Kelly invested in two different funds, Fund F and Fund M. For each $100.00 invested in Fund F, Kelly earned...
GMAT Data Sufficiency : (DS) Questions
Kelly invested in two different funds, Fund F and Fund M. For each \(\$100.00\) invested in Fund F, Kelly earned \(\$8.50\) in interest the first year. For each \(\$100.00\) invested in Fund M, Kelly earned \(\$7.60\) in interest the first year. If Kelly earned \(8.2\%\) in interest from the two investments that year, what dollar amount was invested in Fund M?
- The amount invested in Fund F was \(\$20,000\) more than \(50\%\) of the total amount invested.
- The amount invested in Fund F was \(\$80,000\).
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Understanding the Question
We need to find the exact dollar amount Kelly invested in Fund M.
Given Information
- Fund F earns 8.5% interest (or $8.50 per $100)
- Fund M earns 7.6% interest (or $7.60 per $100)
- The combined portfolio earned 8.2% interest overall
What We Need to Determine
This is a value question - we need a specific dollar amount for Fund M.
Key Insight: The Weighted Average Relationship
Since the overall return of 8.2% falls between the two individual rates (7.6% and 8.5%), this is a weighted average problem. Notice that 8.2% is much closer to 7.6% than to 8.5% - this tells us that more money must be invested in Fund M (the lower-earning fund).
Let's find the exact relationship. If F = amount in Fund F and M = amount in Fund M:
- Total interest earned = \(\mathrm{0.085F + 0.076M}\)
- Total amount invested = \(\mathrm{F + M}\)
- Overall return = Total interest ÷ Total investment = 8.2%
Setting this up: \(\mathrm{(0.085F + 0.076M)/(F + M) = 0.082}\)
Solving for the relationship between F and M:
- \(\mathrm{0.085F + 0.076M = 0.082(F + M)}\)
- \(\mathrm{0.085F + 0.076M = 0.082F + 0.082M}\)
- \(\mathrm{0.003F = 0.006M}\)
- \(\mathrm{F = 2M}\)
This crucial relationship tells us that Kelly invested exactly twice as much in Fund F as in Fund M. Think of it this way: for every $3 total invested, $2 went to Fund F and $1 went to Fund M.
For this question to be sufficient, we need information that allows us to find the unique value of M.
Analyzing Statement 1
Statement 1 tells us: "The amount invested in Fund F was $20,000 more than 50 percent of the total amount invested."
Translating Statement 1
This gives us: \(\mathrm{F = 0.5(F + M) + 20,000}\)
Let's simplify:
- \(\mathrm{F = 0.5F + 0.5M + 20,000}\)
- \(\mathrm{0.5F = 0.5M + 20,000}\)
- \(\mathrm{F = M + 40,000}\)
Finding the Value
Now we have two relationships:
- \(\mathrm{F = 2M}\) (from our weighted average analysis)
- \(\mathrm{F = M + 40,000}\) (from Statement 1)
Since both expressions equal F, we can set them equal:
- \(\mathrm{2M = M + 40,000}\)
- \(\mathrm{M = 40,000}\)
Statement 1 gives us a unique value: M = $40,000
[STOP - Sufficient!]
This 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 tells us: "The amount invested in Fund F was $80,000."
Direct Application
We know directly that F = $80,000.
Using Our Key Relationship
From our weighted average analysis, we know \(\mathrm{F = 2M}\).
Therefore:
- \(\mathrm{80,000 = 2M}\)
- \(\mathrm{M = 40,000}\)
Statement 2 gives us a unique value: M = $40,000
[STOP - Sufficient!]
This is sufficient.
This eliminates choices A, C, and E.
The Answer: D
Since both statements independently give us the exact value of M ($40,000), each statement alone is sufficient.
Answer Choice D: "Each statement alone is sufficient."