Problem 91 - Algebra 1

Question

Find the sum. $$\frac{5}{8}+\frac{1}{3}$$

Step-by-Step Solution

Answer

\(\frac{5}{8}+\frac{1}{3}=\frac{23}{24}\

Step 1: Find the Least Common Denominator (LCD)

First, we need to find the least common denominator (LCD), also known as least common multiple (LCM) for 8 and 3. The LCD for 8 and 3 is 24. This means that we need to convert both fractions so that they have 24 as the denominator, before we can add them

Step 2: Convert Fractions to Have the Same Denominator

To convert \(\frac{5}{8}\) to a fraction with 24 as the denominator, we multiply the numerator and the denominator by 3. Therefore, \(\frac{5}{8}\) becomes \(\frac{15}{24}\). To convert \(\frac{1}{3}\) to a fraction with 24 as the denominator, we multiply the numerator and the denominator by 8. Therefore, \(\frac{1}{3}\) becomes \(\frac{8}{24}\). So, \(\frac{5}{8}+\frac{1}{3}\) becomes \(\frac{15}{24}+\frac{8}{24}\)

Step 3: Add the Fractions

Now that we have the same denominator for both fractions, we can add them. We add the numerators and keep the denominator. Therefore, \(\frac{15}{24}+\frac{8}{24}=\frac{23}{24}\)

Key Concepts

Least Common Denominator

When adding fractions, it's important to ensure that both fractions have the same denominator. This common denominator allows us to combine the fractions easily. The least common denominator (LCD) is the smallest number that both original denominators can divide into without leaving a remainder. For the fractions \(\frac{5}{8}\) and \(\frac{1}{3}\), the denominators are 8 and 3.
To find the LCD, we determine the least common multiple (LCM) of these two numbers.

• First, list the multiples of 8: 8, 16, 24, 32, etc.
• Next, list the multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, etc.

The smallest common multiple here is 24, which becomes our LCD. This allows us to adjust both fractions to have 24 as their denominator, thereby enabling addition. Finding the LCD helps simplify processes when managing different fractions.

Equivalent Fractions

The concept of equivalent fractions is fundamental when adding, comparing, or manipulating fractions. Equivalent fractions represent the same proportion or value, despite having different numerators and denominators. To convert fractions to equivalent ones with a desired denominator, you multiply both the numerator and the denominator by the same number.
For example, to convert \(\frac{5}{8}\) to a fraction with the common denominator of 24, you multiply the numerator and the denominator by 3.

• \(5 \times 3 = 15\), and
• \(8 \times 3 = 24\)

Thus, \(\frac{5}{8}\) is equivalent to \(\frac{15}{24}\). Similarly, for \(\frac{1}{3}\), you multiply both the numerator and the denominator by 8 to get \(\frac{8}{24}\).
Once converted, these fractions can be added to get a sum with the same denominator. Understanding equivalent fractions makes it easy to manipulate fractions for addition and subtraction.

Numerator and Denominator

It's crucial to know the roles of the numerator and denominator when working with fractions. A fraction, in its essence, is a part of a whole.

• The numerator is the top number and represents the number of parts you have.
• The denominator is the bottom number and indicates how many equal parts the whole is divided into.

For example, in \(\frac{5}{8}\), 5 is the numerator, and 8 is the denominator, meaning 5 parts out of a total of 8.
When you add fractions, after ensuring the denominators are the same, you simply add the numerators because the denominators tell you that the parts are now of the same size.
Using our exercise fractions converted to \(\frac{15}{24}\) and \(\frac{8}{24}\), you add 15 and 8 as numerators and keep the denominator 24 the same, resulting in \(\frac{23}{24}\). Understanding these components makes fraction addition straightforward and comprehensible.