The only gift certificates that a certain store sold yesterday were worth either $100 each or $10 each. If the...

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."