Problem 57 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the next section. If \(B=k W,\) find the value of \(k,\) in decimal form, using \(B=5\) and \(W=160\)

Step-by-Step Solution

Answer

The value of \(k\) is \(0.03125\).

Step 1: Substitute the given values into the equation

Substitute \(B=5\) and \(W=160\) into the equation \(B=kW\) to obtain \(5=k*160\).

Step 2: Isolate \(k\)

To obtain \(k\), one will need to isolate it on one side. One can divide both sides of equation by \(W=160\), we get \(k = \frac{5}{160}\).

Step 3: Simplify the expression to find the decimal value of \(k\)

Finally, compute the decimal value which gives \(k = \frac{5}{160} = 0.03125\).

Key Concepts

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. Solving equations is an essential skill in mathematics which involves finding the value of the unknown variable that makes the equation true. In the context of proportional relationships, where one quantity varies directly as another, we often find ourselves solving equations to find the constant of proportionality.

For example, in the equation \( B = k \times W \), where \( B \) is directly proportional to \( W \), our goal is to solve for \( k \). This involves manipulating the equation to isolate \( k \) on one side, which reveals its value. This specific type of equation-solving practice allows us to determine relationships between quantities, which is fundamental in sciences, economics, and various fields of study.

Substitution Method

The substitution method is a technique commonly used in mathematics to solve equations. It refers to the replacement of variables with their corresponding values in an equation, making it easier to solve.

• Identify the known values.
• Substitute these values into the equation.
• Solve for the unknown variable.

In the given exercise, we started with the equation \( B = k \times W \). By substituting \( B = 5 \) and \( W = 160 \), we transform it into a straightforward arithmetic problem: \( 5 = k \times 160 \). This step simplifies our task by reducing the number of variables we need to work with, providing a clear path towards solving for \( k \).

Simplifying Expressions

Simplifying expressions involves the process of making a mathematical expression easier to work with, by performing operations and reducing it to its simplest form. Sometimes, this also involves computations to convert a fraction into a decimal form.

Let's consider the expression \( \frac{5}{160} \). In its fractional form, it's not very intuitive. By dividing 5 by 160, we obtain a decimal \( k = 0.03125 \).

Here are the steps:

• Recognize the fraction \( \frac{5}{160} \).
• Divide the numerator by the denominator.
• Convert the fraction to a decimal.

Simplifying expressions is a fundamental part of solving equations, as it helps clarify the results and make them more useful for interpretation.