Problem 84 - Algebra A Combined Function

Question

In your own words, describe how to add or subtract fractions.

Step-by-Step Solution

Answer

Find a common denominator, convert fractions, perform the operation, and simplify.

Step 1: Understand the Denominator

When adding or subtracting fractions, the first thing you need to do is check if the denominators (the bottom numbers of the fractions) are the same. If they are not the same, you'll need to convert them to a common denominator, which is a multiple of both denominators.

Step 2: Find a Common Denominator

If the fractions have different denominators, find the least common multiple (LCM) of the denominators. This will be the common denominator you use to perform the addition or subtraction.

Step 3: Convert Fractions

Convert each fraction to an equivalent fraction with the common denominator. To do this, divide the common denominator by the original denominator and multiply both the numerator and the denominator of the fraction by this quotient.

Step 4: Add or Subtract the Numerators

With the fractions now having the same denominators, add or subtract their numerators (the top numbers of the fractions) as the problem requires. Keep the common denominator the same.

Step 5: Simplify the Result

Once you have the result from step 4, simplify the fraction if possible. Divide the numerator and the denominator by their greatest common divisor (GCD) to simplify the fraction.

Key Concepts

Common Denominator

When adding or subtracting fractions, having a common denominator is crucial. A common denominator is a shared multiple of the denominators of the fractions you want to work with.
This allows the fractions to be ready for direct addition or subtraction.
Imagine you want to add \(\frac{1}{4}\) and \(\frac{1}{3}\). Both fractions have different denominators - 4 and 3 respectively.

• Without a common denominator, you cannot directly add these fractions.
• You need a denominator that both fractions can "agree" on.
• This shared denominator ensures continuity in the operation across all the fractions involved.

You achieve a common denominator by first finding the least common multiple (LCM) of the denominators. Once both fractions are expressed with this common denominator, only then can you proceed with adding or subtracting the numerators.

Least Common Multiple

Finding the least common multiple (LCM) is an important step when working with fractions. The LCM is the smallest number that is a multiple of two or more numbers. In the context of fractions, it helps to determine the smallest number that can be used as a common denominator for the fractions involved.
For instance, if you need to add \(\frac{2}{5}\) and \(\frac{3}{7}\):

• Identify the denominators: 5 and 7.
• List the multiples of each number: 5 (5, 10, 15, 20, 25, 30, 35, ...), 7 (7, 14, 21, 28, 35, ...).
• Find the smallest common multiple: In this case, 35.

Now, you use 35 as the new common denominator. Once this common denominator is established, each fraction can be converted to an equivalent fraction that can be easily added or subtracted.

Simplifying Fractions

After you've added or subtracted fractions, the resulting fraction may need to be simplified. Simplifying a fraction means reducing it to its simplest form, where the numerator and the denominator have no common factors other than 1.
To simplify:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

For example, consider the fraction \(\frac{8}{12}\). The GCD of 8 and 12 is 4. By dividing both the numerator and the denominator by 4, you simplify \(\frac{8}{12}\) to \(\frac{2}{3}\).
Simplifying fractions is essential for making the final fraction easier to work with and often provides a cleaner, more precise result.

Greatest Common Divisor

The greatest common divisor (GCD), also called the greatest common factor, is the largest number that divides two or more numbers without leaving a remainder. When simplifying fractions, the GCD helps to reduce the fraction to its simplest form. It's a crucial step to ensure the fraction remains equivalent to the original but is more manageable.
Finding the GCD involves identifying common factors.

• List the factors for both the numerator and denominator.
• Determine the largest factor they have in common.

For example, to simplify \(\frac{18}{24}\), find common factors:

• Factors of 18 are: 1, 2, 3, 6, 9, 18.
• Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.

Their greatest common factor is 6. Divide both 18 and 24 by 6 to simplify \(\frac{18}{24}\) to \(\frac{3}{4}\).
Understanding how to find and use the GCD ensures that your fractions are as simple as possible, making mathematical operations easier.