If each of the stamps Carla bought cost 20, 25, text{ or } 30 cents and she bought at least...

Understanding the Question

The question asks us to find the exact number of 25-cent stamps that Carla bought.

Given Information

• Three types of stamps available: \(20¢\), \(25¢\), and \(30¢\)
• Carla bought at least one of each type
• We need a specific value for the number of 25-cent stamps

What We Need to Determine

For sufficiency, we need enough information to arrive at exactly one possible value for the number of 25-cent stamps. If multiple values are possible, the information is not sufficient.

Key Insight

Since Carla must buy at least one of each stamp type, we have built-in constraints. With three unknowns (number of each stamp type), we typically need enough independent constraints to narrow down to a unique solution. The critical question is whether the given information provides those constraints.

Analyzing Statement 1

Statement 1: She spent a total of \(\$1.45\) for stamps.

What Statement 1 Tells Us

We know the total cost is \(\$1.45\) (or \(145\) cents). This gives us one constraint connecting the three unknowns.

Testing Different Scenarios

Let's think about this strategically. With only a total cost constraint and three different stamp denominations, can we have different distributions that yield the same total?

Consider these possibilities:

• More cheap stamps: Many \(20¢\) stamps, fewer expensive ones
• More expensive stamps: Fewer stamps overall, but more \(30¢\) stamps
• Balanced approach: A mix of all three types

Since we have flexibility in how to distribute the stamps while maintaining the same total cost, the number of 25-cent stamps could vary. For instance:

• We could have just one 25-cent stamp with more 20-cent stamps
• We could have several 25-cent stamps with fewer of the others
• Both scenarios could total \(\$1.45\)

Conclusion

Statement 1 alone is NOT sufficient because multiple combinations of stamps can total \(\$1.45\), each with different numbers of 25-cent stamps.

[STOP - 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: She bought exactly 6 stamps.

What Statement 2 Provides

We know the total number of stamps is 6, with at least one of each type.

Logical Analysis

With 6 stamps total and the requirement of at least one of each type (\(20¢\), \(25¢\), \(30¢\)), we've used 3 stamps. This leaves 3 additional stamps that could be of any denomination.

The number of 25-cent stamps could be:

• Minimum: 1 (just the required one)
• Maximum: 4 (the required one plus all 3 additional stamps)
• In between: 2 or 3 (with the additional stamps split among types)

Since the 25-cent stamps can range from 1 to 4, we clearly don't have a unique value.

Conclusion

Statement 2 alone is NOT sufficient because it allows multiple possible values for the number of 25-cent stamps.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Statements

Let's combine both pieces of information:

• Total cost: \(\$1.45\) (\(145\) cents)
• Total stamps: 6
• At least one of each type

Combined Information Analysis

Now we have two constraints for our three unknowns. In general, when we have fewer independent constraints than unknowns, we expect multiple solutions.

Think about it this way: The average price per stamp is approximately \(24¢\) (\(145 ÷ 6 ≈ 24.17\)). This average falls between our stamp prices (\(20¢ < 24.17¢ < 30¢\)), which confirms that we need a mix. But different mixes could achieve both this average price and total of 6 stamps.

Why Together They Aren't Sufficient

Even with both constraints, we still have flexibility in how to distribute the stamps. We could have:

• More \(20¢\) stamps balanced by some \(30¢\) stamps
• More \(30¢\) stamps with fewer total stamps of each type
• Various combinations in between

The key insight is that with only two equations but three unknowns, the system remains underdetermined. Multiple combinations of stamps can satisfy both the cost and quantity requirements, meaning the number of 25-cent stamps is not uniquely determined.

[STOP - Not Sufficient!] This eliminates choice C.

The Answer: E

The statements together are not sufficient because even when combined, they don't provide enough constraints to determine a unique value for the number of 25-cent stamps.

Answer Choice E: "The statements together are not sufficient."