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Of the N candies in a bag, some are peppermint and the rest are spearmint. What is the value of N?
We have N candies total: some peppermint and the rest spearmint. We need to find the exact value of N.
For this value question to be sufficient, we must pinpoint exactly one value for N—not a range, not a relationship, but a specific number.
Each statement gives us a different "snapshot" of what happens to the candy proportions when we remove specific candies. The question is whether these snapshots provide enough information to determine N uniquely.
Statement 1: If we remove 1 peppermint candy, then \(\frac{1}{5}\) of the remaining candies would be peppermint.
Let's think about what this means. After removing one peppermint, the remaining peppermints make up \(\frac{1}{5}\) of the new total. This creates a constraint between the number of peppermints and the total, but does it give us a unique answer?
Scenario 1: Suppose we originally had 6 peppermint candies
Scenario 2: Suppose we originally had 11 peppermint candies
Different starting numbers of peppermints lead to different values of N. We cannot determine a unique value.
Statement 1 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 alone.
Statement 2: If we remove 2 spearmint candies, then \(\frac{1}{4}\) of the remaining candies would be peppermint.
Notice the key difference: we're removing spearmint candies, so the number of peppermints stays constant while the total decreases. Let's test if this pins down a unique value.
Scenario 1: Suppose we have 6 peppermint candies
Scenario 2: Suppose we have 8 peppermint candies
Again, different numbers of peppermints lead to different values of N.
Statement 2 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice B (and confirms D is already eliminated).
Since neither statement alone is sufficient, let's examine what happens when we use both together.
From our analysis:
The crucial insight: Both conditions must be satisfied by the SAME bag of candies.
Think about it conceptually:
These create two different mathematical constraints that must be satisfied simultaneously. It's like having two different equations with two unknowns—they typically intersect at exactly one point.
When we tested Statement 1 alone, we found multiple possibilities (\(\mathrm{N}\) could be 26, 51, etc.)
When we tested Statement 2 alone, we found multiple possibilities (\(\mathrm{N}\) could be 26, 34, etc.)
But notice that \(\mathrm{N} = 26\) appears in both lists. When both conditions must be true, they "lock in" this unique value. The two different "views" of the situation converge on a single answer.
Without calculating the exact value, we can understand why two statements work together:
[STOP - Sufficient!] This eliminates choice E.
Both statements together provide sufficient information to determine N, but neither statement alone is sufficient.
Answer: C - "Both statements together are sufficient, but neither statement alone is sufficient."