Problem 31 - Elementary Algebra

Question

For exercises 27-34, evaluate. $$ \frac{5}{9}+\frac{7}{24} $$

Step-by-Step Solution

Answer

The answer is \( \frac{61}{72} \).

Step 1: Identify the fractions

The given fractions are \( \frac{5}{9} \) and \( \frac{7}{24} \).

Step 2: Find the Least Common Denominator (LCD)

To add the fractions, first find the Least Common Denominator (LCD) of 9 and 24. The LCD of 9 and 24 is 72.

Step 3: Convert to Equivalent Fractions

Convert each fraction to its equivalent form with the LCD as the denominator:1. \( \frac{5}{9} \rightarrow \frac{5 \times 8}{9 \times 8} = \frac{40}{72} \).2. \( \frac{7}{24} \rightarrow \frac{7 \times 3}{24 \times 3} = \frac{21}{72} \).

Step 4: Add the fractions

Add the equivalent fractions: \( \frac{40}{72} + \frac{21}{72} = \frac{40 + 21}{72} = \frac{61}{72} \).

Step 5: Simplify the fraction if possible

Check to see if \( \frac{61}{72} \) can be simplified. Since 61 and 72 have no common factors other than 1, \( \frac{61}{72} \) is already in its simplest form.

Key Concepts

Least Common Denominator

When adding fractions, the first step is to find the Least Common Denominator (LCD). The LCD is the smallest number that is a multiple of all the denominators in the problem.
In the given exercise, we have two fractions: \( \frac{5}{9} \) and \( \frac{7}{24} \). Their denominators are 9 and 24.
To find the LCD, we need to determine the smallest number that both 9 and 24 divide into evenly.
We start by listing the multiples of each number:

• Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72...
• Multiples of 24: 24, 48, 72...

We see that the smallest multiple they have in common is 72. So, the LCD of 9 and 24 is 72.

Equivalent Fractions

Once we find the LCD, we convert the given fractions to equivalent fractions with the LCD as the denominator. This process involves adjusting the numerator by the same factor we used to adjust the denominator.
For \( \frac{5}{9} \), we multiply the numerator (5) and the denominator (9) by 8, because \( 9 \times 8 = 72 \):
\[ \frac{5}{9} \rightarrow \frac{5 \times 8}{9 \times 8} = \frac{40}{72} \]
Similarly, for \( \frac{7}{24} \), we multiply the numerator (7) and the denominator (24) by 3, because \( 24 \times 3 = 72 \):
\[ \frac{7}{24} \rightarrow \frac{7 \times 3}{24 \times 3} = \frac{21}{72} \]
Now, we have two equivalent fractions: \( \frac{40}{72} \) and \( \frac{21}{72} \).

Fraction Addition

With both fractions now having the same denominator, adding them is straightforward. We simply add the numerators while keeping the denominator the same.
In this case, we add \( \frac{40}{72} \) and \( \frac{21}{72} \):
\[ \frac{40}{72} + \frac{21}{72} = \frac{40 + 21}{72} = \frac{61}{72} \]There you go! We have successfully added the fractions.

Simplifying Fractions

The final step is to check if the resulting fraction can be simplified. Simplifying a fraction means reducing it to its lowest terms.
To do that, we need to find the Greatest Common Divisor (GCD) of the numerator and the denominator.
For \( \frac{61}{72} \), we find that 61 and 72 have no common factors other than 1. Therefore, \( \frac{61}{72} \) is already in its simplest form.
So the final answer is: \[ \frac{61}{72} \] .
This is how we add fractions step by step - finding the least common denominator, converting to equivalent fractions, adding them, and simplifying.