Problem 116 - Introductory Algebra for College Students

Question

will help you prepare for the material covered in the first section of the next chapter. Evaluate \(\frac{250 x}{100-x}\) for \(x=60\)

Step-by-Step Solution

Answer

The value of the expression \(\frac{250 x}{100-x}\) for \(x=60\) is 375.

Step 1: Understanding the given expression

The expression to evaluate is \(\frac{250 x}{100-x}\). Here, 'x' is the variable.

Step 2: Substituting 'x'

Substitute 'x' with its given value. So, the expression becomes \(\frac{250 * 60}{100-60}\).

Step 3: Performing the calculations

Carry out the calculations separately in the numerator and denominator. The expression simplifies to \(\frac{15000}{40}\).

Step 4: Simplify further

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor which is 40. The expression becomes \(375\).

Key Concepts

Substitution Method

The substitution method plays a crucial role when evaluating algebraic expressions. It involves replacing the variable in the expression with a specific value, which is provided or known within a problem. For example, if we have an expression like \(\frac{250 x}{100-x}\) and we need to evaluate it for \(x=60\), substitution means we plug \(60\) into the expression wherever \(x\) appears.

The initial step is to understand the structure of the algebraic expression. In our case, \(x\) is in both the numerator and denominator of the fraction. The next move is to perform the substitution: \(\frac{250 \times 60}{100-60}\). This application of the substitution method transforms the algebraic expression into a numerical one, making it easier to solve. Remember, careful substitution is vital to avoid errors in the subsequent calculation steps.

Simplifying Fractions

After substitution, your next objective is to simplify the fraction to its lowest terms. Simplifying fractions essentially means reducing them to their simplest form where the numerator and the denominator have no common factors other than 1.

In our exercise, we perform the calculations to get \(\frac{15000}{40}\). To simplify this, we must identify the Greatest Common Divisor (GCD) of 15000 and 40. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. For 15000 and 40, that GCD is 40. We then divide both terms by 40 to get the simplified result of \(375\).

Simplifying can sometimes involve factoring both the numerator and the denominator and then cancelling out common factors. However, recognizing the GCD directly, as in our case, streamlines the process significantly.

Algebraic Fractions

Algebraic fractions are similar to standard fractions; however, they include variables in the numerator, in the denominator, or in both. Understanding how to manipulate these expressions is key to working efficiently with algebraic equations.

Let's revisit our original problem: \(\frac{250 x}{100-x}\). This is an algebraic fraction with \(x\) in both the numerator and denominator. The principles for simplifying algebraic fractions are comparable to those for numerical fractions. Substituting values for variables helps to convert an algebraic fraction into a numerical one, which can then be simplified.

In some cases, algebraic fractions can also be simplified before substituting values for the variables, if there are common factors in the numerator and denominator that can be cancelled out. But in the given example, simplification occurred after evaluating the expression through substitution.