In Exercises 33–68, add or subtract as indicated. 3x/8 + x/12

Verified step by step guidance

1. Identify the least common denominator (LCD) of the fractions. The denominators are 8 and 12. The LCD is the smallest number that both 8 and 12 divide into evenly. In this case, the LCD is 24.
2. Rewrite each fraction with the LCD as the denominator. To do this, multiply both the numerator and denominator of each fraction by the factor needed to make the denominator equal to 24. For \( \frac{3x}{8} \), multiply by \( \frac{3}{3} \), and for \( \frac{x}{12} \), multiply by \( \frac{2}{2} \).
3. After rewriting, the fractions become \( \frac{9x}{24} \) and \( \frac{2x}{24} \).
4. Since the denominators are now the same, you can add the numerators directly. Combine \( 9x \) and \( 2x \) to get \( 11x \).
5. Write the result as a single fraction: \( \frac{11x}{24} \). This is the simplified form of the sum.

Key Concepts

Finding a Common Denominator

To add or subtract fractions, it is essential to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators involved. In this case, the denominators are 8 and 12, and the LCM is 24. This allows us to rewrite each fraction with the same denominator, making it possible to combine them.

Fraction Addition

Fraction addition involves combining two or more fractions into a single fraction. Once the fractions have a common denominator, you can add the numerators while keeping the common denominator the same. For example, if we convert 3x/8 and x/12 to have a denominator of 24, we can then add the adjusted numerators to find the sum.

Simplifying Fractions

After performing operations on fractions, it is often necessary to simplify the result. Simplifying a fraction involves dividing both the numerator and the denominator by their greatest common divisor (GCD). This process ensures that the fraction is expressed in its simplest form, making it easier to understand and work with.