Problem 66 - Elementary and Intermediate Algebra

Question

Translate each problem to an equation. Do not solve. When 42 is multiplied by a number, the result is 2352 . Find the number.

Step-by-Step Solution

Answer

42n = 2352

Step 1: Identify the variables

Let the unknown number be represented by the variable . This is the number we're trying to find.

Step 2: Set up the problem as an equation

According to the problem, 42 multiplied by the number equals 2352. This can be written as: 42n = 2352

Key Concepts

Algebra

Algebra is a branch of mathematics that uses letters and symbols to represent numbers and quantities in formulas and equations. In algebra, we often use variables to stand in for unknown values we need to solve for.

For example, in our exercise, we're using the letter 'n' to represent the unknown number. Variables are a way to keep track of unknown quantities in math problems. They allow us to write down relationships between numbers in a more general way.

Using algebra, we can take a problem written in words and turn it into a mathematical statement. This helps us use arithmetic operations and properties to find solutions.

Equations

Equations are mathematical statements that assert the equality of two expressions. The statements in the form of equations show a relationship that is true for the specific values of the variables.

In our specific exercise, the equation formed is:
\[ 42n = 2352 \]

Here, the equation represents the statement '42 multiplied by a number equals 2352'. Equations can be easily solved by performing algebraic operations to find the variable's value.

Familiarizing yourself with equations helps significantly in understanding various mathematical contexts and solving a wide range of problems. They form the basis for much of algebra and even advanced math.

Problem Formulation

Problem formulation is the process of translating a word problem into a mathematical representation, like an equation. This involves carefully reading the problem, identifying the unknowns, and expressing the relationships between quantities mathematically.

In our given exercise: 'When 42 is multiplied by a number, the result is 2352. Find the number.', we:

• First identified the unknown number, which we designated as 'n'.
• Then translated the word problem into the equation 42n = 2352.

Accurate problem formulation is crucial in algebra because it sets the stage for solving the problem. It takes verbal and sometimes complex real-world situations and turns them into manageable mathematical equations.

Remember, this first step is about setting up the problem correctly, so always double-check that you've translated the words into math correctly before you proceed to solve it.