Problem 38 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Example 4. $$ \frac{\frac{3}{4}+\frac{1}{y}}{\frac{5}{6}-\frac{1}{y}} $$

Step-by-Step Solution

Answer

The simplified complex fraction is \( \frac{6(3y+4)}{4(5y-6)} \).

Step 1: Combine Numerator Fractions

To simplify the complex fraction, start by combining the fractions in the numerator:\[ \frac{3}{4} + \frac{1}{y} = \frac{3y}{4y} + \frac{4}{4y} = \frac{3y + 4}{4y} \]

Step 2: Combine Denominator Fractions

Next, combine the fractions in the denominator:\[ \frac{5}{6} - \frac{1}{y} = \frac{5y}{6y} - \frac{6}{6y} = \frac{5y - 6}{6y} \]

Step 3: Rewrite the Complex Fraction

Now rewrite the original complex fraction using these new expressions:\[ \frac{\frac{3y+4}{4y}}{\frac{5y-6}{6y}} \]

Step 4: Multiply by the Reciprocal

To simplify, multiply by the reciprocal of the denominator fraction:\[ \frac{3y+4}{4y} \times \frac{6y}{5y-6} = \frac{(3y+4) \cdot 6y}{4y \cdot (5y-6)} \]

Step 5: Cancel Common Factors

Cancel out the common factor \( y \) in the fraction:\[ \frac{(3y+4) \cdot 6}{4 \cdot (5y-6)} \] There are no more common factors remaining to cancel.

Key Concepts

Simplifying Fractions

The journey to simplifying fractions begins by understanding what it really means. A simplified fraction is one where the numerator and denominator are reduced to their smallest possible values while maintaining the same value. To do this, we identify any common factors that the numerator and the denominator have and cancel them out. For instance, if both parts of a fraction are divisible by 2, you divide both by 2. This will not change the value of the fraction, just how it looks.

This is crucial in simplifying complex fractions, as it will result in a simpler form that is easier to understand and work with. Always remember, simplifying is not just reducing numbers but also expressing the fraction in its most efficient and clear form possible, without losing its original value.

Combining Fractions

Combining fractions is an essential step, especially when dealing with complex fractions. This often involves finding a common denominator.

For example, if you have fractions like \(\frac{3}{4}\) and \(\frac{1}{y}\), you must find a common denominator before you can properly add (or subtract) them. Here, your common denominator would be \(4y\). This allows you to rewrite the fractions as \(\frac{3y}{4y}\) and \(\frac{4}{4y}\), and then add them to get \(\frac{3y + 4}{4y}\).

The same approach applies when subtracting fractions in the denominator. Combining fractions requires careful attention to both numerators and denominators to ensure accuracy and proportionality.

Multiplying by the Reciprocal

In a complex fraction, after rewriting with combined numerators and denominators, the next important step is multiplying by the reciprocal of the denominator. The reciprocal, simply put, is what you get when you flip both the numerator and the denominator of a fraction.

For example, if your complex fraction denominator is \(\frac{5y - 6}{6y}\), the reciprocal will be \(\frac{6y}{5y - 6}\). You multiply this reciprocal with the combined numerator fraction to get rid of the fraction in the denominator. This simplifies the expression into a standard fraction that can then be further simplified. Hence, multiplying by the reciprocal helps in translating a complex fraction into a simpler, more manageable form.

Canceling Common Factors

Once you have multiplied by the reciprocal, you will often need to cancel common factors from the numerator and the denominator of the resulting fraction. This is an indispensable step towards simplifying the fraction into its lowest terms.

Imagine you end up with a fraction like \(\frac{(3y+4) \cdot 6y}{4y \cdot (5y-6)}\). Notice that \(y\) is a common factor in both the numerator and the denominator. By canceling this, you simplify your fraction further into \(\frac{(3y+4) \cdot 6}{4 \cdot (5y-6)}\).

Beware, ensuring there are no more cancelable factors is important. Neglecting to cancel can leave fractions unnecessarily complicated, which might lead to errors and less efficient calculations. Therefore, be thorough in this step.