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Difficulty: MediumRatios, Rates, and Percentages

A freight logistics company operates delivery trucks on two distinct transit routes: Route Alpha and Route Beta.

- On Route Alpha, trucks travel at a constant speed of 6060 miles per hour and consume fuel at a rate of 88 gallons per hour.
- On Route Beta, trucks travel at a constant speed of 4545 miles per hour and consume fuel at a rate of 66 gallons per hour.

Yesterday, a fleet of trucks completed a total combined distance of 900900 miles across both routes in a combined total operating time of 1616 hours.

Based on this information, select the option that correctly identifies both the ratio of the distance traveled on Route Alpha to the distance traveled on Route Beta (Column 1) and the percentage of total fuel consumed that was used on Route Alpha (Column 2).

  1. Distance Ratio (Alpha : Beta) = 4 : 1; Fuel Percentage (Alpha) = 80%Answer
  2. B
    Distance Ratio (Alpha : Beta) = 1 : 4; Fuel Percentage (Alpha) = 80%
  3. C
    Distance Ratio (Alpha : Beta) = 4 : 1; Fuel Percentage (Alpha) = 75%
  4. D
    Distance Ratio (Alpha : Beta) = 80%; Fuel Percentage (Alpha) = 4 : 1
  5. E
    Distance Ratio (Alpha : Beta) = 3 : 1; Fuel Percentage (Alpha) = 75%

Answer

Distance Ratio (Alpha : Beta) = 4 : 1; Fuel Percentage (Alpha) = 80%
Solving the system of simultaneous equations gives 12 hours on Route Alpha and 4 hours on Route Beta. Route Alpha accounts for 720 miles while Route Beta accounts for 180 miles, establishing a distance ratio of 720 : 180 = 4 : 1. Fuel consumption is 96 gallons on Route Alpha out of 120 total gallons consumed, which equals 80% of total fuel. The option matching Distance Ratio (Alpha : Beta) = 4 : 1 and Fuel Percentage (Alpha) = 80% is correct.

Step-by-Step Solution

1
Set up a system of linear equations for operating hours on each route
Let tAt_A be the time in hours spent on Route Alpha, and tBt_B be the time in hours spent on Route Beta.
1) Total time equation: tA+tB=16t_A + t_B = 16
2) Total distance equation: 60tA+45tB=90060 t_A + 45 t_B = 900
Distance equals speed multiplied by time for each route.
2
Solve for tAt_A and tBt_B
Substitute tB=16−tAt_B = 16 - t_A into the distance equation:
60tA+45(16−tA)=90060 t_A + 45(16 - t_A) = 900
60tA+720−45tA=90060 t_A + 720 - 45 t_A = 900
15tA=180  ⟹  tA=1215 t_A = 180 \implies t_A = 12 hours.
Then tB=16−12=4t_B = 16 - 12 = 4 hours.
Isolating the single variable tAt_A provides the exact time spent on each route.
3
Calculate distances and determine the distance ratio (Column 1)
Distance on Route Alpha (DAD_A) = 60 mph×12 hours=72060 \text{ mph} \times 12 \text{ hours} = 720 miles.
Distance on Route Beta (DBD_B) = 45 mph×4 hours=18045 \text{ mph} \times 4 \text{ hours} = 180 miles.
Ratio DA:DB=720:180=4:1D_A : D_B = 720 : 180 = 4 : 1.
Computing individual route distances yields the distance ratio required for Column 1.
4
Calculate total fuel consumption and the percentage spent on Route Alpha (Column 2)
Fuel on Route Alpha (FAF_A) = 12 hours×8 gal/hr=9612 \text{ hours} \times 8 \text{ gal/hr} = 96 gallons.
Fuel on Route Beta (FBF_B) = 4 hours×6 gal/hr=244 \text{ hours} \times 6 \text{ gal/hr} = 24 gallons.
Total fuel (FtotalF_{\text{total}}) = 96+24=12096 + 24 = 120 gallons.
Percentage on Route Alpha = 96120×100%=80%\frac{96}{120} \times 100\% = 80\%.
Multiplying time by consumption rate gives fuel volume, from which the relative percentage is calculated.

Key Concept

Simultaneous Rate and System Constraint Modeling
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Topics

Ratios, Rates, and Percentages

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