Problem 49 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$\frac{3}{5}+\frac{1}{3}$$

Step-by-Step Solution

Answer

\( \frac{14}{15} \)

Step 1: Identify the Problem

We are tasked with finding the sum of two fractions: \( \frac{3}{5} \) and \( \frac{1}{3} \). Our goal is to add these fractions together and simplify the result.

Step 2: Find a Common Denominator

In order to add fractions, we need a common denominator. The denominators here are 5 and 3. The least common multiple of 5 and 3 is 15. Therefore, 15 will be our common denominator.

Step 3: Adjust the Fractions

Convert \( \frac{3}{5} \) and \( \frac{1}{3} \) so they both have a denominator of 15. For \( \frac{3}{5} \), multiply numerator and denominator by 3 to get \( \frac{9}{15} \).For \( \frac{1}{3} \), multiply numerator and denominator by 5 to get \( \frac{5}{15} \).

Step 4: Add the Fractions

Now that both fractions have the same denominator, we can add the numerators directly:\[ \frac{9}{15} + \frac{5}{15} = \frac{9 + 5}{15} = \frac{14}{15} \].

Step 5: Simplify the Result

Check if \( \frac{14}{15} \) can be simplified further. Since 14 and 15 have no common factors other than 1, \( \frac{14}{15} \) is already in its simplest form.

Key Concepts

Addition of Fractions

Adding fractions might seem tricky at first, but it becomes much easier once you understand the basic rule: you can only add fractions directly if they have the same denominator. Think of this as adding pieces of a pie. You can only directly sum up the pieces if each piece is of the same size.

To add fractions with different denominators, follow these steps:

• First, find the least common denominator (LCD). This is the smallest number that both denominators divide into.
• Next, convert each fraction to an equivalent fraction with this common denominator. You'll do this by multiplying both the numerator and the denominator by the same number.
• Finally, add the numerators, keeping the denominator the same.

Once you understand these steps, you'll find that adding fractions is just like adding whole numbers, once their types match!

Common Denominator

The term 'common denominator' is crucial when dealing with the addition or subtraction of fractions. It refers to a number that each of the original denominators can divide into without leaving a remainder. The least common denominator (LCD) is the smallest number that can serve this purpose.

Finding the common denominator requires finding the least common multiple (LCM) of the denominators involved. For example, in our exercise with fractions \( \frac{3}{5} \) and \( \frac{1}{3} \), the denominators are 5 and 3. To find their LCM, list their multiples:

• Multiples of 5 are 5, 10, 15, 20...
• Multiples of 3 are 3, 6, 9, 12, 15...

You'll notice the smallest common multiple is 15. Hence, the LCD is 15, which allows us to convert the fractions to a similar form that makes them addable!

Simplifying Fractions

The beauty of fractions is best seen when they are in their simplest form. A fraction is simplified when its numerator and denominator are as small as possible, meaning they are co-prime (they have no common factor other than 1).

After adding fractions with a common denominator, it's essential to check if the resulting fraction can be simplified:

• First, look for the greatest common divisor (GCD) of the numerator and the denominator.
• If the GCD is greater than 1, divide both the numerator and the denominator by this number to simplify.
• If the GCD is 1, the fraction is already as simple as it can get.

In our worked example, the fraction \( \frac{14}{15} \) is the sum. Since 14 and 15 have no common divisors other than 1, \( \frac{14}{15} \) is already in its simplest form. This means you can be confident that you've found the most concise expression for your result!