Problem 33 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ -\frac{3}{8}+\frac{5}{8} $$

Step-by-Step Solution

Answer

The sum is \( \frac{1}{4} \).

Step 1: Identify Common Denominators

Both fractions have the same denominator of 8, which means these fractions can be added directly.

Step 2: Add the Numerators

Since the denominators are the same, you can simply add the numerators together while keeping the denominator constant: \(-\frac{3}{8} + \frac{5}{8} = \frac{-3+5}{8} \) which simplifies to \( \frac{2}{8} \).

Step 3: Simplify the Fraction

The fraction \( \frac{2}{8} \) can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator, which is 2.Divide both the numerator and denominator by their GCD:\(\frac{2\div2}{8\div2} = \frac{1}{4} \).

Key Concepts

Common Denominator

Adding fractions is straightforward when they have a common denominator, which is the shared bottom part of the fractions. A denominator indicates how many equal parts make up a whole. When two fractions have the same denominator, it means they can be combined directly, which simplifies the addition process.

• In our example, both fractions have a denominator of 8. This means they both represent parts of the same-sized whole. Therefore, you can add the numerators of the fractions directly, as they represent the same-sized parts.
• Not having to adjust the denominators simplifies the operation greatly, making it much easier to calculate the sum.

Remember, if fractions have different denominators, you need to find a common denominator before proceeding with the addition.

Simplifying Fractions

Once you've successfully added the numerators using a common denominator, the next step is often to simplify the resulting fraction. Simplification makes a fraction easier to understand and work with, essentially reducing it to its simplest form.

• This involves dividing both the numerator and the denominator by any common factors until no further simplification is possible.
• For instance, the fraction \( \frac{2}{8} \) resulted from adding our fractions, and it can be simplified by realizing both 2 and 8 share common factors.

Remember, a simplified fraction is easier to interpret and often more useful in further calculations. It represents the same value as the original, just using fewer, simpler numbers.

Greatest Common Divisor (GCD)

To simplify a fraction effectively, you need to identify the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the highest number that divides evenly into both the numerator and the denominator.

• In our example, the GCD of 2 and 8 is 2. By dividing both the numerator and the denominator by this number, you reduce the fraction to its simplest form.
• This process involves checking for divisibility by small numbers, usually starting with the smallest primes like 2, 3, or 5.

Once you divide both the numerator and the denominator by the GCD, you reach the simplest version of the fraction. This can make handling calculations or comparing amounts easier.