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Chapter 1: Problem 76

Write each expression in two equivalent forms, as in Example \(6 .\) $$ -\frac{4}{15} $$

Short Answer

Expert verified
Equivalent forms are \ -\frac{4}{15} \ and \ -0.2667 \.

Step by step solution

01

Understanding the Problem

The goal is to write the given expression \(-\frac{4}{15}\) in two equivalent forms. Equivalent forms are simply different representations of the same value.
02

Writing as a Negative Fraction

The given fraction is already in standard negative fraction form: \(-\frac{4}{15}\).
03

Write as an Opposite of a Fraction

Recognize that \(-\frac{4}{15}\) is the same as \frac{-4}{15}\. Both representations denote a negative value of \frac{4}{15}\.
04

Convert to Decimal Form

Another equivalent form is to convert the fraction to decimal. Divide \(-4\) by \15\: \-\frac{4}{15} = -0.2667\ (rounded to four decimal places).

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Key Concepts

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

Negative Fractions
Negative fractions may seem tricky, but they follow simple rules. A negative sign in front of a fraction indicates that the entire fraction鈥檚 value is negative. For example, \(-\frac{4}{15}\) means that we have a negative fraction. This fraction can also be written as \(\frac{-4}{15}\). Both \(\-\frac{4}{15}\) and \(\frac{-4}{15}\) indicate the same value: a negative portion of a whole.

Dealing with negative fractions often involves:
  • Placing the negative sign on the numerator: \(\frac{-a}{b}\)
  • Placing the negative sign on the denominator: \(\frac{a}{-b}\)
  • Placing the negative sign in front of the entire fraction: \(\-\frac{a}{b}\)

Remember, the definition doesn鈥檛 change. It just has multiple representations.
Decimal Conversion
Converting fractions to decimals can be essential for better understanding and comparison. To convert a fraction to a decimal, you divide the numerator by the denominator. For example, to convert \(\-\frac{4}{15}\) to a decimal:

  • Divide 4 by 15 to get approximately 0.2667.
  • Apply the negative sign, making it \-0.2667\.

So, \-0.2667\ is the decimal equivalent of \-\frac{4}{15}\.

Converting to decimal form can make certain operations like addition, subtraction, and comparison more manageable. Always keep in mind that the decimal must accurately represent the fraction's value, including any negative signs.
Fraction Representation
Fractions represent parts of a whole and can be shown in different forms. Understanding these forms is key to mastering fractions. The fraction \(\-\frac{4}{15}\) can be represented as:
  • \(\frac{-4}{15}\): Here, the negative sign is with the numerator.
  • \(\frac{4}{-15}\): Here, the negative sign is with the denominator.

Although these seem different, they represent the same value. Here鈥檚 why:
  • Placing the negative sign on the numerator or the denominator does not change the fraction's value.
  • The definition remains the same, showing that they all indicate -\frac{4}{15}: 4 parts out of 15, but negative.

Different representations can help visualize and solve problems more easily. Understanding this flexibility can improve your math skills significantly.

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Most popular questions from this chapter

Write an equivalent expression without using grouping symbols. $$ -\left(8 x^{3}-6 x+5\right) $$

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Tell whether each statement is true or false for all real numbers m and n. Use various replacements for m and n to support your answer. If \(m>n,\) then \(m-n>0\)

Simplify. $$ 5(2 x-7)-[4(2 x-3)+2] $$
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