Problem 65 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ \frac{2}{3}+\frac{3}{5} $$

Step-by-Step Solution

Answer

The sum is \(\frac{19}{15}\), already in lowest terms.

Step 1: Find the Least Common Denominator

To add fractions, they need to have a common denominator. The denominators here are 3 and 5. The least common denominator (LCD) can be found as the least common multiple (LCM) of these numbers. - List the multiples: - Multiples of 3: 3, 6, 9, 12, 15, ... - Multiples of 5: 5, 10, 15, ... - The least common multiple of 3 and 5 is 15. Therefore, the LCD is 15.

Step 2: Convert Fractions to Common Denominator

Convert each fraction to the common denominator found in Step 1, which is 15.- Convert \(\frac{2}{3}\): Multiply both the numerator and the denominator by 5: \( \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \)- Convert \(\frac{3}{5}\): Multiply both the numerator and the denominator by 3: \( \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \)

Step 3: Add the Fractions

Now that the fractions have the same denominator, add them by adding their numerators:\[\frac{10}{15} + \frac{9}{15} = \frac{10 + 9}{15} = \frac{19}{15}\]

Step 4: Simplify the Fraction

The fraction \(\frac{19}{15}\) cannot be simplified further because 19 and 15 have no common factors other than 1. Thus, \(\frac{19}{15}\) is already in its lowest terms.

Key Concepts

Least Common Denominator

When adding fractions, having the same denominator is crucial. This common denominator is known as the "Least Common Denominator" (LCD). To find the LCD, look for the smallest number that both original denominators can divide into without leaving a remainder. In our example:

• The denominators are 3 and 5.
• We list the multiples of each:
• Multiples of 3: 3, 6, 9, 12, 15, ...
• Multiples of 5: 5, 10, 15, ...
• Compare these lists to find the least common multiple, which is 15.
• So, the LCD for 3 and 5 is 15.

Starting with a common denominator allows for straightforward addition of numerators, simplifying the entire process.

Least Common Multiple

We touched on the Least Common Multiple (LCM) briefly while finding the LCD. The LCM is essential not just for fractions but in various math problems. Here’s a bit more on how we find it for two numbers, say for our denominators 3 and 5:

• Write out the multiples of each number:
• The multiples of 3 include 3, 6, 9, 12, 15, etc.
• The multiples of 5 include 5, 10, 15, etc.
• Identify the smallest number that appears in both lists. For 3 and 5, this is 15.

This process finds the LCM, which is critical for aligning the denominators in fractions. LCM helps in making sure fractions are easily comparable and addable.

Simplifying Fractions

Once you've added fractions, sometimes the result might need simplifying. This means reducing the fraction to the smallest possible equivalent fraction. The aim is to make it as simple as possible.For a fraction like \( \frac{19}{15} \):

• Check if the numerator (19) and the denominator (15) share any common factors.
• Factors of 19 are 1 and 19, and factors of 15 are 1, 3, 5, and 15.
• The only common factor is 1, so \( \frac{19}{15} \) is already in its simplest form.

Simplifying fractions means finding common factors and dividing them out, which helps uphold neatness and clarity in mathematical work.

Converting Fractions

To add fractions effectively, converting them to have the same denominator is necessary. This involves rewriting each fraction with the Least Common Denominator we found. Converting the fractions from the original exercise:

• For \( \frac{2}{3} \):
  • Multiply numerator (2) and denominator (3) by 5, since 5 is needed to make 15.
  • This gives \( \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \).
• For \( \frac{3}{5} \):
  • Multiply numerator (3) and denominator (5) by 3, which changes it to 15.
  • This gives \( \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \).

Converting fractions, by adjusting them to the LCD, ensures they can be easily added or compared by aligning them optimally.