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Chapter 2: Problem 8

Write each ratio in simplest form. \(\frac{18}{27}\)

Short Answer

Expert verified
The simplest form of \( \frac{18}{27} \) is \( \frac{2}{3} \).

Step by step solution

01

Identify Common Factors

To simplify the given ratio \( \frac{18}{27} \), first identify the greatest common factor (GCF) of the numerator (18) and the denominator (27). Both 18 and 27 have common factors, including 1, 3, and 9.
02

Find the Greatest Common Factor

Determine the greatest of these factors, which is 9. This is the largest number that divides both 18 and 27 without leaving a remainder.
03

Divide by the Greatest Common Factor

Divide both the numerator and the denominator by the GCF (9). Therefore, \( \frac{18}{27} \) becomes \( \frac{18 \div 9}{27 \div 9} = \frac{2}{3} \).
04

Confirm the Simplification

Check that the new fraction \( \frac{2}{3} \) is in its simplest form. Since 2 and 3 have no common factors other than 1, \( \frac{2}{3} \) is indeed the simplest form.

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Key Concepts

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

Greatest Common Factor
When simplifying ratios or fractions, finding the greatest common factor (GCF) is a key step. The GCF is the largest number that can divide both the numerator and the denominator without any remainder.

For example, if we have 18 and 27, we look for numbers that divide both evenly. The factors of 18 are {1, 2, 3, 6, 9, 18}, and the factors of 27 are {1, 3, 9, 27}. The largest factor common to both sets is 9. This means 9 is the GCF.

To simplify a ratio or a fraction, divide both the top and bottom numbers by their GCF. This reduces the numbers to their smallest possible form while maintaining the same value.
Numerator and Denominator
In fractions and ratios, numbers are written with one number over the other. These are called the numerator and the denominator.

  • The numerator is the top number. It represents how many parts of the whole are being considered. For the ratio \( \frac{18}{27} \), 18 is the numerator.
  • The denominator is the bottom number. It shows the total number of equal parts the whole is divided into. In our example, 27 is the denominator.
Understanding these roles is crucial since simplifying involves changing these numbers without altering the overall relationship. The numerator and denominator are reduced by dividing using their greatest common factor.
Simplest Form
A ratio or fraction reaches its simplest form when the numerator and the denominator have no common factors other than 1. This means they cannot be divided further by any other numbers but 1.

To express \( \frac{18}{27} \) in simplest form, we identified the GCF as 9. Then we divided both the numerator and the denominator by 9:

\[ \frac{18 \div 9}{27 \div 9} = \frac{2}{3} \]

Checking the simplified fraction \( \frac{2}{3} \) confirms it's in simplest form because the numbers 2 and 3 have no shared factors apart from 1. Ensuring a fraction is in simplest form makes it easier to understand and use in calculations.

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Most popular questions from this chapter

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{2 a}{a+2} \cdot \frac{a^{2}-4}{4 a^{2}} \div \frac{a-2}{a} $$

In \(21-24,\) the length and width of a rectangle are expressed in terms of a variable. a. Express each perimeter in terms of the variable. b. Express each area in terms of the variable. $$ l=\frac{x}{x+1} \text { and } w=\frac{x}{x+2} $$

Simplify each complex rational expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{3-\frac{3}{b}}{b-1}\)

In \(3-20,\) solve each equation and check. $$ 0.2 a=0.05 a+3 $$

In \(13-24,\) divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{c^{2}-6 c+9}{5 c-15} \div \frac{c-3}{5} $$
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