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Chapter 5: Problem 71

Write each decimal as a fraction or mixed number in simplest form. $$-0.2$$

Short Answer

Expert verified
The decimal \(-0.2\) is \(-\frac{1}{5}\) as a fraction.

Step by step solution

01

Identify the Place Value

The decimal \(-0.2\) has one digit after the decimal point, which means it is in the tenths place. We can rewrite it as a fraction where the denominator is 10.
02

Write as Fraction

Now, write the decimal \(-0.2\) as the fraction \(-\frac{2}{10}\).
03

Simplify the Fraction

To simplify \(-\frac{2}{10}\), find the greatest common divisor (GCD) of 2 and 10, which is 2. Divide both the numerator and denominator by 2: \(-\frac{2}{10} = -\frac{1}{5}\).
04

Final Step: Check Simplicity

Ensure the fraction \(-\frac{1}{5}\) is in its simplest form. Since 1 and 5 have no common divisors other than 1, \(-\frac{1}{5}\) is fully simplified.

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Key Concepts

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

Simplifying Fractions
When you convert a decimal to a fraction, you aim to express it in the simplest form. This means reducing the fraction so that the numerator and denominator have no common factors other than 1. It makes the fraction easier to understand and work with. To simplify, follow these steps:
  • Find any common factors of the numerator and denominator.
  • Divide both by their greatest common divisor (GCD) to reduce the fraction.
  • Check if the fraction can be simplified further.
For example, in converting \( -\frac{2}{10} \) to \( -\frac{1}{5} \), you divide both 2 (numerator) and 10 (denominator) by their GCD, which is 2. This leaves you with \( -\frac{1}{5} \), which is already simplified.
Place Value
Understanding place value is crucial when converting a decimal to a fraction. Each digit in a decimal has a position, and its value depends on that position. For decimals:
  • The first place after the decimal is the tenths place.
  • The second place is the hundredths place.
So, for \( -0.2 \), the 2 is in the tenths position. This signifies that it is equal to \( -\frac{2}{10} \) when converted to a fraction. Knowing this allows you to represent any decimal as a fraction with appropriate denominator based on its place value.
Greatest Common Divisor (GCD)
Finding the greatest common divisor (GCD) is key to simplifying fractions. The GCD is the largest number that can divide both the numerator and the denominator without any remainder.
  • List the factors of both numbers.
  • Identify the largest factor that both share.
In our case, for the numbers 2 and 10, the factors of 2 are 1 and 2, and the factors of 10 are 1, 2, 5, and 10. The greatest common factor they both share is 2, hence, it is the GCD. Using the GCD lets you simplify fractions effectively, as seen with \( -\frac{2}{10} \) simplified to \( -\frac{1}{5} \).

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Most popular questions from this chapter

Find each product. Write in simplest form. $$-\frac{2}{9} \cdot-\frac{15}{16}$$

A newspaper is \(12 \frac{1}{4}\) inches wide and 22 inches long. This is \(1 \frac{1}{4}\) inches narrower and one-half inch longer than the old edition. What were the previous dimensions of the newspaper?

Choose two rational numbers whose product is a number between 0 and 1.

Find each sum. Write in simplest form. $$\frac{1}{10}+\frac{1}{3}$$

Find each product. Use an area model if necessary. $$-\frac{5}{12} \cdot 3 \frac{1}{9}$$
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