Problem 57 - Algebra 2

Question

Simplify each expression. $$ \frac{3 m+2}{m+n}+\frac{4}{2 m+2 n} $$

Step-by-Step Solution

Answer

\(\frac{3m+4}{m+n}\)

Step 1: Identify a common denominator

To add two fractions, both fractions need a common denominator. The denominators given are \(m+n\) and \(2m+2n\). Notice that \(2m+2n\) can be factored to \(2(m+n)\). Hence, the least common denominator (LCD) between \(m+n\) and \(2(m+n)\) is \(2(m+n)\).

Step 2: Adjust the first fraction to match the LCD

Multiply both the numerator and the denominator of the first fraction, \(\frac{3m+2}{m+n}\), by 2 to make the denominator \(2(m+n)\). This gives us the adjusted fraction: \(\frac{2(3m+2)}{2(m+n)} = \frac{6m+4}{2(m+n)}\).

Step 3: Combine the fractions

Now that both fractions \(\frac{6m+4}{2(m+n)}\) and \(\frac{4}{2(m+n)}\) have the same denominator, we can add them directly. Add the numerators: \((6m+4) + 4 = 6m+8\). Thus, the combined fraction is \(\frac{6m+8}{2(m+n)}\).

Step 4: Simplify the combined fraction

Factor the numerator \(6m + 8\). Notice \(6m + 8 = 2(3m + 4)\). Therefore, the fraction becomes \(\frac{2(3m+4)}{2(m+n)}\). Cancel out the common factor of 2 in the numerator and the denominator, resulting in the simplified fraction \(\frac{3m+4}{m+n}\).

Key Concepts

Common Denominators

When working with fractions, a common denominator is essential for adding or subtracting them. The denominator is the bottom part of a fraction, and it shows into how many equal parts the whole is divided. To easily add two fractions, they must share the same denominator.

In our exercise, the fractions have different denominators: \(m+n\) and \(2m+2n\). Instead of directly attempting to add these fractions, we need to find a common denominator that allows us to combine them seamlessly. This process aligns the denominators, preparing them for straightforward addition.

Factoring Polynomials

Factoring polynomials involves breaking down complex expressions into simpler multipliers that can be conveniently managed. Polynomials are algebraic expressions made up of variables and coefficients. To factor them means to express them as a product of simpler polynomials.

In the original problem, the denominator \(2m+2n\) can be factored. Observe that both terms in \(2m+2n\) are divisible by 2. Factor out the 2 to simplify the expression to \(2(m+n)\). Factoring is vital as it helps in simplifying the problem, allowing us to identify the least common denominator with ease.

Adding Fractions

Adding fractions is straightforward once both fractions have a common denominator. After aligning denominators, you can directly add the numerators, simplifying the addition process.

For example, once both fractions in our problem have the denominator \(2(m+n)\), we add the top parts (numerators) of these fractions. Initially, the first fraction is adjusted to \(\frac{6m+4}{2(m+n)}\), and the second fraction is already \(\frac{4}{2(m+n)}\). Adding these numerators yields \((6m+4) + 4 = 6m+8\). Thus, the resulting fraction is \(\frac{6m+8}{2(m+n)}\).

Least Common Denominator

The least common denominator (LCD) is the smallest common multiple shared by the denominators of multiple fractions. It is essential because it simplifies the process of adding or subtracting fractions by providing them a shared footing.

In our exercise, the denominators \(m+n\) and \(2(m+n)\) already suggest the LCD to be \(2(m+n)\). This is because \(2(m+n)\) is exactly the product of the factorization of the larger denominator and already accommodates \((m+n)\). By establishing this common base, you ensure that fractions are equivalent and ready for direct addition, leading directly to the simplified result.