Question

Luciano measured the amount of water that evaporated over a period of time from a container holding \(w\) ounces of water, where \(w\) is greater than 12. By the end of the first day, the cup had lost 2 ounces of water. By the end of the 7th day, the cup had lost an additional 8 ounces of water. By the end of the 11th day, the cup had lost half of the water that remained after the 7th day. Which of the following represents the remaining amount of water, in ounces, in Luciano's container at the end of the 11th day? A) \(\frac{w-2}{8}\) B) \(\frac{w-2}{2}-10\) C) \(\frac{1}{2} w-10\) D) \(\frac{w-10}{2}\)

Short answer

\(\frac{1}{2}w - 5\)

Step-by-step solution

Determine the amount of water after 1st day

To find the amount of water after the 1st day, subtract 2 ounces from w (the initial amount of water): \(w-2\)

Determine the amount of water after 7th day

By the end of the 7th day, the container loses an additional 8 ounces of water. So, subtract 8 ounces from the amount of water after the 1st day: \(w-2-8\) Simplify it: \(w-10\)

Determine the amount of water evaporated after 11th day

By the end of the 11th day, the container loses half of the water remaining after the 7th day. So, we need to find half of the amount after the 7th day: \(\frac{1}{2}(w-10)\)

Determine the remaining amount of water after 11th day

Now, we need to subtract the amount of water evaporated after the 11th day from the amount of water after the 7th day: \((w-10) - \frac{1}{2}(w-10)\)

Simplify the expression

To simplify the expression, we can distribute the negative sign on the right side and then combine like terms: \(w-10 - \frac{1}{2}w + \frac{1}{2}10\) Now, combine like terms: \(\frac{1}{2}w - 10+5\) Simplify: \(\frac{1}{2}w - 5\) So, the remaining amount of water in Luciano's container at the end of the 11th day is: \(\boxed{\frac{1}{2}w - 5}\) However, none of the given options match with the above result. Check the exercise to make sure it is well posed and has a unique solution.

Key concepts

Algebraic Expressions

Algebraic expressions form the foundation of many mathematical problems and solutions. They consist of numbers, variables like \( w \), and arithmetic operations such as addition and subtraction. In the given exercise, we using the variable \( w \) to represent the initial amount of water in ounces. Throughout the solution, this expression is manipulated to represent various stages of water evaporation. Learning to manage algebraic expressions lets you translate real-world situations into mathematical formulas. This skill is important because it allows you to see connections and relationships between variables. When faced with a problem involving algebraic expressions, remember to carefully follow the order of operations and ensure each step logically leads to the next.

Step-by-Step Solutions

When solving complex math problems, breaking them into smaller steps can make the process much easier. Each step involves a certain amount of calculation, ensuring that every stage is clearly understood. In our exercise, the solution was divided into several steps:

• Step 1 involves finding the water quantity after the first day.
• Step 2 calculates the remaining water after seven days.
• The focus of Step 3 is finding how much evaporates by the 11th day.
• Finally, Step 4 determines the water back in day eleven.

By following these steps, you can systematically approach the problem without feeling overwhelmed. Step-by-step solutions help you track your work and identify errors, ensuring you find the right answer efficiently.

Mathematical Reasoning

Mathematical reasoning is the ability to think logically about numbers and operations. It allows you to make connections between different parts of a problem. For the evaporation problem, mathematical reasoning helps us see that losing water over time is a sequential process: first 2 ounces, then 8, and then half of the remaining amount. Each loss impacts the total amount of water at the end. To build strong mathematical reasoning skills:

• Look for patterns and relationships between numbers.
• Think about how each operation affects the overall outcome.
• Check your results by considering if they make sense in the context of the problem.

An excellent way to enhance your reasoning is by practicing similar problems and finding connections in how they are structured and solved.

Problem-Solving Strategies

Problem-solving in math involves looking at various strategies and applying the most effective one to reach a solution. Here, some strategies can be particularly effective:

• Understanding the problem: Skim through the problem to grasp the main idea.
• Breaking it down: Divide the problem into smaller, more manageable parts.
• Checking your work: Always verify if your final answer make sense.

In our evaporation problem, breaking down the problem into days and stages was critical. By simplifying each part, you control the complexity and find the core of the solution. With time and practice, problem-solving strategies become a natural part of your approach to math exercises, enhancing confidence and effectiveness in finding solutions.