Problem 42 - Elementary Algebra Within Reach

Question

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ \frac{5 x}{8} \cdot \frac{16}{5} $$

Step-by-Step Solution

Answer

The simplified form of \(\frac{5x}{8} \cdot \frac{16}{5}\) is \(2x\).

Step 1: Identify the fractions

The given math problem consists of two fractions, which are \(\frac{5x}{8}\) and \(\frac{16}{5}\) to be multiplied

Step 2: Multiply Numerators and Denominators

Multiply the two numerators together and the two denominators together separately. This results in \(\frac{(5x * 16)}{(8 * 5)}\) which simplifies to \(\frac{80x}{40}\)

Step 3: Simplify the Fraction

Now, law of cancellation can be applied to simplify the fraction. Both, 80 and 40, have the common factor of 40 which can be divided out of the numerator and the denominator. That simplification results in the simplest fraction form: \(\frac{2x}{1}\)

Step 4: Final Simplification

A fraction with 1 as the denominator is simply the numerator itself. Thus, \(\frac{2x}{1}\) will be simplified to \(2x\)

Key Concepts

Multiplication of Fractions

Understanding multiplication of fractions is essential in algebra and calculus. It allows for the combination of fractional values in calculations. When you multiply fractions, you perform two main actions:

• Multiply the numerators (the top numbers of the fractions) together.
• Multiply the denominators (the bottom numbers of the fractions) together.

By performing these actions, you effectively "scale" one fraction by the other, combining both into one.Let's consider the example from the exercise: \[ \frac{5x}{8} \cdot \frac{16}{5} \]This operation involves multiplying the numerators \(5x\) and \(16\) to get \(80x\) and the denominators \(8\) and \(5\) to get \(40\). Hence, the result is \[ \frac{80x}{40} \].

This result can be further simplified, which we'll explore next, but remember: to multiply fractions, just perform these two steps of multiplying across the numerators and denominators.

Fraction Simplification

Once you have multiplied the fractions, the next step is simplification. Simplifying a fraction means reducing it to its simplest form, so it is easier to understand and use in further calculations.

To simplify a fraction, you need to find the greatest common factor (GCF) of the numerator and the denominator, and divide both by this number. Let's look at our previous result:\[\frac{80x}{40}\]Both 80 and 40 share a common factor of 40. By dividing each by 40, you get:\[\frac{80x \div 40}{40 \div 40} = \frac{2x}{1}\]Simplifying fractions also involves recognizing when a fraction can no longer be reduced without changing its value. Here, the fraction reduces to \(2x\) since any fraction with a denominator of 1 is simply the numerator itself.

This simplification is crucial in making expressions easier to handle.

Algebraic Fractions

Algebraic fractions are fractions where the numerator or the denominator includes a variable (like \(x\) in our example). Understanding how to work with these is vital in algebra, as they frequently appear in equations and expressions.Dealing with algebraic fractions is much like dealing with numerical fractions, but requires a few additional considerations:

• Always check for opportunities to simplify both before and after performing operations like multiplication.
• Be mindful of variables in the expressions and ensure that any simplifications or operations don't end up altering the variable's value.

In \[\frac{5x}{8} \cdot \frac{16}{5}\]we see the variable \(x\), and while simplifying, it's crucial to remember that \(2x\) is an equivalent, but more manageable form of the expression \(\frac{2x}{1}\).Recognizing these principles will aid in solving complex algebraic problems elegantly and accurately. Keep these tips in mind, and you’ll find algebraic fractions much less daunting.