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Chapter 9: Problem 25

Reduce each of the following fractions as completely as possible. $$\frac{5 y}{10 y+20}$$

Short Answer

Expert verified
\(\frac{y}{2(y+2)}\)

Step by step solution

01

Identify the common factor

To simplify the fraction, we first need to identify the common factor in the numerator and the denominator. We see that in the fraction \(\frac{5y}{10y + 20}\), the numerator is \(5y\) and the denominator is \(10y + 20\).
02

Factor the denominator

Next, factor the denominator. \[10y + 20 = 10(y + 2)\]Now the fraction is \(\frac{5y}{10(y+2)}\).
03

Simplify by canceling the common factor

The common factor in the numerator and the denominator is 5. Cancel out the common factor to simplify the fraction.\[\frac{5y}{10(y+2)} = \frac{y}{2(y+2)}\]

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Key Concepts

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

Common Factor
When simplifying fractions, a crucial first step is to identify the common factor. A common factor is a number or variable that evenly divides both the numerator and the denominator.
For the given fraction \(\frac{5y}{10y+20}\), observe the terms: the numerator is \(\text{5y}\) and the denominator is \(10y + 20\).
To simplify the fraction, we look for any common factors between \(5y\) and \(10y + 20\). Once identified, we use these common factors to reduce the fraction.
In this case, both \(5y\) and \(10y + 20\) have the common factor of 5. This insight will allow us to simplify the fraction.
Numerator
The numerator is the top part of a fraction. It indicates how many parts of the whole are being considered. In the fraction \(\frac{5y}{10y+20}\), the numerator is \(\text{5y}\).
The numerator plays a vital role in determining the value of the fraction. When simplifying fractions, we often look for factors that the numerator and denominator share.
By factoring the numerator, we can make the simplification process more straightforward.
Denominator
The denominator is the bottom part of a fraction. It shows how many equal parts the whole is divided into. In the fraction \(\frac{5y}{10y+20}\), the denominator is \(10y + 20\).
Simplifying fractions often involves factoring the denominator to find common factors with the numerator.
For instance, we can rewrite \(10y + 20\) as \(10(y+2)\). This factoring step is necessary to identify shared factors, making it easier to simplify the fraction.
Factoring
Factoring means breaking down an expression into a product of simpler elements. This technique is essential for simplifying fractions.
For our example fraction \(\frac{5y}{10y+20}\), let's factor the denominator: \(10y + 20\) factors into \(10(y + 2)\).
Factoring reveals common factors which allow us to reduce the fraction. After factoring, the fraction becomes \(\frac{5y}{10(y+2)}\).
By canceling out the common factor of 5, we further reduce the fraction to \(\frac{y}{2(y+2)}\). This simpler form is the result of successful factoring and simplifying.

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Most popular questions from this chapter

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$3(m+2 p)=4(p-m) \text { for } m$$

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$4(x-y)-\frac{x-y}{2}=3-(y-x) \text { for } x$$

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In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$y=\frac{u+1}{u-1} \quad \text { for } u$$

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