Problem 174 - Intermediate Algebra

Question

In the following exercises, add or subtract. $$ \frac{5}{12}+\frac{3}{8} $$

Step-by-Step Solution

Answer

\[\frac{19}{24}\]

Step 1: Find the Least Common Denominator (LCD)

To add the fractions, first find the least common denominator of the two fractions' denominators. The denominators are 12 and 8. The least common multiple of 12 and 8 is 24.

Step 2: Convert Fractions to Equivalent Fractions with the LCD

Convert \(\frac{5}{12}\) and \(\frac{3}{8}\) to fractions with the denominator 24. Multiply the numerator and denominator of \(\frac{5}{12}\) by 2 to get \(\frac{10}{24}\). Multiply the numerator and denominator of \(\frac{3}{8}\) by 3 to get \(\frac{9}{24}\).

Step 3: Add the Equivalent Fractions

With the fractions now having a common denominator, add them together: \(\frac{10}{24} + \frac{9}{24} = \frac{19}{24}\).

Key Concepts

headline of the respective core concept

When adding fractions, the first crucial step is finding the Least Common Denominator (LCD). The LCD is the smallest number that both denominators can divide into evenly. This ensures we compare like terms. For example, with fractions \(\frac{5}{12}\) and \(\frac{3}{8}\), the denominators are 12 and 8. To find the LCD, you must identify the smallest multiple they share. In this case, 24 is the first number both 12 and 8 divide into fully. Knowing how to find the LCD is essential in fraction addition and helps make other steps much simpler.

headline of the respective core concept

The next important step in adding fractions is converting them into Equivalent Fractions. Equivalent fractions are fractions that look different but represent the same value. To convert two fractions to equivalent fractions with the same common denominator, we need to adjust their numerators and denominators. For \(\frac{5}{12}\) and \(\frac{3}{8}\), we convert them so they have the LCD, which is 24. To do this: Multiply the numerator and denominator of \(\frac{5}{12}\) by 2, resulting in \(\frac{10}{24}\). Similarly, multiply the numerator and denominator of \(\frac{3}{8}\) by 3, giving us \(\frac{9}{24}\). Now, both fractions are ready for addition.

headline of the respective core concept

Finally, it's time to Add the Equivalent Fractions. Once we have the fractions with a common denominator, adding them is straightforward. Now our fractions to add are \(\frac{10}{24}\) and \(\frac{9}{24}\). To add these: Simply add their numerators while keeping the common denominator: \(\frac{10}{24} + \frac{9}{24}\) results in \(\frac{19}{24}\). And that’s your final answer. It's always a good idea to check if the resulting fraction can be simplified further. In this case, \(\frac{19}{24}\) is already in its simplest form.