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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

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Data Sufficiency
DS - Money
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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?

  1. The amount invested in Fund F was \(\$20,000\) more than \(50\%\) of the total amount invested.
  2. The amount invested in Fund F was \(\$80,000\).
A
Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
B
Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are not sufficient.
Solution
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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:

  1. \(\mathrm{F = 2M}\) (from our weighted average analysis)
  2. \(\mathrm{F = M + 40,000}\) (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."

Answer Choices Explained
A
Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
B
Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are not sufficient.
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