Problem 63 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{2}{15}+\frac{7}{9} $$

Step-by-Step Solution

Answer

The result is \( \frac{41}{45} \).

Step 1: Finding a Common Denominator

To add fractions, they must have the same denominator. The denominators 15 and 9 have a least common denominator (LCD) of 45. We'll convert each fraction to an equivalent fraction with this common denominator.

Step 2: Converting the First Fraction

Convert \( \frac{2}{15} \) to an equivalent fraction with denominator 45 by multiplying both the numerator and the denominator by 3:\[\frac{2}{15} = \frac{2 \times 3}{15 \times 3} = \frac{6}{45}\]

Step 3: Converting the Second Fraction

Convert \( \frac{7}{9} \) to an equivalent fraction with denominator 45 by multiplying both the numerator and the denominator by 5:\[\frac{7}{9} = \frac{7 \times 5}{9 \times 5} = \frac{35}{45}\]

Step 4: Adding the Fractions

Now that both fractions have the same denominator, add the numerators while keeping the denominator the same:\[\frac{6}{45} + \frac{35}{45} = \frac{6 + 35}{45} = \frac{41}{45}\]

Step 5: Simplifying the Result

Check if \( \frac{41}{45} \) can be simplified. Since 41 is a prime number and does not divide evenly into 45, the fraction is already in simplest form.

Key Concepts

Common Denominator

When adding fractions, you can't do it directly if they have different denominators. Think of the denominator as the size of the pieces you are dealing with. For the pieces to fit together nicely, they all need to be the same size.
The process requires finding a shared denominator, known as the common denominator, so that both fractions talk the same math language.
Here's a quick guide on how to find a common denominator:

• Identify the denominators of the fractions you are working with. In our example, we have 15 and 9.
• Aim to find a number that both denominators can divide into evenly. This magical number they both share is a common denominator.
• Sometimes, the least common denominator (LCD), which is the smallest such number, can make calculations simpler. For 15 and 9, the LCD is 45.

By converting each fraction to have this shared denominator, you make it possible to add them smoothly.

Simplifying Fractions

Once the fractions are added together, you might end up with a fraction that looks a bit complicated. Simplifying fractions is like cleaning up, so everything looks nicer and easier to understand.
Here’s how you can simplify fractions:

• Check if there is a number, greater than 1, that divides both the numerator and the denominator evenly. This number is called the greatest common divisor (GCD).
• Divide both the top and bottom numbers of the fraction by this GCD.
• If no such number exists, then your fraction is already as simple as it gets.

In the exercise, our result \( \frac{41}{45} \) was already in its simplest form because 41 is a prime number and doesn't share factors with 45.

Least Common Denominator

Finding the least common denominator is the strategic part of dealing with fraction problems. It’s all about making fractions easier to handle by finding the smallest possible common denominator.
Why is it important? Because using the smallest number possible keeps your calculations neat and manageable. Here's how to find the LCD:

• List the multiples of each denominator.
• Identify the smallest multiple that appears in both lists.

The LCD is not always obvious at first glance, but practice makes this process quicker. In our example, to find the LCD for 15 and 9, we do:

• Multiples of 15: 15, 30, 45, 60,...
• Multiples of 9: 9, 18, 27, 36, 45,...

The smallest shared number, 45, is chosen as the LCD. Once fractions are rewritten with the LCD, they can be easily added or subtracted!