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Chapter 1: Problem 278

In the following exercises, write each decimal as a fraction. $$ -0.375 $$

Short Answer

Expert verified
``-0.375 = -3/8``.

Step by step solution

01

- Identify the place value

Recognize the place value of the last digit in the decimal. For ``-0.375`, the last digit (5) is in the thousandths place (0.001).
02

- Write the decimal Over 1

Express the decimal as ``\(-0.375/1\) and understand that the decimal must be converted to a fraction.
03

- Multiply to Eliminate the Decimal

Multiply the numerator and the denominator by 1000 to move the decimal to the right three places: ``\(-0.375/1 \times 1000/1000 = -375/1000\)
04

- Simplify the Fraction

Determine the greatest common divisor (GCD) of 375 and 1000, which is 125. Divide both the numerator and denominator by their GCD: ``\(-375/1000 \div 125/125 = -3/8\).
05

- Write the Final Answer

The simplified fraction is ``\(-3/8\)

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Key Concepts

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

Place Value
Place value is crucial when converting decimals to fractions. It helps us understand the position of each digit in a number. For instance, in the decimal \(-0.375\), the digit 5 is in the thousandths place, meaning it represents \(0.001\). The place value tells us how to express the decimal as a fraction over 1. In this case, \(-0.375\) can be written as \(-0.375/1\). Recognizing place value helps lay the foundation for accurately converting and simplifying decimals to fractions.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This process involves dividing the numerator and the denominator by their greatest common divisor (GCD). For instance, the fraction \(-375/1000\) can be simplified by first identifying their GCD, which is 125. By dividing both the numerator and the denominator by 125, you get: \(-375/1000 \div 125/125 = -3/8\). This simplified form is easier to understand and often required in math assignments and real-world applications.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator without leaving a remainder. Finding the GCD is essential for simplifying fractions. For example, in converting \(-0.375\) to a fraction, we found \(-375/1000\). Identifying that 125 is the GCD of 375 and 1000 allows us to simplify the fraction to \(-3/8\). To find the GCD, you can use the prime factorization method or the Euclidean algorithm. Understanding how to find and use the GCD is a fundamental skill in reducing fractions to their simplest forms.

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