Problem 24 - College Algebra

Question

Multiply or divide as indicated. $$ \frac{x+5}{7} \div \frac{4 x+20}{9} $$

Step-by-Step Solution

Answer

The answer is \( \frac{9}{28} \).

Step 1: Rewrite Division as Multiplication

Division by a fraction is the same as multiplication by its reciprocal. Reciprocal of \( \frac{4x+20}{9} \) is \( \frac{9}{4x+20} \). So, the original expression can be rewritten as \( \frac{x+5}{7} \times \frac{9}{4x+20} \).

Step 2: Simplify Factors

The numerator in second fraction can be simplified. \( 4x + 20 \) is same as \( 4(x + 5) \). So, our expression now is \( \frac{x+5}{7} \times \frac{9}{4(x+5)} \).

Step 3: Cancel Common Factors

The term \( x+5 \) in the numerator of the first fraction and the denominator of the second fraction can be canceled out. This leads to \( \frac{1}{7} \times \frac{9}{4} \).

Step 4: Perform Multiplication

Now, just multiply the numerators together and the denominators together to get \( \frac{1 \times 9}{7 \times 4} \) which simplifies to \( \frac{9}{28} \).

Key Concepts

Multiplication by Reciprocal

When it comes to dividing fractions, one common method is multiplication by the reciprocal. Essentially, this means that instead of dividing by a fraction, you multiply by its opposite—the reciprocal. The reciprocal of a fraction flips the numerator and the denominator, so the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). For instance, if you are given \( \frac{x+5}{7} \div \frac{4x+20}{9} \), you multiply \( \frac{x+5}{7} \) by the reciprocal of \( \frac{4x+20}{9} \) which is \( \frac{9}{4x+20} \).
By converting division into multiplication in this way, the operation becomes straightforward since multiplying fractions merely involves multiplying the numerators together and the denominators together.

Simplifying Expressions

Once you've rewritten the problem using multiplication by reciprocal, the next step is simplifying expressions. This process often involves breaking down algebraic expressions to their simplest form. In our example, we notice that \(4x + 20\) in the second fraction can be expressed as \(4(x + 5)\), since \(20\) is \(4 \times 5\). Simplifying helps to reveal factors that may cancel out during multiplication, making the expressions easier to work with. It's important to always look out for common factors like this, which take the form of numerical coefficients or algebraic terms.

Canceling Common Factors

With the expression simplified, the next technique is canceling common factors. This strategy involves identifying and eliminating factors that appear in both the numerator and denominator of a fraction.

In our case, after simplifying, we have \( \frac{x+5}{7} \times \frac{9}{4(x+5)} \). Here, \(x+5\) is both in the numerator of the first fraction and the denominator of the second, which means we can cancel it out. What remains is \( \frac{1}{7} \times \frac{9}{4} \). Canceling common factors is a powerful tool to reduce fractions and is especially handy when working with more complex algebraic expressions.

Rational Expressions

The entire process we have been discussing involves manipulating rational expressions. These are fractions where both the numerator and the denominator are polynomials. In algebra, rational expressions require special attention as they can often be simplified before performing operations like addition, subtraction, multiplication, and division.

In our example, the rational expressions become particularly manageable after canceling the common factor \(x+5\). It's crucial to work with expressions in their simplest form to avoid errors and to streamline calculations. Whether you're multiplying or dividing rational expressions, the process becomes much easier when you use these algebraic techniques.