The only gift certificates that a certain store sold yesterday were worth either $100 each or $10 each. If the...
GMAT Data Sufficiency : (DS) Questions
The only gift certificates that a certain store sold yesterday were worth either \(\$100\) each or \(\$10\) each. If the store sold a total of \(20\) gift certificates yesterday, how many gift certificates worth \(\$10\) each did the store sell yesterday?
- The gift certificates sold by the store yesterday were worth a total of between \(\$1,650\) and \(\$1,800\).
- Yesterday the store sold more than \(15\) gift certificates worth \(\$100\) each.
Understanding the Question
We need to find the exact number of \(\$10\) gift certificates sold yesterday.
Given Information
- The store sells only two types of gift certificates: \(\$100\) each or \(\$10\) each
- Total number of gift certificates sold = 20
- All certificates sold yesterday were one of these two types
What We Need to Determine
Since there are exactly 20 certificates total, and each must be either \(\$100\) or \(\$10\), we need to find a unique split between these two types. If we know how many \(\$100\) certificates were sold, we automatically know how many \(\$10\) certificates were sold (since they must add up to 20).
For sufficiency, we need information that pins down exactly one possible value for the number of \(\$10\) certificates.
Analyzing Statement 1
Statement 1 tells us the total value of all certificates sold was between \(\$1,650\) and \(\$1,800\).
Strategic Approach
Here's a key insight: swapping one \(\$10\) certificate for one \(\$100\) certificate increases the total value by exactly \(\$90\) (since \(\$100 - \$10 = \$90\)). This means we can test a few strategic values rather than solving complex inequalities.
Testing Different Scenarios
Let's start near the upper bound and work our way down:
- 18 certificates at \(\$100\) and 2 at \(\$10\):
Total = \(18(\$100) + 2(\$10) = \$1,800 + \$20 = \$1,820\) ✗ (exceeds \(\$1,800\)) - 17 certificates at \(\$100\) and 3 at \(\$10\):
Total = \(17(\$100) + 3(\$10) = \$1,700 + \$30 = \$1,730\) ✓ (within range!) - 16 certificates at \(\$100\) and 4 at \(\$10\):
Total = \(16(\$100) + 4(\$10) = \$1,600 + \$40 = \$1,640\) ✗ (below \(\$1,650\))
Conclusion
Only one combination falls within the given range: 17 certificates at \(\$100\) and 3 certificates at \(\$10\).
Therefore, exactly 3 gift certificates worth \(\$10\) were sold.
[STOP - Statement 1 is Sufficient!]
Statement 1 is SUFFICIENT.
This eliminates answer choices B, C, and E.
Analyzing Statement 2
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us the store sold more than 15 gift certificates worth \(\$100\) each.
What This Means
"More than 15" means at least 16. Since we have exactly 20 certificates total, the possible scenarios are:
- 16 certificates at \(\$100\) → 4 certificates at \(\$10\)
- 17 certificates at \(\$100\) → 3 certificates at \(\$10\)
- 18 certificates at \(\$100\) → 2 certificates at \(\$10\)
- 19 certificates at \(\$100\) → 1 certificate at \(\$10\)
- 20 certificates at \(\$100\) → 0 certificates at \(\$10\)
Conclusion
We have five different possible values for the number of \(\$10\) certificates (0, 1, 2, 3, or 4). Without additional information, we cannot determine which possibility is correct.
Statement 2 is NOT SUFFICIENT.
This eliminates answer choices B and D.
The Answer: A
Statement 1 alone gives us exactly one possible value (3 certificates at \(\$10\)), while Statement 2 alone leaves us with multiple possibilities.
Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."