Rounded to the nearest 10,000, the populations of San Pepe and of its suburb Maple Beach are 140,000 and 80,000,...

Understanding the Question

We need to find the combined population of San Pepe and Maple Beach, rounded to the nearest 10,000.

Given Information

• San Pepe population (rounded to nearest 10,000): 140,000
• Maple Beach population (rounded to nearest 10,000): 80,000
• We want the COMBINED population, rounded to nearest 10,000

What We Need to Determine

The key insight is understanding how rounding to the nearest 10,000 works:

• When a number rounds to 140,000, the actual value is between 135,000 and 144,999
• When a number rounds to 80,000, the actual value is between 75,000 and 84,999

This means the combined population could range from:

• Minimum: \(135,000 + 75,000 = 210,000\)
• Maximum: \(144,999 + 84,999 = 229,998\)

Critical Rounding Thresholds

When we round the combined population to the nearest 10,000:

• If the sum is less than 215,000 → rounds to 210,000
• If the sum is between 215,000 and 224,999 → rounds to 220,000
• If the sum is 225,000 or more → rounds to 230,000

So we need information that tells us which of these intervals contains the actual combined population.

Analyzing Statement 1

Statement 1: The exact population figures of San Pepe and Maple Beach are each within 2% of the rounded figures above.

Let's calculate what this means:

• San Pepe's actual population is within 2% of 140,000
  • 2% of 140,000 = \(0.02 \times 140,000 = 2,800\)
  • So the actual population is between \(140,000 - 2,800 = 137,200\) and \(140,000 + 2,800 = 142,800\)

• Maple Beach's actual population is within 2% of 80,000
  • 2% of 80,000 = \(0.02 \times 80,000 = 1,600\)
  • So the actual population is between \(80,000 - 1,600 = 78,400\) and \(80,000 + 1,600 = 81,600\)

The combined population must be between:

• Minimum: \(137,200 + 78,400 = 215,600\)
• Maximum: \(142,800 + 81,600 = 224,400\)

Key insight: This entire range (215,600 to 224,400) falls within the interval \([215,000, 225,000)\), which means the combined population would always round to 220,000.

[STOP - Statement 1 is 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: The approximate population figure for San Pepe was rounded up, whereas the figure for Maple Beach was rounded down.

What does this tell us?

• San Pepe was rounded UP to 140,000 → its actual population must be less than 140,000
  • Since it rounds to 140,000, the actual value is in \([135,000, 140,000)\)

• Maple Beach was rounded DOWN to 80,000 → its actual population must be greater than 80,000
  • Since it rounds to 80,000, the actual value is in \((80,000, 85,000)\)

Note: The parenthesis ")" means "not including" and the bracket "[" means "including".

Let's find the range of the combined population:

• Minimum: Just over 135,000 + just over 80,000 = just over 215,000
• Maximum: Just under 140,000 + just under 85,000 = just under 225,000

Key insight: The entire possible range is again between 215,000 and 225,000. This means the combined population would always round to 220,000.

[STOP - Statement 2 is Sufficient!]

The Answer: D

Both statements independently guarantee that the combined population rounds to 220,000.

Answer Choice D: "Each statement alone is sufficient."