Question

A water jug with a capacity of 20 gallons is 20 percent full. At the end of every third day, water is added to the jug. If the amount of water added is equal to 50 percent of the water in the jug at the beginning of that day, how many days does it take for the jug to be at least $85\mathrm{\%}$ full? A. $$4$$ B. $$6$$ C. $$12$$ D. $$15$$ E. $$20$$

Short answer

The correct answer is option D, which is $15$ days.

Step-by-step solution

Calculate Initial Water Level

The water jug is initially $20\mathrm{\%}$ full of its $20$ gallon capacity. This means there are initially $20\mathrm{\%}$ of $20$ gallons = $20/100\times 20=4$ gallons of water in the jug.

Calculate Water Added Every Third Day

Every third day, water is added to the jug. The amount of water added is $50\mathrm{\%}$ of the water in the jug at the beginning of that day. So the amount of water in the jug after the third day = $150\mathrm{\%}=1.5$ times the amount of water at the beginning of the day.

Calculate Water Level After Each Third Day

Calculate the amount of water in the jug after every third day, by multiplying the existing amount of water by $1.5$, until it is $85\mathrm{\%}$ full of its $20$ gallon capacity.

Continuation of Step 3

Apply the calculation until the jug is at least $85\mathrm{\%}$ full, that is minimum $85\mathrm{\%}$ of $20$ gallons = $85/100\times 20=17$ gallons.

Count the Days Until Jug Is 85% Full

Count the number of third days it takes until the amount of water in the jug is at least $17$ gallons.

Key concepts

Percentage Calculations

Understanding percentage calculations is crucial when solving problems involving changes in quantity. A percentage represents a part of a whole, expressed as a fraction of 100. In the given exercise, the water jug is initially 20% full, which means it contains 20% of its total 20-gallon capacity.
To convert this percentage into actual gallons, you multiply the percentage by the total capacity and divide by 100. Mathematically, it can be expressed as:$$\text{Amount of water}=\frac{20}{100}\times 20=4\text{ gallons}$$
You will also see additional percentage calculations when water is added to the jug every three days based on the initial amount it already contains. This systematic use of percentage helps in progressively increasing the water volume towards the set goal of 85% full.

Sequence of Operations

The sequence of operations is the order in which calculations are performed. It is important to follow this order to ensure accuracy. In this problem:

• First, we calculate the initial amount of water in the jug.
• Next, every third day, we determine how much water is added by calculating 50% of the current amount in the jug.
• Finally, this amount is added to the existing water to evaluate if it reaches at least 85% capacity.

Every step follows a logical order, carefully applying the calculations to reach the final volume. If you skip steps or rearrange them, the solution may be incorrect.

Capacity and Volume Calculations

Calculating capacity and volume involves understanding the total space available and the current volume occupied. A water jug with a 20-gallon capacity needs to be 85% full to meet the problem's condition.
Initially, we have 4 gallons from a 20-gallon jug, which leaves room for more water to be added. With every third day, more water is added based on the current quantity in the jug. The target is to reach 17 gallons ($85\mathrm{\%}$ of 20 gallons).
Performing these calculations step by step ensures the jug is precisely filled according to the given percentages.

Step-by-Step Solutions

Step-by-step solutions break down complex problems into manageable parts. This process outlines each calculation clearly, so no detail is overlooked when reaching the correct answer.

• Step 1: Determine the initial water level.
• Step 2: Calculate how much water is added at each interval.
• Step 3: Iterate this process until the jug reaches the desired fullness.
• Step 4: Keep track of the total days elapsed in increments of three.

By following each step in order, you maintain control over variables and ensure the accuracy of results.

Algebraic Reasoning

Algebraic reasoning allows us to solve problems by forming equations and finding unknown values. In this problem, the operations described can be reasoned through algebra to systematically show how the jug's water volume approaches 85% over time.
You can set it up as:Let $x$ be the number of intervals (days/3).Each cycle adds $(1.5{)}^x$ times the initial water quantity of 4 gallons until the total volume reaches or exceeds 17 gallons.
This helps visualize the progression as:$$4\times (1.5{)}^x\ge 17$$
Solving this inequality provides the answer, demonstrating how algebra bridges concepts to find the solution effectively.