Problem 63 - Prealgebra and Introductory Algebra

Question

Divide. $$\frac{28 x+14}{45 x-30} \div \frac{14 x+7}{30 x-20}$$

Step-by-Step Solution

Answer

The simplified answer to the given division of fractions problem is \( \frac{4}{3} \)

Step 1: Converting the Division as Multiplication

Convert the division into multiplication by taking the reciprocal of the second fraction. \( \frac{28 x+14}{45 x-30} \div \frac{14 x+7}{30 x-20} = \frac{28 x+14}{45 x-30} \times \frac{30 x-20}{14 x+7} \)

Step 2: Simplify

Simplify the fractions as much as possible. This can be done by factoring out common factors in the numerators and denominators and cancelling out. On simplification we get: \( \frac{2(14x+7)}{15(3x-2)} \times \frac{10(3x-2)}{(14x + 7)} \)

Step 3: Cancel out Common terms

Cancel out common terms that appear in both the numerator and denominator. In this case, \(14x+7\) and \(3x-2\) are present in both and hence can be cancelled out. So, it simplifies to: \( \frac{2}{15} \times \frac{10}{1} \)

Step 4: Multiply remaining fractions

Multiply the remaining fractions to find the final answer. \( \frac{2*10}{15*1} = \frac{20}{15} \)

Step 5: Further Simplify

The fraction \( \frac{20}{15} \) can still be simplified by dividing both numerator and denominator by their Greatest Common Divisor (GCD), which is 5. So, it becomes \( \frac{20/5}{15/5} = \frac{4}{3} \)

Key Concepts

Division of Rational Expressions

To divide rational expressions, you follow a process that is somewhat like dividing regular fractions. The key difference is that these involve polynomial expressions in their numerators and denominators. When dividing rational expressions, start by converting the division into multiplication. This is done by taking the reciprocal of the polynomial in the denominator of the division expression.
For example, if you have \( \frac{a}{b} \div \frac{c}{d} \), it is converted to \( \frac{a}{b} \times \frac{d}{c} \). Here, the second fraction, \( \frac{c}{d} \), is flipped to become \( \frac{d}{c} \).
This step simplifies the problem into a straightforward multiplication task, but remember: the order in which you perform operations like multiplication and cancellation hinges heavily on correctly flipping for multiplication.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing the expression to its simplest form. The first step is factoring both numerator and denominator to find and cancel out common factors.
For instance, if you have an expression like \( \frac{28x + 14}{45x - 30} \), notice anything common? Factor out the greatest common factors from each part:

• Numerator: \( 28x + 14 = 2(14x + 7) \)
• Denominator: \( 45x - 30 = 15(3x - 2) \)

You'll find that common factors are easier to spot once every term is properly factored. After factoring, cancel any common factors shared between the numerator and the denominator. This reduces the fraction and simplifies the expression.

Factoring Polynomials

Factoring is the key to simplifying algebraic expressions, especially polynomials. It involves breaking down the polynomial into a product of its simplest polynomials.
When dealing with expressions like \( 28x + 14 \) or \( 45x - 30 \), recognize patterns such as the common factors in each term.

• The expression \( 28x + 14 \) factors into \( 2(14x + 7) \), showing a common factor of 2.
• Similarly, \( 45x - 30 \) breaks down to \( 15(3x - 2) \), highlighting the factor of 15.

Breaking down polynomials into their factors not only simplifies the expression but also eases operations like addition, subtraction, and in this case, division of rational expressions. Always look for common terms, perfect squares, or grouping methods to achieve this effectively.