A certain military vehicle can run on pure Fuel X, pure Fuel Y, or any mixture of X and Y....

GMAT Data Sufficiency : (DS) Questions

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Data Sufficiency

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A certain military vehicle can run on pure Fuel X, pure Fuel Y, or any mixture of X and Y. Fuel X costs $\$3$ per gallon; the vehicle can go $20$ miles on a gallon of Fuel X. In contrast, Fuel Y costs $\$5$ per gallon, but the vehicle can go $40$ miles on a gallon of Fuel Y. What is the cost per gallon of the fuel mixture currently in the vehicle's tank?

Using fuel currently in its tank, the vehicle burned $8$ gallons to cover $200$ miles.
The vehicle can cover $7\frac{1}{7}$ miles for every dollar of fuel currently in its tank

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

Understanding the Question

We need to determine the cost per gallon of the fuel mixture currently in the vehicle's tank.

Given Information

Fuel X: $\$3/\mathrm{gallon}$, gives $20 \, \mathrm{miles/gallon}$
Fuel Y: $\$5/\mathrm{gallon}$, gives $40 \, \mathrm{miles/gallon}$
The vehicle can use pure X, pure Y, or any mixture

What We Need to Find

The exact cost per gallon of whatever mixture is currently in the tank. To answer this, we need to know the exact proportion of Fuel X and Fuel Y in the mixture.

Key Insight

Here's the crucial understanding: When you mix two fuels with different properties, any performance metric (like miles per gallon or miles per dollar) will fall somewhere between the pure fuel values. More importantly, that metric's exact value tells us the precise mixture ratio - it's like mixing hot and cold water where the final temperature reveals exactly how much of each you used.

For a statement to be sufficient, it must give us information that determines the exact mixture ratio, which then gives us the exact cost per gallon.

Analyzing Statement 1

Statement 1: Using fuel currently in its tank, the vehicle burned 8 gallons to cover 200 miles.

What This Tells Us

The current mixture's fuel efficiency: $200 \, \mathrm{miles} \div 8 \, \mathrm{gallons} = \mathbf{25 \, \mathrm{miles \, per \, gallon}}$

Logical Analysis

Let's see what this efficiency reveals:

Pure Fuel X: $20 \, \mathrm{miles/gallon}$
Our mixture: $25 \, \mathrm{miles/gallon}$
Pure Fuel Y: $40 \, \mathrm{miles/gallon}$

Notice that 25 is exactly $\frac{1}{4}$ of the way from 20 to 40.

Distance from X to mixture: $25 - 20 = 5$
Total distance from X to Y: $40 - 20 = 20$
Fraction: $\frac{5}{20} = \frac{1}{4}$

This means the mixture contains $\frac{3}{4}$ Fuel X and $\frac{1}{4}$ Fuel Y.

With these exact proportions:

$\frac{3}{4}$ of the mixture costs $\$3/\mathrm{gallon}$
$\frac{1}{4}$ of the mixture costs $\$5/\mathrm{gallon}$
Cost per gallon = $\left(\frac{3}{4}\right) \times \$3 + \left(\frac{1}{4}\right) \times \$5 = \$2.25 + \$1.25 = \$3.50$

Conclusion

Statement 1 is sufficient because the fuel efficiency of 25 mpg uniquely identifies the mixture as $\frac{3}{4}$ Fuel X and $\frac{1}{4}$ Fuel Y, allowing us to calculate the exact cost per gallon.

[STOP - Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now we analyze Statement 2 independently, forgetting Statement 1 completely.

Statement 2: The vehicle can cover 7 and $\frac{1}{7}$ miles for every dollar of fuel currently in its tank.

What This Tells Us

The mixture's "bang for buck": $7\frac{1}{7}$ miles per dollar (which equals $\frac{50}{7} \approx 7.14$ miles per dollar)

Logical Analysis

Let's examine the extremes:

Pure Fuel X: $20 \, \mathrm{miles/gallon} \div \$3/\mathrm{gallon} = \frac{20}{3} \approx 6.67 \, \mathrm{miles/dollar}$
Pure Fuel Y: $40 \, \mathrm{miles/gallon} \div \$5/\mathrm{gallon} = 8 \, \mathrm{miles/dollar}$
Our mixture: $7.14 \, \mathrm{miles/dollar}$

The mixture's value (7.14) falls between the two pure fuels' values (6.67 and 8), which makes sense. But here's the key insight: each possible mixture ratio would produce a different miles-per-dollar value. Since we know this value is exactly $7\frac{1}{7}$, there's only one possible mixture ratio that produces this result.

Think of it this way: if we had mostly Fuel X, we'd be closer to 6.67 miles/dollar. If we had mostly Fuel Y, we'd be closer to 8 miles/dollar. The fact that we get exactly $7\frac{1}{7}$ miles/dollar pins down the precise mixture.

Once we know the unique mixture ratio, we can determine the cost per gallon.

Conclusion

Statement 2 is sufficient because the miles-per-dollar value of $7\frac{1}{7}$ uniquely identifies one specific mixture composition, allowing us to determine the exact cost per gallon.

[STOP - Sufficient!]

This eliminates choices A, C, and E.

The Answer: D

Both statements independently provide a unique performance metric that allows us to determine the exact fuel mixture ratio, and therefore the exact cost per gallon.

The answer is 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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