Problem 9 - Algebra for College Students

Question

For Problems 1-12, perform the indicated operations involving rational numbers. Be sure to express your answers in reduced form. \(\frac{1}{5}+\frac{5}{6}-\frac{7}{15}\)

Step-by-Step Solution

Answer

\( \frac{17}{30} \)

Step 1: Find a Common Denominator

To add or subtract fractions, they must have a common denominator. The fractions \( \frac{1}{5} \), \( \frac{5}{6} \), and \( \frac{7}{15} \) have denominators 5, 6, and 15. The least common multiple (LCM) of these numbers is 30. We will convert each fraction to have a denominator of 30.

Step 2: Convert Each Fraction

Convert \( \frac{1}{5} \) by multiplying both the numerator and denominator by 6 to get \( \frac{6}{30} \). Convert \( \frac{5}{6} \) by multiplying the numerator and denominator by 5 to get \( \frac{25}{30} \). Convert \( \frac{7}{15} \) by multiplying the numerator and denominator by 2 to get \( \frac{14}{30} \). The expression now is \( \frac{6}{30} + \frac{25}{30} - \frac{14}{30} \).

Step 3: Perform the Addition and Subtraction

Add \( \frac{6}{30} \) and \( \frac{25}{30} \) first: \( \frac{6}{30} + \frac{25}{30} = \frac{31}{30} \). Then subtract \( \frac{14}{30} \) from \( \frac{31}{30} \): \( \frac{31}{30} - \frac{14}{30} = \frac{17}{30} \).

Step 4: Simplify the Fraction

The fraction \( \frac{17}{30} \) is already in its simplest form because 17 is a prime number and does not divide 30. Therefore, no further simplification is needed.

Key Concepts

Least Common Denominator

When working with fractions, finding a common denominator is crucial for easy calculations. In this case, the least common denominator (LCD) is particularly useful.
The LCD is the smallest multiple that is common to all the denominators in a given set of fractions.
This ensures the fractions have like denominators, making addition or subtraction possible.To determine the LCD for fractions such as \( \frac{1}{5} \), \( \frac{5}{6} \), and \( \frac{7}{15} \), we first identify the denominators: 5, 6, and 15. Next, calculate the least common multiple (LCM) of these numbers.

• The multiples of 5 are 5, 10, 15, 20, 25, etc.
• The multiples of 6 are 6, 12, 18, 24, 30, etc.
• The multiples of 15 are 15, 30, 45, 60, etc.

The smallest number found in all these lists is 30. Therefore, the LCD for our fractions is 30. By converting each fraction to have this common denominator, we can easily perform addition and subtraction.

Addition and Subtraction of Fractions

Once fractions have a common denominator, the addition and subtraction processes become straightforward.
This is because you only need to deal with the numerators directly.In our example, the fractions \( \frac{1}{5} \), \( \frac{5}{6} \), and \( \frac{7}{15} \) were converted to \( \frac{6}{30} \), \( \frac{25}{30} \), and \( \frac{14}{30} \) respectively. This was done by multiplying the numerators and denominators by specific factors to equal the LCD.Here's how to handle the operation

• Add \( \frac{6}{30} \) and \( \frac{25}{30} \) to get \( \frac{31}{30} \).
• Next, subtract \( \frac{14}{30} \) from \( \frac{31}{30} \) to arrive at \( \frac{17}{30} \).

The key idea is that once the denominators are the same, simply focus on the numerators and follow basic addition or subtraction rules.
Remember that the resulting fraction may need simplification.

Simplifying Fractions

Simplifying fractions is the act of reducing them to their simplest form. This makes fractions more compact and easier to understand.
To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD).In the problem at hand, after performing the addition and subtraction, the fraction \( \frac{17}{30} \) is obtained.
Initially, you should check if both the numerator and denominator have any common factors other than 1.

• Here, 17 is a prime number, which means its only divisors are 1 and itself.
• Since 17 does not divide evenly into 30, the fraction is already in its simplest form.

When simplifying, always look for any common factors. If none exist, the fraction is already simplified.
This step ensures that answers are presented in their most reduced form, which is typically preferred for clarity and ease of understanding.