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?

The amount invested in Fund F was $\$20,000$ more than $50\%$ of the total amount invested.
The amount invested in Fund F was $\$80,000$.

Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

Solution

markdown
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."

Answer Choices Explained

Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

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markdown 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."