For each order, a certain company charges a delivery fee d that depends on the total price x of the...
GMAT Data Sufficiency : (DS) Questions
For each order, a certain company charges a delivery fee \(\mathrm{d}\) that depends on the total price \(\mathrm{x}\) of the merchandise in the order, where \(\mathrm{d}\) and \(\mathrm{x}\) are in dollars and
\(\mathrm{d = 3}\), if \(\mathrm{0 < x \leq 100}\)
\(\mathrm{d = 3 + \frac{x - 100}{100}}\), if \(\mathrm{100 < x \leq 500}\)
\(\mathrm{d = 7}\), if \(\mathrm{x > 500}\)
If George placed two separate orders with the company, was the total price of the merchandise in the two orders greater than $499?
- The delivery fee for one of the two orders was $3.
- The sum of the delivery fees for the two orders was $10.
Understanding the Question
We need to determine whether the total price of merchandise in George's two orders was greater than \(\$499\).
Let's first understand the delivery fee structure:
- For merchandise price \(\$0 < \mathrm{x} ≤ \$100\): delivery fee = \(\$3\)
- For merchandise price \(\$100 < \mathrm{x} ≤ \$500\): delivery fee = \(\$3 + (\mathrm{x}-100)/100\)
- For merchandise price \(\mathrm{x} > \$500\): delivery fee = \(\$7\)
Notice how the fee increases gradually in the middle range. At \(\mathrm{x} = \$200\), the fee is \(\$4\); at \(\mathrm{x} = \$300\), it's \(\$5\); and at \(\mathrm{x} = \$500\), it reaches \(\$7\).
What we need: To know if \(\mathrm{x₁} + \mathrm{x₂} > \$499\), where \(\mathrm{x₁}\) and \(\mathrm{x₂}\) are the merchandise prices of the two orders.
For this yes/no question to be sufficient, we need information that leads to a definitive "yes" or "no" answer about whether the total exceeds \(\$499\).
Analyzing Statement 1
Statement 1 tells us that the delivery fee for one of the two orders was \(\$3\).
A delivery fee of \(\$3\) means that order had merchandise price between \(\$0\) and \(\$100\) (inclusive).
However, we know nothing about the second order. Let's test extreme scenarios to see if we can get different answers:
- Scenario 1: If the second order has \(\mathrm{x₂} = \$50\) (fee = \(\$3\)), then total = \(\$100 + \$50 = \$150\), which is NOT > \(\$499\)
- Scenario 2: If the second order has \(\mathrm{x₂} = \$450\) (fee = \(\$6.50\)), then total = \(\$100 + \$450 = \$550\), which IS > \(\$499\)
Since we can get both "yes" and "no" answers, Statement 1 is NOT sufficient.
This eliminates choices A and D.
Analyzing Statement 2
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us that the sum of the delivery fees for the two orders was \(\$10\).
Let's think systematically about what combinations of fees could sum to \(\$10\):
- Two orders at minimum fee: \(\$3 + \$3 = \$6\) (too low)
- Two orders at maximum fee: \(\$7 + \$7 = \$14\) (too high)
- One minimum, one maximum: \(\$3 + \$7 = \$10\) ✓
Key insight: The only way to get exactly \(\$10\) is if one order has a \(\$3\) fee and the other has a \(\$7\) fee.
Now, what does this tell us about merchandise prices?
- The \(\$3\) fee order has merchandise price ≤ \(\$100\)
- The \(\$7\) fee order must have merchandise price > \(\$500\) (since the fee only reaches \(\$7\) when \(\mathrm{x} ≥ \$500\))
Therefore, the total merchandise price must be at least \(\$100 + \$500 = \$600\), which is definitely greater than \(\$499\).
[STOP - Sufficient!]
Statement 2 is sufficient.
The Answer: B
Statement 2 alone is sufficient because it forces one order to be over \(\$500\) and guarantees a total exceeding \(\$499\), while Statement 1 alone leaves the second order's value completely unknown.
Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."