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Difficulty: HardPercentage Calculations

A fuel tank at a military base is initially filled to 80%80\% of its total capacity. After a training exercise, the fuel level decreases by 35%35\% of its current volume. If 260260 gallons of fuel remain in the tank after the exercise, what is the total capacity of the fuel tank in gallons?

Answer: 500 gallons

Answer

500
To find the total capacity, we must track the percentage of fuel at each stage relative to the total capacity. Initially, the tank holds 80%80\% of its capacity, or 0.800.80. A decrease of 35%35\% means 65%65\% of that fuel remains. Multiplying the initial proportion (0.800.80) by the remaining proportion (0.650.65) yields 0.520.52. This indicates that the final 260260 gallons represent exactly 52%52\% of the tank's total capacity. Setting up the equation 0.52C=2600.52C = 260 and dividing 260260 by 0.520.52 gives a total capacity of 500500 gallons.

Step-by-Step Solution

1
Define a variable for the total capacity and express the initial fuel volume.
Initial volume = 0.80C0.80C
The tank is starting at 80%80\% of its total capacity (CC).
2
Determine the percentage of fuel remaining after the exercise.
Remaining percentage = 100%−35%=65%100\% - 35\% = 65\%
A 35%35\% decrease means that 65%65\% of the fuel that was in the tank before the exercise is still there.
3
Express the final remaining volume in terms of the total capacity (CC).
0.65×0.80C=0.52C0.65 \times 0.80C = 0.52C
Applying the 65%65\% remaining proportion to the initial 0.80C0.80C volume gives the final percentage of the total capacity.
4
Set up an equation using the known final volume and solve for CC.
0.52C=260⇒C=2600.52=5000.52C = 260 \Rightarrow C = \frac{260}{0.52} = 500
Equating the algebraic expression to the 260260 gallons given in the problem isolates the unknown base capacity.

Key Concept

Calculating successive percentages and identifying an unknown base value.
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Topics

Percentage Calculations

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