Problem 68 - Prealgebra

Question

These problems review the four basic operations with fractions from this chapter. Perform the indicated operations. $$\frac{2}{3}-\frac{3}{5}$$

Step-by-Step Solution

Answer

The result of \( \frac{2}{3} - \frac{3}{5} \) is \( \frac{1}{15} \).

Step 1: Find a common denominator

To subtract fractions, they need to have the same denominator. For \( \frac{2}{3} \) and \( \frac{3}{5} \), we first find the least common multiple (LCM) of 3 and 5, which is 15.

Step 2: Convert to equivalent fractions

Convert \( \frac{2}{3} \) to an equivalent fraction with a denominator of 15. Multiply both the numerator and the denominator by 5, resulting in \( \frac{10}{15} \). Similarly, convert \( \frac{3}{5} \) by multiplying both the numerator and the denominator by 3, resulting in \( \frac{9}{15} \).

Step 3: Subtract the fractions

Subtract \( \frac{9}{15} \) from \( \frac{10}{15} \): \( \frac{10}{15} - \frac{9}{15} = \frac{1}{15} \).

Step 4: Final result

The simplest form of the result is \( \frac{1}{15} \). Since the numerator and denominator have no common factors other than 1, \( \frac{1}{15} \) is already in its simplest form.

Key Concepts

Subtracting Fractions

When you subtract fractions, the primary rule is to have the same denominators. Think of the denominator as a common "language" for the fractions to communicate.
To subtract fractions such as \( \frac{2}{3} \) and \( \frac{3}{5} \), we need them both to speak in terms of a common denominator. In this case, we look for the least common multiple (LCM) of their denominators (3 and 5).

• Find the least common multiple of the denominators.
• Convert each fraction to an equivalent fraction with the common denominator.
• Subtract the numerators, keeping the denominator the same.

The result is a new fraction, which may need further simplification.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest form, which makes them easier to work with. In the context of our exercise, after subtracting \( \frac{9}{15} \) from \( \frac{10}{15} \), we get \( \frac{1}{15} \). Simplifying this involves...
Checking if there are any common factors of the numerator and denominator.

• If there are, divide both by the greatest common factor (GCF).
• For \( \frac{1}{15} \), the GCF is 1 for both the numerator and denominator, meaning it's already in its simplest form.

This process ensures you have the most efficient version of your fraction, with no unnecessary complexity.

Least Common Multiple

The least common multiple (LCM) is crucial when adding or subtracting fractions. The LCM is the smallest number that is a multiple of the denominators you are dealing with.
For denominators 3 and 5, as with \( \frac{2}{3} \) and \( \frac{3}{5} \), the LCM is 15.

• List the multiples of each number until a common multiple is found.
• For example, multiples of 3 are 3, 6, 9, 12, 15, 18, and so on, while multiples of 5 are 5, 10, 15, 20, etc.
• The first shared multiple is 15, making it the LCM.

Finding the LCM is essential to create equivalent fractions, thereby allowing you to perform operations like subtraction or addition efficiently.