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Chapter 2: Problem 65

For Problems \(65-74\), reduce each fraction to simplest form. $$ \frac{9 x y}{15 x} $$

Short Answer

Expert verified
\( \frac{3y}{5} \)

Step by step solution

01

Identify Common Factors

Identify the common factors in the numerator and the denominator. Here, the common factor is \(x\).
02

Cancel Common Factors

Cancel out the common factors from both the numerator and the denominator. This gives us: \[ \frac{9y}{15} \] after canceling \(x\).
03

Simplify the Remaining Fraction

Now, simplify the fraction \( \frac{9y}{15} \) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3. This gives \( \frac{3y}{5} \).

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Key Concepts

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

Common Factors
When working with fractions, it's crucial to identify common factors in the numerator and the denominator. A factor is a number or variable that divides another number or expression completely, without a remainder. For example, in the expression \( \frac{9xy}{15x} \), both the numerator, \(9xy\), and the denominator, \(15x\), share common factors. These include numbers or variables, and in this exercise, \(x\) is a common factor present in both.
  • First, break down the numerator and denominator into their individual components or factors.
  • Look for numbers or variables that appear in both the numerator and the denominator.
  • These shared elements are known as common factors.
Recognizing common factors is the first step in simplifying fractions because it allows you to cancel these factors out, leading to a simplified expression.
Greatest Common Divisor
Once common factors are identified and canceled out, the next step often involves simplifying the remaining fraction further. Here, the key tool is the Greatest Common Divisor (GCD). The GCD of two numbers is the largest number that can divide both numbers without leaving a remainder. To simplify, let's take the fraction \(\frac{9y}{15}\) that results from our previous cancellation:
  • List the divisors of the remaining numbers: for 9 (1, 3, 9) and for 15 (1, 3, 5, 15).
  • Identify the largest number present in both lists, which in this case is 3.
  • Divide both the numerator and the denominator by this GCD.
Using the GCD ensures that the fraction is reduced to its simplest form, which is its most concise and readable form.
Fraction Reduction
Fraction reduction is the process of simplifying a fraction to its simplest form by removing common factors and using the greatest common divisor. The goal is to make the fraction as simple as possible while keeping it equivalent to the original.After identifying and canceling out common factors, you focus on reducing what's left:
  • In the expression \(\frac{9y}{15}\), the simplified fraction after using the GCD of 3 gives us \(\frac{3y}{5}\).
  • This fraction cannot be simplified further as there are no more common factors between 3 and 5, and the variables are already separated.
  • The process is complete when no further simplification can be achieved.
Fraction reduction not only makes calculations easier but also aids in clearer visualization of the problem.

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Most popular questions from this chapter

Evaluate each algebraic expression for the given values of the variables. Don't forget that for some problems it might be helpful to combine similar terms first and then to evaluate. \(0.7 x+0.6 y \quad\) for \(x=-2\) and \(y=6\)

Do you think that \(2 \sqrt{2}\) is a rational or an irrational number? Defend your answer.

Marcus is decorating his room and plans on hanging. three prints that are each \(13 \frac{3}{8}\) inches wide. He is goang to hang the prints side by side with \(2 \frac{1}{4}\) inches between the prints. What width of wall space is needed to display the three prints?

Evaluate each algebraic expression for the given values of the variables. Don't forget that for some problems it might be helpful to combine similar terms first and then to evaluate. \(\frac{3}{5} y-\frac{2}{3} y-\frac{7}{15} y \quad\) for \(y=-\frac{5}{2} \quad \frac{4}{3}\)

Simplify each algebraic expression by combining similar terms. $$6.2 n-7.8 n-1.3 n$$
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