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Difficulty: HardFractions and Decimals

A local water reservoir was initially filled to exactly 0.3‾0.\overline{3} of its maximum capacity. After several days of heavy rainfall, an additional 14.414.4 million liters of water flowed into the reservoir, bringing the water level to 0.60.6 of its maximum capacity.

Currently, a nearby town consumes water from the reservoir at a steady rate of 1121\frac{1}{2} million liters per day. Additionally, an undiscovered leak at the base of the reservoir drains water at a constant rate of 0.050.05 million liters per hour.

If the reservoir were completely full and no further water was added, in how many days would it become completely empty?

Answer: 20 days

Answer

20
The total capacity of the reservoir is found by equating the fractional increase (35−13=415\frac{3}{5} - \frac{1}{3} = \frac{4}{15}) to the volume added (14.414.4 million liters), yielding 5454 million liters. The total daily water depletion is the sum of the town's daily usage (1.51.5 million liters) and the leak converted to a daily rate (0.05×24=1.20.05 \times 24 = 1.2 million liters), giving 2.72.7 million liters per day. Dividing 5454 by 2.72.7 gives exactly 2020 days.

Step-by-Step Solution

1
Convert the decimal fill levels into fractions and compute their difference.
Initial level = 13\frac{1}{3}. Final level = 35\frac{3}{5}. Difference = 35−13=415\frac{3}{5} - \frac{1}{3} = \frac{4}{15}.
Working with exact fractions avoids rounding errors from recurring decimals and simplifies finding the proportion of water added.
2
Calculate the maximum capacity of the reservoir.
415×Capacity=14.4  ⟹  Capacity=14.4×154=54\frac{4}{15} \times \text{Capacity} = 14.4 \implies \text{Capacity} = 14.4 \times \frac{15}{4} = 54 million liters.
The difference in the fractional water level represents exactly the volume of rain added.
3
Calculate the total volume of water lost per day.
Town = 1.51.5 million L/day. Leak = 0.05×24=1.20.05 \times 24 = 1.2 million L/day. Total = 2.72.7 million L/day.
The rates must be in the same time unit (days) before they can be accurately combined.
4
Determine the time required to empty the completely full reservoir.
54÷2.7=2054 \div 2.7 = 20 days.
Dividing the total capacity by the combined daily rate of depletion gives the time in days.

Key Concept

Fractions and Decimals
Estimated Time:3m 0s
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Topics

Fractions and Decimals

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