A certain economics report defines a middle-class family as a family whose income is at least half, but no more...

Understanding the Question

We need to determine if a family earning $73,000 is considered middle-class.

Given Information

• Middle-class definition: Income must be between \(0.5 \times\) and \(2 \times\) the median family income
• Family in question earns: $73,000
• We need: A definitive YES or NO answer

What We Need to Determine

To answer this question, we need to know if $73,000 falls within the middle-class range. Since the range spans from \(0.5\mathrm{M}\) to \(2\mathrm{M}\) (where M is the median), this creates a \(4 \times\) span from minimum to maximum.

Key Insight

Rather than calculating exact boundaries, we can think proportionally. If we know either the median or one boundary, we can determine where $73,000 sits relative to the middle-class range.

Analyzing Statement 1

Statement 1: The median family income is $37,152

What Statement 1 Tells Us

With the median at $37,152, we can think about where $73,000 sits proportionally. Notice that $73,000 is approximately twice $37,152 (just under \(2 \times\)).

Logical Analysis

Since middle-class extends from \(0.5 \times\) to \(2 \times\) the median:

• The upper boundary is \(2 \times\) $37,152 (which would be about $74,000)
• The lower boundary is \(0.5 \times\) $37,152 (which would be about $18,500)
• $73,000 is clearly more than half the median and just under twice the median

Therefore, $73,000 falls within the middle-class range.

Conclusion

Statement 1 is SUFFICIENT to answer with a definitive YES.

[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: The minimum income of a middle-class family is $18,576

What Statement 2 Provides

We know the lower boundary of middle-class is $18,576. Since this represents \(0.5\mathrm{M}\), the upper boundary must be \(4 \times\) this amount (because \(2\mathrm{M} \div 0.5\mathrm{M} = 4\)).

Logical Analysis

The maximum middle-class income = \(4 \times\) $18,576 ≈ $74,000+

Now we can see that:

• $73,000 is well above the minimum of $18,576
• $73,000 is just below \(4 \times\) the minimum (approximately $74,000)

Therefore, $73,000 falls within the middle-class range.

Conclusion

Statement 2 is SUFFICIENT to answer with a definitive YES.

[STOP - Sufficient!]

The Answer: D

Both statements independently allow us to determine that $73,000 falls within the middle-class income range.

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