A drainage system is clearing a flooded trench. During the first hour of operation, the sy… — Armed Services Vocational Aptitude Battery (ASVAB)

A drainage system is clearing a flooded trench. During the first hour of operation, the system removes $30\%$ of the initial volume of water. During the second hour, it removes $40\%$ of the water that remained after the first hour. If there are exactly $2,100$ gallons of water still left in the trench after the second hour, calculate the initial volume of water and the exact amount of water removed during the second hour.

Answer:The initial volume of water in the trench was 【5000】 gallons, and the system removed exactly 【1400】 gallons during the second hour.

Answer

The initial volume was 5,000 gallons, and 1,400 gallons were removed during the second hour.

The problem requires tracking successive percentage changes relative to an unknown original base. Because

30\%

was removed initially,

70\%

of the original volume remained. During the second hour,

40\%

of that

70\%

was removed, which equates to removing an additional

28\%

of the original volume. Subtracting

28\%

from the

70\%

leaves

42\%

of the original volume. We are told this

42\%

equals

2,100

gallons. By dividing

2,100

0.42

, we find the initial volume was

5,000

gallons. The water removed during the second hour is

28\%

5,000

, which equals

1,400

gallons.

Step-by-Step Solution

Calculate the percentage of the initial volume remaining after the first hour.

100\% - 30\% = 70\%

of the initial volume remains.

The second hour's removal rate is applied to the remaining water, so establishing this new base is required.

Determine the percentage of the initial volume that is removed during the second hour.

40\%

70\%

0.40 \times 0.70 = 0.28

, or

28\%

of the initial volume.

This converts the second hour's removal rate into a percentage relative to the overall initial volume.

Calculate the final percentage of the initial volume remaining after the second hour.

70\% - 28\% = 42\%

of the initial volume remains.

Subtracting the second hour's removed percentage from the first hour's remaining percentage yields the final remaining proportion.

Set up an equation using the final remaining volume to find the initial volume (

V

0.42V = 2,100

, which solves to

V = \frac{2,100}{0.42} = 5,000

gallons.

This algebraic equation equates the final percentage representation to the actual known volume to isolate the initial total.

Calculate the exact volume removed during the second hour.

0.28 \times 5,000 = 1,400

gallons.

Applying the percentage removed during the second hour to the solved initial volume provides the specific amount removed.

Key Concept

Successive Percentages and Reverse Base Calculations

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