understanding the Question

We need to find the total cost of all processing and shipping fees for Rajeev's five orders.

Given Information

• Each order has two cost components:
  • Processing fee: Fixed amount per order (same regardless of books)
  • Shipping fee: Charged per book
• Rajeev placed 5 orders: 1 book (Jan), 2 books (Feb), 3 books (Mar), 4 books (Apr), 5 books (May)
• Total books ordered: \(1 + 2 + 3 + 4 + 5 = 15\) books

What We Need to Determine

To find the total cost uniquely, we need to know BOTH:

• The processing fee per order (let's call it P)
• The shipping fee per book (let's call it S)

Total cost = \((5 \text{ orders} \times \mathrm{P}) + (15 \text{ books} \times \mathrm{S}) = 5\mathrm{P} + 15\mathrm{S}\)

For sufficiency, we need enough information to determine exactly one value for this total.

Analyzing Statement 1

Statement 1: The March order (3 books) cost $1 more than the January order (1 book).

Let's think about what creates this $1 difference:

• Both orders have the same processing fee (P)
• March has 2 more books than January
• So the $1 difference comes entirely from shipping those 2 extra books

This tells us: \(2 \text{ books} \times \text{shipping fee} = \$1\)
Therefore: Shipping fee = $0.50 per book

But here's the critical issue: Statement 1 tells us nothing about the processing fee. Let's see why this matters:

Different processing fees → Different total costs → Multiple answers possible

Statement 1 alone is NOT sufficient.

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

Analyzing Statement 2

Now we completely forget Statement 1 and analyze Statement 2 independently.

Statement 2: The total shipping fees for all five orders was $7.50.

Since shipping is charged per book and we ordered 15 books total:

• \(15 \text{ books} \times \text{shipping fee per book} = \$7.50\)
• Therefore: Shipping fee = $0.50 per book

Once again, we know the shipping fee but have no information about the processing fee. Let's test different scenarios:

Again, different processing fees produce different total costs.

Statement 2 alone is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Statements

When we use both statements together:

• Statement 1 reveals: shipping fee = $0.50 per book
• Statement 2 reveals: shipping fee = $0.50 per book

Both statements give us the same shipping fee (which is reassuring—the problem is internally consistent!). However, neither statement provides any information about the processing fee.

With both statements combined:

• Total shipping costs = \(15 \times \$0.50 = \$7.50\) ✓
• Total processing costs = \(5 \times \mathrm{P} = \text{?}\)

Since we still cannot determine P, we cannot find a unique total cost. Whether the processing fee is $1, $10, or $100 per order, we'll get different answers:

• If P = $2: Total = \(5(\$2) + \$7.50 = \$17.50\)
• If P = $15: Total = \(5(\$15) + \$7.50 = \$82.50\)

Both statements together are NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice C.

The Answer: E

The statements together are not sufficient because we have a two-variable problem but only constraints on one variable. Both statements tell us about shipping fees but neither addresses processing fees.

Key Insight: In problems with multiple cost components, check whether the statements constrain ALL components. If any component remains unconstrained, the answer will be E.

Answer: E - The statements together are not sufficient.