Joanna bought only $0.15 stamps and $0.29 stamps. How many $0.15 stamps did she buy? She bought $4.40 worth of...

Understanding the Question

We need to find the exact number of \(\$0.15\) stamps that Joanna bought.

Given Information

• Joanna bought only two types of stamps: \(\$0.15\) stamps and \(\$0.29\) stamps
• We need a specific number (not a range or yes/no answer)

What We Need to Determine

For information to be sufficient, it must lead to exactly ONE possible value for the number of \(\$0.15\) stamps. If multiple values are possible, the information is NOT sufficient.

Key Insight

When dealing with stamp combinations where the values share no common factors (15 cents and 29 cents are coprime), valid integer combinations for specific totals are often unique or very limited. This makes strategic testing more efficient than complex algebra.

Analyzing Statement 1

Statement 1 tells us: She bought \(\$4.40\) worth of stamps.

What Statement 1 Tells Us

The total value of all stamps is \(\$4.40\). This constrains the possible combinations of \(\$0.15\) and \(\$0.29\) stamps.

Testing for Uniqueness

Let's find the boundaries first:

• Maximum \(\$0.29\) stamps: \(\$4.40 \div \$0.29 \approx 15.2\), so at most 15 stamps
• Maximum \(\$0.15\) stamps: \(\$4.40 \div \$0.15 \approx 29.3\), so at most 29 stamps

Now let's test a round number combination:

• Try 10 of each: \(10 \times \$0.15 + 10 \times \$0.29 = \$1.50 + \$2.90 = \$4.40\) ✓

This works! But is it the only combination?

Checking Other Possibilities

Let's test a few other scenarios to ensure uniqueness:

• With 5 of the \(\$0.29\) stamps: \(5 \times \$0.29 = \$1.45\)
  • Remaining needed: \(\$4.40 - \$1.45 = \$2.95\)
  • Number of \(\$0.15\) stamps: \(\$2.95 \div \$0.15 = 19.67...\)
  • Not a whole number ✗
• With 15 of the \(\$0.29\) stamps: \(15 \times \$0.29 = \$4.35\)
  • Remaining needed: \(\$4.40 - \$4.35 = \$0.05\)
  • Can't make \(\$0.05\) with \(\$0.15\) stamps ✗

The key mathematical fact: 15 and 29 share no common factors (15 = 3×5, and 29 is prime). This severely restricts valid combinations that can sum to exactly \(\$4.40\).

Conclusion for Statement 1

Our testing reveals that 10 of each stamp type is the unique solution. Therefore, she bought exactly 10 stamps of the \(\$0.15\) type.

[STOP - Statement 1 is Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: She bought an equal number of \(\$0.15\) stamps and \(\$0.29\) stamps.

What Statement 2 Provides

If she bought x stamps of the \(\$0.15\) type, she also bought x stamps of the \(\$0.29\) type.

The total cost would be: \(x \times (\$0.15 + \$0.29) = x \times \$0.44\)

Why This Isn't Sufficient

Statement 2 doesn't tell us what x actually is. She could have bought:

• 1 of each type (total: \(\$0.44\))
• 2 of each type (total: \(\$0.88\))
• 10 of each type (total: \(\$4.40\))
• 100 of each type (total: \(\$44.00\))

Without knowing the total amount spent or any other constraint, we cannot determine the specific value of x, and therefore cannot determine the exact number of \(\$0.15\) stamps.

Conclusion for Statement 2

Statement 2 is NOT sufficient because it allows for multiple possible values.

This eliminates choices B and D.

The Answer: A

Since only Statement 1 provides enough information to determine the exact number of \(\$0.15\) stamps (which is 10), while Statement 2 leaves multiple possibilities open, the answer is A.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."