Problem 16 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{-\frac{7}{8 y}}{\frac{21}{4 y}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{-1}{6} \).

Step 1: Understand the Complex Fraction

The given expression is a complex fraction, which is a fraction where the numerator and/or the denominator are also fractions. Our task is to simplify the expression \( \frac{-\frac{7}{8y}}{\frac{21}{4y}} \).

Step 2: Rewrite the Division

A complex fraction can be rewritten as a division of the numerator by the denominator. This means that we can rewrite the expression as: \( -\frac{7}{8y} \div \frac{21}{4y} \).

Step 3: Apply Invert and Multiply

To simplify \( -\frac{7}{8y} \div \frac{21}{4y} \), we convert the division into multiplication by inverting the second fraction (denominator) and multiplying. So it becomes: \( -\frac{7}{8y} \times \frac{4y}{21} \).

Step 4: Simplify the Expression

Start by multiplying the numerators together and the denominators together: \( \frac{-7 \times 4y}{8y \times 21} = \frac{-28y}{168y} \). Next, simplify this fraction.

Step 5: Cancel Common Terms

Notice that \( y \) is a common factor in the numerator and the denominator, so you can cancel it out: \( \frac{-28}{168} \).

Step 6: Simplify the Resulting Fraction

To simplify \( \frac{-28}{168} \), find the greatest common divisor (GCD) of 28 and 168, which is 28. Divide both the numerator and the denominator by 28: \( \frac{-28 \div 28}{168 \div 28} = \frac{-1}{6} \).

Key Concepts

Simplify Fractions

When dealing with complex fractions, the goal is to make them simpler and easier to work with. A complex fraction is essentially a fraction where either or both the numerator and the denominator are themselves fractions. To simplify these, it often helps to rewrite the expression so it's not so overwhelming.

Let's take a close look at the given problem. We have \[\frac{-\frac{7}{8y}}{\frac{21}{4y}}\]Rewriting this complex fraction as a division directly reflects the operation: \[-\frac{7}{8y} \div \frac{21}{4y}\]

This step is crucial because it sets up our ability to simply "invert and multiply," making the expression much easier to tackle. Remember, rethinking the structure of complex fractions opens the door to simple arithmetic that’s much easier to handle than the intimidating look of nested fractions.

Invert and Multiply

In algebra, when you divide by a fraction, you actually "invert and multiply." This means flipping the second fraction (denominator) upside down and changing the operation to multiplication.

In the original exercise, we rewrote the complex fraction as follows:\[-\frac{7}{8y} \div \frac{21}{4y}\]

This division can then be transformed:\[-\frac{7}{8y} \times \frac{4y}{21}\]

Now, multiplication is generally simpler to handle because you just multiply the numerators together and the denominators together. This straightforward action makes even seemingly complex operations clear. This is a vital strategy for simplifying complex fractions, as it breaks down a complex operation into a series of simpler ones. So remember, when you see a division with fractions, think "invert and multiply!" as your go-to method.

Greatest Common Divisor (GCD)

Finding the Greatest Common Divisor (GCD) is an important step in simplifying fractions further. This is because the GCD helps you identify how you can scale down your numbers to the simplest possible terms.

In our scenario, after canceling out common factors like \( y \), we ended up with:\[\frac{-28}{168}\]

To find the GCD, look at both numbers and determine the largest number that divides both exactly. Here, 28 is the GCD of both the numerator and denominator. Simplifying by this common divisor, you divide:

• \( -28 \div 28 = -1 \)
• \( 168 \div 28 = 6 \)

This gives you the fraction:\[\frac{-1}{6}\]

Using the GCD not only makes fractions smaller but also makes subsequent calculations and comparisons easier. This technique is essential for simplifying results to their most understandable and representative form.

Canceling Common Factors

Canceling out common factors is a handy trick for making fractions simpler. By identifying and eliminating unnecessary factors that appear in both the numerator and the denominator, you genuinely simplify your expression.

While solving our exercise, we reached the expression:\[\frac{-28y}{168y}\]

Here, \( y \) appears in both the numerator and the denominator, so you simply "cancel" it:\[\frac{-28}{168}\]

This drastically simplifies the expression by ridding it of "extra" components that do not affect the value, akin to peeling layers off something to get to its core. Always look for these opportunities when simplifying fractions because they can drastically reduce calculation complexity, transforming an initially complex problem into an easily manageable solution.