Of the N candies in a bag, some are peppermint and the rest are spearmint. What is the value of...

Understanding the Question

We have N candies total: some peppermint and the rest spearmint. We need to find the exact value of N.

What We Need to Determine

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.

Given Information

• Total candies = N
• Some are peppermint, the rest are spearmint
• No other constraints given initially

Key Insight

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.

Analyzing Statement 1

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?

Testing Different Scenarios

Scenario 1: Suppose we originally had 6 peppermint candies

• After removing 1 peppermint: 5 remain
• These 5 must equal \(\frac{1}{5}\) of the new total
• So new total = 25 candies
• Therefore \(\mathrm{N} = 26\) (before removal)

Scenario 2: Suppose we originally had 11 peppermint candies

• After removing 1 peppermint: 10 remain
• These 10 must equal \(\frac{1}{5}\) of the new total
• So new total = 50 candies
• Therefore \(\mathrm{N} = 51\) (before removal)

Different starting numbers of peppermints lead to different values of N. We cannot determine a unique value.

Conclusion

Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

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.

Testing Different Scenarios

Scenario 1: Suppose we have 6 peppermint candies

• After removing 2 spearmints: still 6 peppermints
• These 6 must equal \(\frac{1}{4}\) of the new total
• So new total = 24 candies
• Therefore \(\mathrm{N} = 26\) (before removal)

Scenario 2: Suppose we have 8 peppermint candies

• After removing 2 spearmints: still 8 peppermints
• These 8 must equal \(\frac{1}{4}\) of the new total
• So new total = 32 candies
• Therefore \(\mathrm{N} = 34\) (before removal)

Again, different numbers of peppermints lead to different values of N.

Conclusion

Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B (and confirms D is already eliminated).

Combining Statements

Since neither statement alone is sufficient, let's examine what happens when we use both together.

Combined Information

From our analysis:

• Statement 1 creates one relationship between peppermints and total
• Statement 2 creates a different relationship between the same quantities

The crucial insight: Both conditions must be satisfied by the SAME bag of candies.

Why Together They Are Sufficient

Think about it conceptually:

• Statement 1 involves removing a peppermint (changing the peppermint count in our fraction)
• Statement 2 involves removing spearmints (keeping the peppermint count constant)

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.

Mathematical Intuition

Without calculating the exact value, we can understand why two statements work together:

• Each statement creates a line of possible (peppermint, total) combinations
• These two lines intersect at exactly one point
• That unique intersection gives us our answer

[STOP - Sufficient!] This eliminates choice E.

The Answer: C

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."