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A certain economics report defines a middle-class family as a family whose income is at least half, but no more than twice, the median family income. According to this report, is a family whose income is \(\$73,000\) considered a middle-class family?
We need to determine if a family earning $73,000 is considered middle-class.
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.
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.
Statement 1: The median family income is $37,152
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 \(2 \times\)).
Since middle-class extends from \(0.5 \times\) to \(2 \times\) the median:
Therefore, $73,000 falls within the middle-class range.
Statement 1 is SUFFICIENT to answer with a definitive YES.
[STOP - Sufficient!]
This eliminates choices B, C, and E.
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
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 \(2\mathrm{M} \div 0.5\mathrm{M} = 4\)).
The maximum middle-class income = \(4 \times\) $18,576 ≈ $74,000+
Now we can see that:
Therefore, $73,000 falls within the middle-class range.
Statement 2 is SUFFICIENT to answer with a definitive YES.
[STOP - Sufficient!]
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."