Problem 29 - Algebra and Trigonometry

Question

Perform the indicated operations. \(\frac{3}{10}+\frac{4}{15}\)

Step-by-Step Solution

Answer

The sum is \(\frac{17}{30}\).

Step 1: Find a Common Denominator

To add fractions, we need a common denominator. For \(\frac{3}{10}\) and \(\frac{4}{15}\), the least common multiple (LCM) of 10 and 15 is 30.

Step 2: Convert Fractions to Equivalent Fractions

Convert each fraction to have the common denominator of 30. \(\frac{3}{10}\) becomes \(\frac{9}{30}\) (since \(3 \times 3 = 9\)) and \(\frac{4}{15}\) becomes \(\frac{8}{30}\) (since \(4 \times 2 = 8\)).

Step 3: Add the Fractions

Now add the fractions with the common denominator: \(\frac{9}{30} + \frac{8}{30} = \frac{17}{30}\).

Step 4: Simplify if Necessary

Check if \(\frac{17}{30}\) can be simplified. Since 17 is a prime number and doesn't divide 30, the fraction is already in simplest form.

Key Concepts

Understanding the Least Common Multiple

When adding fractions, it's essential to have the same denominator for each fraction before you can add them together. This is where the concept of the Least Common Multiple (LCM) becomes important. The LCM of two numbers is the smallest number that is a multiple of both numbers. For example:

• To find the LCM of 10 and 15, list the multiples: 10 (10, 20, 30, ...) and 15 (15, 30, 45, ...).
• The smallest multiple they have in common is 30.

Finding the LCM is valuable because it allows us to rewrite the fractions being added so that they have the same denominators, making addition straightforward. Without the LCM, adding fractions becomes a cumbersome process.

The Role of Equivalent Fractions

Equivalent fractions are fractions that have different numerators and denominators but represent the same value. When you find a common denominator for adding or subtracting fractions, you create equivalent fractions. Here's how it works:

• For the fraction \(\frac{3}{10}\), to change its denominator to 30 (our LCM), you multiply both the numerator and the denominator by 3, getting \(\frac{9}{30}\).
• Similarly, for \(\frac{4}{15}\), you multiply both the numerator and the denominator by 2 to switch to the LCM denominator, resulting in \(\frac{8}{30}\).

Both of these newly formed fractions, \(\frac{9}{30}\) and \(\frac{8}{30}\), look different from \(\frac{3}{10}\) and \(\frac{4}{15}\), but they are equivalent in value to their original fractions. Creating equivalent fractions is a crucial step in the process of adding or subtracting fractions.

Simplifying Fractions to their Simplest Form

After adding fractions, the resulting fraction may often need to be simplified, although sometimes it is already in its simplest form. Simplifying a fraction means breaking it down into the smallest possible whole numbers for its numerator and denominator, ensuring the numbers are co-prime (having no common factors other than 1). Consider:

• If you have \(\frac{17}{30}\), you would check whether 17 and 30 have any common factors. Here, 17 is a prime number.
• Since 17 does not evenly divide into 30 or share any common factors with 30, \(\frac{17}{30}\) is already in its simplest form.

Simplifying fractions ensures that they are easy to read and understand, and can also make mathematical calculations, like further operations, more straightforward.