Problem 74 - Intermediate Algebra

Question

Data Analysis. Use the data in the table to find the average measurement for the three-trial experiment. Express the answer as a simplified rational expression. $$ \begin{array}{|c|c|c|} \hline \text { Trial 1 } & \text { Trial 2 } & \text { Trial 3 } \\ \hline \frac{k}{3} & \frac{k}{5} & \frac{k}{6} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The average measurement is \(\frac{7k}{30}\).

Step 1: Write the expression for average

The average of the three measurements is given by the formula: \[ \text{Average} = \frac{\text{Trial 1} + \text{Trial 2} + \text{Trial 3}}{3} \] Substitute the expressions for each trial into this formula: \[ \text{Average} = \frac{\frac{k}{3} + \frac{k}{5} + \frac{k}{6}}{3} \]

Step 2: Find a common denominator

To add the fractions, find the least common denominator (LCD) of 3, 5, and 6. The LCD is 30. Express each fraction with a denominator of 30: \[ \frac{k}{3} = \frac{10k}{30}, \quad \frac{k}{5} = \frac{6k}{30}, \quad \text{and} \quad \frac{k}{6} = \frac{5k}{30} \]

Step 3: Add the fractions

Add the fractions with the common denominator: \[ \frac{10k}{30} + \frac{6k}{30} + \frac{5k}{30} = \frac{10k + 6k + 5k}{30} = \frac{21k}{30} \]

Step 4: Calculate the average

Now, divide the sum of the fractions by 3 to find the average: \[ \text{Average} = \frac{\frac{21k}{30}}{3} = \frac{21k}{30 \times 3} = \frac{21k}{90} \]

Step 5: Simplify the rational expression

Simplify \(\frac{21k}{90}\). The greatest common divisor (GCD) of 21 and 90 is 3, so divide both the numerator and the denominator by 3: \[ \frac{21k}{90} = \frac{7k}{30} \] Thus, the average is \(\frac{7k}{30}\).

Key Concepts

Average Calculation

Calculating the average is a fundamental part of mathematical data analysis. When you compute an average, you're trying to find a central value of a set of numbers or expressions.
To find the average, add together all of the values and then divide by the number of values. In this problem, we were given the rational expressions for three trials:

• Trial 1: \(\frac{k}{3}\)
• Trial 2: \(\frac{k}{5}\)
• Trial 3: \(\frac{k}{6}\)

For the average, the formula is:\[\text{Average} = \frac{\text{Trial 1} + \text{Trial 2} + \text{Trial 3}}{3}\] You add these expressions together and then divide by 3. Remember, when adding fractions, ensure they have a common denominator first for easy addition. After finding the sum, dividing by the total number of trials gives you the average.

Rational Expression

A rational expression is simply a fraction in which the numerator and/or the denominator are polynomials. In many math problems, especially in algebra, you'll encounter rational expressions.
The expressions given in the problem for each trial are examples of rational expressions, such as \(\frac{k}{3}\), \(\frac{k}{5}\), and \(\frac{k}{6}\). These involve variables (in this case \(k\)) and are in the form of a fraction.
When working with rational expressions, it's important to remember that:

• You need a least common denominator (LCD) to add or subtract them.
• Simplifying them might involve factoring both the numerator and denominator.

Ultimately, in this problem, the goal was to simplify the resulting rational expression after finding their average, \(\frac{7k}{30}\), ensuring we have the expression in its simplest form.

Least Common Denominator

The least common denominator (LCD) is essential when dealing with multiple fractions. It's the smallest number that each of the denominators can divide into evenly.
When adding or subtracting fractions, they must have the same denominator to combine them. In our example, we needed to figure out the LCD for the denominators 3, 5, and 6.
To find the LCD:

• List the multiples of each denominator, or use prime factorization.
• The LCD of 3, 5, and 6 is 30 because it is the smallest number that is divisible by all three.

Once you have the LCD, convert each fraction to an equivalent fraction with this common denominator: \(\frac{10k}{30}\), \(\frac{6k}{30}\), and \(\frac{5k}{30}\).
This allows you to easily add the fractions together, which is a crucial step when calculating averages of rational expressions.