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Chapter 7: Problem 3

In Exercises 3-8, find the sum or difference. \(\frac{15}{4 x}+\frac{5}{4 x}\)

Short Answer

Expert verified
\(\frac{5}{x}\)

Step by step solution

01

Identify the Fraction Type

Both of the fractions are of the type \(\frac{a}{b}\) where \(a\) is the numerator and \(b\) is the denominator. Since both of the fractions have the same denominator which is \(4x\), it makes this addition problem simpler.
02

Add the Fractions

When the denominators are the same, you can add the numerators directly. So, we add the numerators \(15+5\) and obtain \(20\). The denominator remains as \(4x\). So the sum of the given fractions is \(\frac{20}{4x}\).
03

Simplify the Fraction

The obtained fraction \(\frac{20}{4x}\) can be simplified by dividing the numerator and the denominator by \(4\). After simplification, the fraction becomes \(\frac{5}{x}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerator and Denominator
Understanding the terms numerator and denominator is crucial when working with fractions. The fraction \( \frac{a}{b} \) is made up of two main parts:
  • The numerator is the number above the fraction line. It indicates how many parts of a whole are being considered. In our exercise, the numerators are 15 and 5.

  • The denominator is the number below the fraction line. It shows into how many parts the whole is divided. Both fractions share a denominator in the exercise, namely \(4x\).
A strong understanding of these components is indispensable when performing operations like addition or subtraction of fractions.
By focusing on the numerators, particularly when denominators are the same, you can easily find the sum or difference of fractions.
Simplifying Fractions
Simplifying fractions makes them easier to understand and use. Simplicity involves reducing the fraction to its smallest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).
This doesn't change the value of the fraction, just its appearance, making it neater. In our example, after adding the numerators, we arrive at \(\frac{20}{4x}\), which needs simplifying. The steps to simplify:
  • Find the GCD of the numerator and denominator. For 20 and 4, the GCD is 4.

  • Divide both the numerator and the denominator by the GCD. Dividing 20 and 4 by 4 gives us the simplified fraction \(\frac{5}{x}\).
Simplifying fractions helps in clearer communication of quantities and often makes further calculations easier.
Common Denominator
A common denominator is key when adding or subtracting fractions. It refers to a shared divisor among different fractions that we use to make calculations straightforward.
When fractions have a common denominator, you only need to add or subtract the numerators, while the denominator stays the same.In the exercise, both fractions had \(4x\) as their denominator, making them easier to work with. This commonality eliminates the need for further adjustments or finding equivalent fractions, streamlining the process:
  • With denominators aligned, directly combine or differentiate the numerators. Here, adding 15 and 5 gives 20.

  • The denominator, \(4x\), isn't manipulated during this addition step.
A shared denominator is essential for fluid and efficient fraction operations.

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Most popular questions from this chapter

Find the sum or difference. \(\frac{4 x^2}{2 x-1}-\frac{1}{2 x-1}\)

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{15}{x} $$

In Exercises 25鈥�32, graph the function. State the domain and range. $$ y=\frac{-2 x+3}{-x+10} $$

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