In the figure above, if A, B, and C are the areas, respectively, of the three nonoverlapping regions formed by...

Understanding the Question

We have two circles of equal area that overlap, creating three distinct regions: A, B, and C. Let's visualize this:

• Region A: The part of the first circle that doesn't overlap with the second
• Region B: The overlapping area (intersection of both circles)
• Region C: The part of the second circle that doesn't overlap with the first

Since both circles have equal total area, we can write:

• Area of Circle 1 = \(\mathrm{A + B}\)
• Area of Circle 2 = \(\mathrm{B + C}\)

Here's the key insight: Since these areas are equal, we have \(\mathrm{A + B = B + C}\), which simplifies to \(\mathrm{A = C}\).

Our goal is to find the specific value of \(\mathrm{B + C}\). For a statement to be sufficient, it must give us exactly one value for \(\mathrm{B + C}\).

Analyzing Statement 1

Statement 1: \(\mathrm{A + 2B + C = 24}\)

Since we know \(\mathrm{A = C}\), let's substitute:

• \(\mathrm{C + 2B + C = 24}\)
• \(\mathrm{2C + 2B = 24}\)
• \(\mathrm{2(B + C) = 24}\)
• \(\mathrm{B + C = 12}\)

We get a unique value: \(\mathrm{B + C = 12}\).

[STOP - 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: \(\mathrm{A + C = 18}\) and \(\mathrm{B = 3}\)

We know from our initial analysis that \(\mathrm{A = C}\). Using the first part:

• \(\mathrm{A + A = 18}\) (since \(\mathrm{A = C}\))
• \(\mathrm{2A = 18}\)
• \(\mathrm{A = 9}\)

Therefore, \(\mathrm{C = 9}\) as well.

With \(\mathrm{B = 3}\) and \(\mathrm{C = 9}\):

• \(\mathrm{B + C = 3 + 9 = 12}\)

We get a unique value: \(\mathrm{B + C = 12}\).

[STOP - Sufficient!]

This eliminates choice A.

The Answer

Both statements independently give us \(\mathrm{B + C = 12}\).

Answer: D - Each statement alone is sufficient.