Problem 19 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{4 n+3}{8}+\frac{6 n+5}{8}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{5n + 4}{4} \).

Step 1: Identify Like Denominators

Notice that both fractions, \( \frac{4n+3}{8} \) and \( \frac{6n+5}{8} \), have the same denominator, which is 8. This means you can directly add the numerators.

Step 2: Add the Numerators

Add the numerators of the fractions while keeping the denominator the same: \[(4n + 3) + (6n + 5) = 4n + 6n + 3 + 5\]This simplifies to:\[10n + 8\]

Step 3: Simplify the Fraction

Now that the numerators are added, place the result over the common denominator:\[\frac{10n + 8}{8}\]Check if this fraction can be simplified by dividing each term in the numerator by their greatest common factor, which in this case is 2:\[\frac{10n + 8}{8} = \frac{2(5n + 4)}{8} = \frac{5n + 4}{4}\]

Step 4: Final Expression

The simplified form of the expression is \( \frac{5n + 4}{4} \).

Key Concepts

Simplifying Fractions

Simplifying fractions means making them as simple as possible. It involves reducing the fraction so that the numerator and the denominator have no common factors other than 1. When a fraction is in its simplest form, it can't be made smaller without changing its value. To simplify a fraction, find the greatest common factor (GCF) of both the numerator and the denominator and divide them by this number.

• In our case, we have the fraction \( \frac{10n + 8}{8} \).
• Calculate the GCF. Here, the greatest number that divides both 10 and 8 is 2.
• Divide both the numerator and the denominator by this GCF.
• This gives us \( \frac{5n + 4}{4} \), the simplest form of the original expression.

Always ensure your final answer is simplified, especially in exams or any kind of competition. Simplified expressions are easier to read and work with.

Common Denominator

Understanding common denominators is crucial when adding or subtracting fractions. A common denominator is a shared multiple of the denominators of two or more fractions. When fractions have the same denominator, they are known as like fractions and can be easily added or subtracted by working with the numerators.

In the problem, both fractions \( \frac{4n+3}{8} \) and \( \frac{6n+5}{8} \) share a common denominator of 8. This allows us to seamlessly add the numerators:

• The same denominator (8) remains in the answer.
• Only the numerators change when we perform operations on like fractions.

It’s much easier to work with a common denominator, often eliminating complex calculations when the same numerators are involved. Always check for a common denominator first.

Numerator

The numerator is the top number in a fraction. It represents how many parts of the whole are considered. In the given problem, we work with the numerators to perform addition or subtraction.

### Steps to Simplify with the Numerator1. **Identify the Numerators**: The problem involves two numerators, \(4n + 3\) and \(6n + 5\).2. **Perform Operations**: Add these numerators together: - \((4n + 3) + (6n + 5) = 4n + 6n + 3 + 5\) - The simplified expression becomes \(10n + 8\).3. **Place over Common Denominator**: Once combined, this simplified numerator is placed over the common denominator, providing \(\frac{10n + 8}{8}\).
Always manipulate the numerators appropriately. This simplifies the overall task of managing fractions significantly and aids in clear and precise solutions.