Problem 29 - Algebra A Combined Function

Question

Solve. If needed, round money amounts to two decimal places and all other amounts to one decimal place. The number 40 is \(80 \%\) of what number?

Step-by-Step Solution

Answer

The number is 50.

Step 1: Set Up the Equation

To find the number where 40 is 80% of it, let the unknown number be represented by the variable \( x \). The equation models the relationship as: \[ 40 = 0.80 \times x \] This equation states that 40 is equal to 80% of \( x \).

Step 2: Solve for x

To isolate \( x \), divide both sides of the equation by 0.80. This step will help you find the value of \( x \):\[ x = \frac{40}{0.80} \] Calculate the division to determine the value of \( x \).

Step 3: Compute the Result

Perform the division to solve for \( x \): \[ x = \frac{40}{0.80} = 50 \] Hence, the number where 40 is 80% is 50.

Key Concepts

Algebra

Algebra forms a vital part of mathematics that helps us express mathematical relationships using symbols, usually letters, to represent unknown values. In solving percentage problems, algebra simplifies complex expressions by making use of equations and variables. In the given exercise, we denoted the unknown number as \( x \). Setting up such expressions allows us to manipulate the variables and solve for the unknown number. This method transforms real-world problems into a form that can be calculated using logical operations. Algebra acts like a toolbox, where different tools are equations and formulas that help decode mathematical relationships.

Equation Solving

Solving equations is the process of finding the unknown value in an expression. It’s like unlocking a mystery by logically deducing the answer through systematic steps.
In the original exercise, we set up the equation based on the relationship provided by the percentage problem where 40 is \( 80\% \) of an unknown number \( x \):\[ 40 = 0.80 \times x \] In this arrangement, the task is to isolate \( x \) to reveal its value. Breaking down the steps:

• First, ensure that the equation correctly represents all given information.
• To isolate \( x \), divide both sides of the equation by the numerical coefficient of \( x \), i.e., \( 0.80 \), thus simplifying the equation.

Performing these steps logically leads us to the final calculation of \( x \). Each move in solving the equation is deliberate, working towards simplifying and eventually solving the problem.

Arithmetic Operations

Arithmetic operations include basic mathematical processes such as addition, subtraction, multiplication, and division. These operations are crucial and foundational in solving equations and calculations.
In this exercise, division was the primary operation used to find the unknown number from the provided percentage equation. After having established that \( 40 = 0.80 \times x \), we executed division to solve for \( x \):\[ x = \frac{40}{0.80} \]This operation effectively "undoes" the multiplication by \( 0.80 \) to provide the value for \( x \). When performing division, it's important to ensure that the divisor is not zero, as division by zero is undefined in mathematics. Additionally, when dealing with calculations, it's essential to check precision, especially when involving decimals or rounding, as precision can impact the accuracy of the result.