A logistics company operates a fleet of 14 vans consisting solely of Model X and Model Y v… — GRE General Test

A logistics company operates a fleet of $14$ vans consisting solely of Model $X$ and Model $Y$ vans.

- The ratio of the cargo volume capacity of one Model $X$ van to one Model $Y$ van is $3 : 5$ .
- The ratio of the energy consumption rate (in kWh per mile) of one Model $X$ van to one Model $Y$ van is $4 : 5$ .
- When fully loaded, a Model $X$ van carries $30\text{ m}^3$ of cargo and a Model $Y$ van carries $50\text{ m}^3$ of cargo.
- The entire fleet of $14$ vans, when fully loaded, transports a total of $620\text{ m}^3$ of cargo in one trip.
- A Model $X$ van consumes $2\text{ kWh}$ of energy per mile.

Which of the following statements must be true? Select all such statements.

There are exactly $4$ Model $X$ vans in the fleet.Answer
B
Model $X$ vans account for $\frac{3}{8}$ of the total cargo capacity transported by the fleet.
The total energy consumed by the entire fleet per mile is $33\text{ kWh}$ .Answer
D
The average energy consumption rate per van in the fleet is $2.25\text{ kWh}$ per mile.
E
If the cargo capacity of each Model $Y$ van is increased by $20\%$ , the total cargo capacity of the entire fleet would increase by $20\%$ .

Answer

The correct statements are that there are exactly 4 Model X vans in the fleet and that the total energy consumed by the entire fleet per mile is 33 kWh.

The statement specifying that there are exactly 4 Model X vans is correct because solving the system

n_X + n_Y = 14

and

30 n_X + 50 n_Y = 620

gives

n_X = 4

. The statement asserting that total fleet energy consumption per mile is 33 kWh is also correct because

4(2\text{ kWh}) + 10(2.5\text{ kWh}) = 33\text{ kWh}

Step-by-Step Solution

Determine the number of Model X and Model Y vans using fleet totals.

Let

n_X

be the number of Model X vans and

n_Y

be the number of Model Y vans. Given

n_X + n_Y = 14

, substitute

n_Y = 14 - n_X

into the total cargo equation:

30 n_X + 50(14 - n_X) = 620 \implies 700 - 20 n_X = 620 \implies 20 n_X = 80 \implies n_X = 4

. Thus,

n_Y = 10

This establishes the exact composition of the fleet.

Evaluate the fraction of cargo carried by Model X vans.

Total cargo carried by Model X vans =

4 \times 30 = 120\text{ m}^3

. Fleet total cargo =

620\text{ m}^3

. Fraction =

\frac{120}{620} = \frac{6}{31}

Shows that the fleet cargo share depends on the van counts, not just the single-van ratio of 3 to 5.

Calculate individual consumption rates and total fleet consumption per mile.

The consumption rate ratio of Model X to Model Y is

4 : 5

. Given Model X rate is

2\text{ kWh/mile}

, Model Y rate is

2 \times \frac{5}{4} = 2.5\text{ kWh/mile}

. Total fleet consumption per mile =

4(2) + 10(2.5) = 8 + 25 = 33\text{ kWh}

Applies unit rates to the known number of each van model to find total rate.

Calculate the true weighted average consumption rate per van.

Average consumption rate per van =

\frac{\text{Total energy per mile}}{\text{Total vans}} = \frac{33\text{ kWh}}{14\text{ vans}} \approx 2.36\text{ kWh/mile}

Demonstrates why taking an unweighted mean

(2.0 + 2.5)/2 = 2.25

is incorrect when group sizes are unequal.

Analyze the impact of a 20% increase in Model Y capacity.

Initial Model Y total capacity =

10 \times 50 = 500\text{ m}^3

. A

20\%

increase adds

0.20 \times 500 = 100\text{ m}^3

. Percent increase in total fleet capacity =

\frac{100}{620} \approx 16.13\%

A percentage change in a part does not equal the percentage change in the whole.

Key Concept

Weighted averages and component vs. whole relationships in multi-part ratios and rates.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic