Problem 51 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{3 n-1}{3}+\frac{2 n+5}{4}$$

Step-by-Step Solution

Answer

\( \frac{18n + 11}{12} \)

Step 1: Find a Common Denominator

To add the fractions \( \frac{3n-1}{3} \) and \( \frac{2n+5}{4} \), we first need a common denominator. The denominators are 3 and 4, so their least common multiple is 12. We will convert each fraction to have a denominator of 12.

Step 2: Convert Fractions to Common Denominator

Convert \( \frac{3n-1}{3} \) to \( \frac{12}{12} \):\[ \frac{3n-1}{3} \times \frac{4}{4} = \frac{(4)(3n-1)}{12} = \frac{12n-4}{12} \]Convert \( \frac{2n+5}{4} \) to \( \frac{12}{12} \):\[ \frac{2n+5}{4} \times \frac{3}{3} = \frac{(3)(2n+5)}{12} = \frac{6n+15}{12} \]

Step 3: Add the Numerators

With the fractions now having a common denominator, add the numerators:\[ \frac{12n-4}{12} + \frac{6n+15}{12} = \frac{(12n-4) + (6n+15)}{12} \].

Step 4: Simplify the Expression

Combine like terms in the numerator:\[ (12n - 4) + (6n + 15) = 12n + 6n - 4 + 15 = 18n + 11 \]Thus,\[ \frac{18n + 11}{12} \] is our expression with the combined numerator over the common denominator.

Key Concepts

Common Denominator

When adding fractions, it's important that they have the same denominator. This is called the common denominator. Think of it like trying to add apples and oranges — you need a common language to combine them, like converting both into units of fruit.
To find this common denominator, you'll often need to use the least common multiple (LCM) of the denominators. This ensures that you only make minimal adjustments to the fractions while bringing them to a common base.
If you have denominators, like 3 and 4, you find their LCM. The smallest number that both 3 and 4 divide into evenly is 12.

• For 3, the multiples are 3, 6, 9, 12...
• For 4, the multiples are 4, 8, 12...

By aligning these fractions to have the common denominator of 12, you make it much easier to add or subtract them correctly.

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. It is very helpful when adding or subtracting fractions with different denominators.
To find the LCM of two numbers, it's useful to start listing the multiples of each number until you find the smallest one they both share. For instance, with the denominators 3 and 4, as seen before, their LCM turned out to be 12.
This process involves:

• Writing out the multiples of each number.
• Finding the smallest multiple they have in common.

Using the LCM helps in turning different fractions into versions that are easy to work with by giving them that much-needed common denominator.

Simplifying Expressions

Simplifying expressions is like cleaning up your room - making it tidy and more understandable. It involves removing any unnecessary numbers or terms while ensuring the expression remains equivalent.
In mathematical contexts, this often means combining like terms, eliminating complex fractions, and even factoring when necessary.
For example, combining terms like \( (12n - 4) + (6n + 15) \) involves:

• First, sum the 'n' terms: \( 12n + 6n = 18n \).
• Then, add the constants: \( -4 + 15 = 11 \).
• Put it all together into one neat expression: \( 18n + 11 \).

After simplification, what you have is a neater, easier-to-read format. This not only makes the expression easier to understand but also more convenient for further calculations or integration into larger mathematical problems.