Question

A rainfall doubled the original amount of water in a reservoir in 1 day and quadrupled the original amount in 5 days. Which of the following expressions represents the approximate amount of water in the reservoir after the 5 days of rain, if there were \(x\) gallons of water in the reservoir before the rainfall? A. \(x+4\) B. \(x+6\) C. \(4 x\) D. \(5 x\)

Short answer

The correct expression for the approximate amount of water in the reservoir after the 5 days of rain is C. \(4 x\).

Step-by-step solution

Amount of Water after 1 Day

After 1 day, the rainfall doubled the original amount of water in the reservoir. So, the amount of water in the reservoir after the 1st day is \( 2x\), where \(x\) is the original amount.

Find the Rainfall Amount

To find the rainfall amount, we can subtract the amount of water in the reservoir after 1 day from the original amount. Since we have doubled the original amount after 1 day, we can write it as: \(2x - x\) This simplifies to: \(x\)

Amount of Water after 5 Days

In 5 days, the rainfall quadrupled the original amount of water in the reservoir. So the total amount of water in the reservoir after the 5 days is \( 4x\).

Identify the Correct Expression

We are given four expressions to choose from: A. \(x + 4\) B. \(x + 6\) C. \(4 x\) D. \(5 x\) Looking at our findings from the previous steps and comparing it with the options, we can already determine that the correct answer is \(4x\), since it matches what we found in Step 3. Therefore, the correct expression for the approximate amount of water in the reservoir after the 5 days of rain is: C. \(4 x\)

Key concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain numbers, variables, and operation signs, which are used to represent real-world quantities in a concise and general form. For example, in the context of the provided rainfall problem, the variable 'x' represents the original amount of water in the reservoir. The expression '4x' then signifies that the water has been quadrupled after five days.

Understanding how algebraic expressions work is crucial when facing problems like these. Variables, such as 'x', can stand for any number, and the operations applied to them, such as multiplication or addition, model what happens to these quantities in a real-world scenario. In our rainfall example, the algebraic expression captures the growth of the water amount without needing to specify its exact volume. Expressions can get more complex with more variables and operations, but the fundamental concept remains the same—using symbols to represent quantities and their relationships.

Problem-Solving Strategies

When approaching mathematical problems, especially word problems, it's essential to have a set of problem-solving strategies to guide you through the process. The exercise on rainfall demonstrates a common approach:

1. Identification: Recognize the quantities involved, which are represented by variables.
2. Operation Determination: Decide on the mathematical operations that relate these variables.
3. Equation Setup: Formulate the relationships as algebraic expressions or equations.
4. Execution: Perform the calculations as indicated by the expressions or equations.
5. Validation: Verify your answer against the information given and the question asked in the problem.

Use these steps sequentially to break down the problem into manageable parts. For example, in our problem, we identified 'x' as the original water amount, determined we are dealing with doubling and quadrupling, set up the expressions '2x' and '4x', executed the comparison with given options, and validated that '4x' was indeed the correct expression. This structured approach helps in systematically arriving at the solution.

Word Problems in Mathematics

Word problems in mathematics are essentially stories or real-life scenarios that require math to solve. They connect abstract mathematical theories to the real world by presenting numerical information in a narrative format. To solve word problems, comprehension is key—you must first understand the scenario before applying mathematical concepts.

Firstly, read the entire problem carefully. Identify the knowns and unknowns, and look out for keywords that indicate mathematical operations (e.g., 'doubled' implies multiplication by 2). Next, translate the words into an algebraic expression or equation that represents the situation. Finally, once you have an expression or equation, use suitable mathematical methods to solve for the unknowns.

In essence, tackling word problems demands both literacy and numeracy skills. Students are often required to sketch out a visual aid, such as a diagram, to help understand the problem better. Also, keeping in mind units of measurement and contextual implications is vital to ensure the solutions are realistic and applicable. In the rainfall problem, for instance, understanding the words 'doubled' and 'quadrupled' allowed us to translate the situation into the algebraic expressions '2x' and '4x' and reach the solution effectively.