A sum of $200,000 from a certain estate was divided among a spouse and three children. How much did the...

Understanding the Question

A sum of \(\$200,000\) from a certain estate was divided among a spouse and three children. We need to determine exactly how much the youngest child received.

Given Information

• Total estate: \(\$200,000\)
• Recipients: 1 spouse + 3 children (4 people total)
• All money was distributed among these 4 people

What We Need to Determine

For sufficiency, we need to find one unique value for the youngest child's share. If multiple distributions are possible, we don't have sufficiency.

Key Insight

We have 4 unknowns (spouse's share, oldest child's share, middle child's share, youngest child's share) that must sum to \(\$200,000\). To find one specific share uniquely, we need enough relationships to eliminate all ambiguity about how the money was divided.

Analyzing Statement 1

Statement 1: The spouse received 1/2 of the sum from the estate, and the oldest child received 1/4 of the remainder.

What Statement 1 Tells Us

• Spouse's share: \(\frac{1}{2} \times \$200,000 = \$100,000\)
• After spouse takes their share: \(\$200,000 - \$100,000 = \$100,000\) remains
• Oldest child's share: \(\frac{1}{4} \times \$100,000 = \$25,000\)
• Amount left for the two younger children: \(\$200,000 - \$100,000 - \$25,000 = \$75,000\)

Testing Different Scenarios

Since we have \(\$75,000\) to split between two younger children, let's see if the youngest child could receive different amounts:

• Scenario 1: Middle child gets \(\$50,000\), youngest gets \(\$25,000\)
• Scenario 2: Middle child gets \(\$25,000\), youngest gets \(\$50,000\)
• Scenario 3: Each younger child gets \(\$37,500\)

All three scenarios are mathematically valid since they all total \(\$75,000\). Statement 1 doesn't tell us how to split this \(\$75,000\) between the two younger children.

Conclusion

Since we can identify multiple possible values for the youngest child's share (\(\$25,000\), \(\$50,000\), or \(\$37,500\)), 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 independently.

Statement 2: Each of the two younger children received \(\$12,500\) more than the oldest child and \(\$62,500\) less than the spouse.

What Statement 2 Provides

This statement gives us two crucial pieces of information:

1. Both younger children received the same amount (the word "each" tells us this)
2. This amount equals:
   • Oldest child + \(\$12,500\)
   • Spouse - \(\$62,500\)

The Key Relationship

Since both expressions equal the same amount (what each younger child received), we can write:

Oldest child + \(\$12,500\) = Spouse - \(\$62,500\)

Rearranging: Spouse = Oldest child + \(\$75,000\)

Building the Complete Picture

Now we can express everyone's share in terms of the oldest child's share (let's call it "O"):

• Oldest child: \(\mathrm{O}\)
• Spouse: \(\mathrm{O} + \$75,000\)
• Middle child: \(\mathrm{O} + \$12,500\)
• Youngest child: \(\mathrm{O} + \$12,500\)

Since all shares must sum to \(\$200,000\):

\(\mathrm{O} + (\mathrm{O} + \$75,000) + (\mathrm{O} + \$12,500) + (\mathrm{O} + \$12,500) = \$200,000\)

Simplifying: \(4\mathrm{O} + \$100,000 = \$200,000\)

Therefore: \(4\mathrm{O} = \$100,000\), which means \(\mathrm{O} = \$25,000\)

The Unique Solution

This gives us exactly one possible distribution:

• Oldest child: \(\$25,000\)
• Each younger child: \(\$25,000 + \$12,500 = \$37,500\)
• Spouse: \(\$25,000 + \$75,000 = \$100,000\)
• Total check: \(\$25,000 + \$37,500 + \$37,500 + \$100,000 = \$200,000\) ✓

Conclusion

Statement 2 provides enough constraints to determine exactly one distribution. The youngest child received \(\$37,500\).

Statement 2 is sufficient.

[STOP - Sufficient!] This confirms our answer is B.

The Answer: B

Since Statement 1 alone is not sufficient but Statement 2 alone is sufficient, the answer is B.

Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."