Question

Simplify. Leave your answers as improper fractions. $$\frac{\frac{a b}{7}-3 d}{3 c-\frac{a b}{d}}$$

Short answer

\frac{d(a b - 21 d)}{7(3 c d - a b)}

Step-by-step solution

Find a common denominator

Search for a common denominator within each fraction to combine terms. In this case, the common denominator for the terms \(\frac{a b}{7}\) and \(3 d\) is \(7d\), and for the terms \(3 c\) and \(\frac{a b}{d}\) it is \(d\).

Combine terms over the common denominator

Rewrite each term over the common denominator so you can combine them easily. For the numerator, multiply \(3 d\) by \(\frac{7}{7}\) to get \(\frac{21 d}{7}\). For the denominator, multiply \(3 c\) by \(\frac{d}{d}\) to get \(\frac{3 c d}{d}\). Now the expression looks like \(\frac{\frac{a b}{7}-\frac{21 d}{7}}{\frac{3 c d}{d}-\frac{a b}{d}}\).

Simplify the numerator and denominator

Combine the terms in the numerator and denominator separately to get \(\frac{\frac{a b - 21 d}{7}}{\frac{3 c d - a b}{d}}\).

Multiply by the reciprocal of the denominator

To divide by a fraction, multiply by its reciprocal. Here, you multiply \(\frac{a b - 21 d}{7}\) by \(\frac{d}{3 c d - a b}\), resulting in \(\frac{(a b - 21 d) \cdot d}{7 \cdot (3 c d - a b)}\).

Simplify the expression

If any simplification is possible, do it now. In this case, there is no further simplification possible without specific values for \(a\), \(b\), \(c\), and \(d\), so the final answer is \(\frac{d(a b - 21 d)}{7(3 c d - a b)}\).

Key concepts

Common Denominator

Understanding the concept of a common denominator is key to simplifying complex fractions. A common denominator is a shared multiple of the denominators of two or more fractions. It's the 'common ground' that allows us to add, subtract, or compare fractions easily. Finding a common denominator involves looking for the least common multiple (LCM) of the original denominators. In our exercise, we searched for a common denominator for the fractions \(\frac{ab}{7}\) and \(3d\), and separately for \(3c\) and \(\frac{ab}{d}\). By identifying \(7d\) and \(d\) as common denominators, respectively, we can then rewrite the fractions to have the same denominator, making it possible to perform the necessary algebraic operations.

For example, if you have fractions with denominators 7 and d, the LCM is \(7d\), so to convert \(3d\) to have the common denominator, you would multiply both the numerator and denominator by 7, resulting in \(\frac{21d}{7d}\). This step simplifies the process of combining terms because it aligns the fractions on a common base.

Combining Terms

Once we have a common denominator, the next step is combining terms. This is much like gathering like terms in an algebraic expression. By expressing all terms with the common denominator, we can simply add or subtract the numerators to combine them. In our textbook exercise, after finding the common denominator, we combined the terms \(\frac{ab}{7}\) and \(\frac{21d}{7}\) by subtracting their numerators, resulting in a new numerator \(\frac{ab - 21d}{7}\). Similarly, for the denominator, we combined \(\frac{3cd}{d}\) and \(\frac{ab}{d}\) to get \(\frac{3cd - ab}{d}\).

It's important to keep in mind that we only combine the numerators, as the common denominator remains the same. The process is equivalent to adding or subtracting whole numbers, but instead of a singular value, we're dealing with expressions.

Reciprocal of a Fraction

The reciprocal of a fraction is simply a way to flip a fraction over. To find a reciprocal, you exchange the numerator and the denominator. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). The concept of reciprocal is fundamental when you divide by a fraction. This is because dividing by a fraction is equivalent to multiplying by its reciprocal. In the solution process for the textbook exercise, after simplifying both the numerator and the denominator separately, we find ourselves needing to divide one fraction by another. To do this, we multiply by the reciprocal. For example, dividing \(\frac{ab - 21d}{7}\) by \(\frac{3cd - ab}{d}\) means multiplying \(\frac{ab - 21d}{7}\) by the reciprocal of the denominator, which is \(\frac{d}{3cd - ab}\).

Improper Fractions

An improper fraction is a fraction where the numerator is equal to or greater than the denominator, meaning it could be expressed as a whole number with a fraction (or simply as a whole number in some cases). However, in many mathematical contexts, it’s useful to leave these as improper fractions for ease of calculation and for representing ratios or quotients exactly. In our practice problem, the end result is presented as an improper fraction, following the instruction that the answer should be left as such. This avoids any potential confusion that might arise from mixed numbers in further algebraic operations and provides a precise ratio of the quantities involved.

Working with improper fractions can be less intuitive than with proper fractions (where the numerator is less than the denominator), but they follow all the same rules. Being comfortable using and simplifying improper fractions is a valuable skill in algebra and higher mathematics.