Question

Evaluate each expression. $$ \frac{8}{9}+\frac{15}{2} $$

Short answer

The result is \(\frac{151}{18}\).

Step-by-step solution

Identify the problem

The problem is to add the two fractions \(\frac{8}{9} + \frac{15}{2}\).

Find a common denominator

To add the fractions, first find a common denominator. The denominators are 9 and 2. The least common multiple (LCM) of 9 and 2 is 18.

Convert fractions to have a common denominator

Convert each fraction to an equivalent fraction with a denominator of 18: \(\frac{8}{9} = \frac{8 \times 2}{9 \times 2} = \frac{16}{18}\) and \(\frac{15}{2} = \frac{15 \times 9}{2 \times 9} = \frac{135}{18}\).

Add the fractions

Now add the fractions with the common denominator: \(\frac{16}{18} + \frac{135}{18} = \frac{16 + 135}{18} = \frac{151}{18}\).

Simplify the result (if possible)

The fraction \(\frac{151}{18}\) cannot be simplified further as 151 and 18 have no common factors. Thus, the final result is \(\frac{151}{18}\).

Key concepts

common denominator

In order to effectively add fractions, finding a common denominator is essential. The denominator is the number found at the bottom of each fraction and using a common one makes addition straightforward. For example, in our problem, we have the fractions \(\frac{8}{9} \) and \(\frac{15}{2}\). Here, the denominators are 9 and 2.

The idea is to transform these fractions so that both have the same denominator. This requires finding the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators divide into without leaving a remainder.

Using a common denominator, we can convert both fractions to have the same bottom number. This step smoothes the way for adding the fractions together, which we’ll explore in further detail in the next sections.

least common multiple

The least common multiple (LCM) plays an intricate role in adding fractions. It helps in determining the smallest shared multiple of the denominators. To find the LCM of two numbers, you can list the multiples of each number and then find the smallest multiple they both share.

For our problem, the denominators are 9 and 2. Here’s how we find the LCM:

• List the multiples of 9: 9, 18, 27, 36, ...
• List the multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, ...

We observe that 18 is the smallest number that appears in both lists. Thus, the LCM of 9 and 2 is 18.This means that 18 will be our common denominator for adding these fractions.

With the LCM found, we now need to adjust each fraction to have this common denominator before we can add them.

equivalent fractions

To add fractions with different denominators, we convert them into equivalent fractions that have the same denominator. Equivalent fractions represent the same value but look different. For instance, \(\frac{8}{9} \) can become \(\frac{16}{18}\) if both the numerator and denominator are multiplied by 2.

Let's consider our problem again. We found the common denominator to be 18. Now, we convert each fraction:

• \(\frac{8}{9} \) needs to be converted. Since 9 times 2 equals 18, we also multiply the numerator 8 by 2, which gives us \(\frac{16}{18}\).
• \(\frac{15}{2} \) needs to be converted. Since 2 times 9 equals 18, we also multiply the numerator 15 by 9, resulting in \(\frac{135}{18}\).

Now both fractions have a common denominator of 18, enabling us to add them together.

simplifying fractions

After adding the fractions, the result might need simplifying. Simplifying a fraction means making it as simple as possible by ensuring the numerator and the denominator are as small as possible. This is done by dividing both by their greatest common divisor (GCD).

Suppose you have \(\frac{151}{18}\). To check if it can be simplified, look for a common factor between 151 and 18. In this case, 151 and 18 have no common factors other than 1. This means the fraction is already in its simplest form.

Let's summarize the steps:

• First, find a common denominator, usually via the LCM.
• Convert the fractions into equivalent fractions with this common denominator.
• Add the fractions together.
• Finally, simplify the resulting fraction if needed.

Simplifying fractions is an essential final step to ensure your answers are neat and as simple as they can be.