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For each order, a mail order bookseller charges a fixed processing fee and an additional shipping fee for each book in the order. Rajeev placed five different orders with this bookseller - an order for 1 book in January, an order for 2 books in February, an order for 3 books in March, an order for 4 books in April, and an order for 5 books in May. What was the total of Rajeev's processing and shipping fee for these five orders?
We need to find the total cost of all processing and shipping fees for Rajeev's five orders.
To find the total cost uniquely, we need to know BOTH:
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.
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:
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:
| Processing Fee | Shipping per Book | Total Cost (5P + 15S) |
| $1 per order | $0.50 | \(5(\$1) + 15(\$0.50) = \$12.50\) |
| $10 per order | $0.50 | \(5(\$10) + 15(\$0.50) = \$57.50\) |
Different processing fees → Different total costs → Multiple answers possible
Statement 1 alone is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
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:
Once again, we know the shipping fee but have no information about the processing fee. Let's test different scenarios:
| Processing Fee | Shipping per Book | Total Cost (5P + 15S) |
| $5 per order | $0.50 | \(5(\$5) + 15(\$0.50) = \$32.50\) |
| $20 per order | $0.50 | \(5(\$20) + 15(\$0.50) = \$107.50\) |
Again, different processing fees produce different total costs.
Statement 2 alone is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice B.
When we use both statements together:
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:
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:
Both statements together are NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice C.
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.