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If each of the stamps Carla bought cost \(20, 25, \text{ or } 30\) cents and she bought at least one of each denomination, what is the number of \(25\)-cent stamps that she bought?
The question asks us to find the exact number of 25-cent stamps that Carla bought.
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.
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.
Statement 1: She spent a total of \(\$1.45\) for stamps.
We know the total cost is \(\$1.45\) (or \(145\) cents). This gives us one constraint connecting the three unknowns.
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:
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:
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.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: She bought exactly 6 stamps.
We know the total number of stamps is 6, with at least one of each type.
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:
Since the 25-cent stamps can range from 1 to 4, we clearly don't have a unique value.
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.
Let's combine both pieces of information:
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.
Even with both constraints, we still have flexibility in how to distribute the stamps. We could have:
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 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."