Problem 38 - Beginning and Intermediate Algebra

Question

Add or subcract as indicared. Write the answer in lower terms. See Erample 5 $$ \frac{18}{35}-\frac{11}{35} $$

Step-by-Step Solution

Answer

\( \frac{1}{5} \)

Step 1: Understand the Problem

We need to subtract two fractions, \( \frac{18}{35} \) and \( \frac{11}{35} \). Since the fractions have the same denominator, we can subtract the numerators directly.

Step 2: Subtract the Numerators

Subtract the numerators of the fractions: \( 18 - 11 = 7 \). Keep the same denominator. So, the result of the subtraction is \( \frac{7}{35} \).

Step 3: Simplify the Fraction

To simplify \( \frac{7}{35} \), find the greatest common divisor (GCD) of 7 and 35. The GCD is 7. Divide both the numerator and the denominator by the GCD: \( \frac{7 \div 7}{35 \div 7} = \frac{1}{5} \).

Step 4: Write the Answer in Simplest Form

The simplified form of \( \frac{18}{35} - \frac{11}{35} \) is \( \frac{1}{5} \).

Key Concepts

Subtracting Fractions

When it comes to subtracting fractions, one of the first key points is to ensure that both fractions have the same denominator. This shared number makes it easy to directly subtract their numerators. For example, with fractions like \( \frac{18}{35} \) and \( \frac{11}{35} \), since both share the denominator 35, you can simply subtract the numerator of the second fraction from the numerator of the first: \( 18 - 11 = 7 \). Thus, your answer becomes \( \frac{7}{35} \).

• Step 1: Ensure both fractions have the same denominator. If not, you'll need to find a common denominator.
• Step 2: Subtract the numerators and keep the common denominator.
• Step 3: Simplify the result if possible.

The process of subtracting numerators while keeping the same denominator is what makes handling fractions manageable. Remember, the denominator tells you the number of equal parts the whole is divided into, so it stays constant when subtracting fractions with a common denominator.

Simplifying Fractions

Simplifying fractions can make them easier to understand and work with. It means reducing the fraction to its smallest form, where the numerator and denominator have no common factors other than 1.In our example, after subtracting, we obtained \( \frac{7}{35} \). To simplify it, we divide both the numerator and denominator by their greatest common divisor. This leads to a cleaner, simpler representation of the fraction.The procedure involves:

• Finding the greatest common divisor (GCD) of the numerator and denominator.
• Dividing both the numerator and the denominator by the GCD.

By simplifying fractions, you not only tidy up your answer but also make it more intuitive to interpret. In the case of \( \frac{7}{35} \), simplifying gives us \( \frac{1}{5} \), showing that the fraction represents 1 part of 5 equal parts of a whole.

Greatest Common Divisor

The greatest common divisor (GCD) is crucial for simplifying fractions. It's the largest number that can perfectly divide both the numerator and the denominator without leaving a remainder.Let's take a look at our previous example where we found \( \frac{7}{35} \). To simplify, you need to find the GCD of 7 and 35. The number 7 is the GCD here because:

• 7 divides 7 perfectly: \( 7 \div 7 = 1 \)
• 7 divides 35 perfectly: \( 35 \div 7 = 5 \)

To find the GCD:

• List the factors of both numbers.
• Identify the largest factor that appears in both lists.

Once the GCD is found, dividing the numerator and the denominator by this number gives the fraction in its simplest form. The GCD ensures we are reducing the fraction to its lowest terms, making \( \frac{1}{5} \) our answer for the subtraction \( \frac{18}{35} - \frac{11}{35} \).