Problem 40
Question
In Exercises 39-42, write an algebraic equation. Do not solve the equation. With the \(1.2\)-inch rainfall today, the total for the month is \(4.5\) inches. How much had been recorded for the month before today's rainfall?
Step-by-Step Solution
Verified Answer
The algebraic equation from the problem is: \(1.2 + x = 4.5\)
1Step 1: Define the Variables
Let's denote 'x' as the amount of rainfall recorded for the month before today's rainfall.
2Step 2: Formulate the Equation
From the problem, we have that the total rainfall for the month is 4.5 inches. This total consists of the 1.2 inches that fell today and the x inches of rain that had already been recorded before today. Thus, the situation forms the following equation: \(1.2 + x = 4.5\).
Key Concepts
Defining VariablesFormulating EquationsRainfall Problems
Defining Variables
Defining variables is the foundation of solving any algebraic word problem. Variables are symbols that represent unknown values, and they are crucial in translating real-world problems into mathematical equations. In most cases, we use letters like 'x', 'y', or 'z' to depict these unknowns.
In the context of rainfall problems, defining variables helps us simplify and effectively manage the information we have. For instance, in our exercise, we identified that 'x' is the variable representing the amount of rainfall recorded for the month before today's rainfall. This step is essential because it breaks down the problem into manageable parts and helps to focus on what's necessary for finding the solution. By defining 'x', we explicitly state what we are looking for, which is that unknown quantity recorded before today's additional rainfall.
In the context of rainfall problems, defining variables helps us simplify and effectively manage the information we have. For instance, in our exercise, we identified that 'x' is the variable representing the amount of rainfall recorded for the month before today's rainfall. This step is essential because it breaks down the problem into manageable parts and helps to focus on what's necessary for finding the solution. By defining 'x', we explicitly state what we are looking for, which is that unknown quantity recorded before today's additional rainfall.
Formulating Equations
Once we have defined our variables, the next critical step is formulating equations. This process involves translating the problem statement into a mathematical form that can be analyzed and solved. For our rainfall problem, formulating the equation involves recognizing how the known and unknown quantities relate to each other.
We know from the exercise that the month's total rainfall amounted to 4.5 inches, including today's 1.2 inches. Thus, the equation should express this relationship. When we state that the sum of today's rainfall and the previously recorded rainfall equals 4.5 inches, we form the equation:
\[1.2 + x = 4.5\]
This equation succinctly captures the situation as described. It enables us to see what needs to be solved once we decide to tackle the equation for 'x'. Formulating equations is a skill that's particularly important in algebra because it helps transform word problems into a mathematical framework that is easier to manipulate.
We know from the exercise that the month's total rainfall amounted to 4.5 inches, including today's 1.2 inches. Thus, the equation should express this relationship. When we state that the sum of today's rainfall and the previously recorded rainfall equals 4.5 inches, we form the equation:
\[1.2 + x = 4.5\]
This equation succinctly captures the situation as described. It enables us to see what needs to be solved once we decide to tackle the equation for 'x'. Formulating equations is a skill that's particularly important in algebra because it helps transform word problems into a mathematical framework that is easier to manipulate.
Rainfall Problems
Rainfall problems are a specific type of algebraic word problem that utilizes rainfall measurements to create equations. These problems often involve determining unknown quantities, such as rainfall amounts over a period of time.
In this scenario, understanding how rainfall problems are structured can help immensely. Typically, they involve adding or subtracting rainfall amounts to or from a total to find missing data. The equation in our original problem focuses on the fact that today's 1.2 inches of rainfall bring the monthly total to 4.5 inches, suggesting the need to subtract the prior rain accumulations from the total to find the missing piece.
When tackling such problems, it’s essential to:
In this scenario, understanding how rainfall problems are structured can help immensely. Typically, they involve adding or subtracting rainfall amounts to or from a total to find missing data. The equation in our original problem focuses on the fact that today's 1.2 inches of rainfall bring the monthly total to 4.5 inches, suggesting the need to subtract the prior rain accumulations from the total to find the missing piece.
When tackling such problems, it’s essential to:
- Read the problem statement carefully.
- Identify the variables and known quantities.
- Formulate the equation based on the relationships described.
Other exercises in this chapter
Problem 39
$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ (-5 z)\left(2 z^{2}\right) $$
View solution Problem 39
In Exercises 37-44, evaluate the algebraic expression for the given values of the variable(s). \(2 x^{2}-5\) (a) \(x=2\) (b) \(x=3\)
View solution Problem 40
The height of a rectangular picture frame is \(1.5\) times the width \(w\). Write an algebraic expression that represents the perimeter of the picture frame.
View solution Problem 40
$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ (10 t)\left(-4 t^{2}\right) $$
View solution