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Chapter 5: Problem 31

In the following exercises, convert each decimal to a fraction or mixed number. $$0.239$$

Short Answer

Expert verified
0.239 as a fraction is \( \frac{239}{1000} \).

Step by step solution

01

Understanding Decimal Place Value

The number 0.239 is a decimal number where 2 is in the tenths place, 3 is in the hundredths place, and 9 is in the thousandths place.
02

Convert Decimal to Fraction

Rewrite 0.239 as a fraction: \(0.239 = \frac{239}{1000}\).
03

Simplify the Fraction

We need to find the greatest common divisor (GCD) of 239 and 1000 to simplify the fraction. The GCD of 239 and 1000 is 1, so \( \frac{239}{1000} \) is already in its simplest form.

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Key Concepts

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

decimal place value
Decimal place value is a way to understand the position of each digit in a decimal number. For instance, in the number 0.239, each digit has a specific place value:
  • The digit 2 is in the tenths place, meaning its value is 0.2 (or 2/10).
  • The digit 3 is in the hundredths place, meaning its value is 0.03 (or 3/100).
  • The digit 9 is in the thousandths place, meaning its value is 0.009 (or 9/1000).
Understanding place value is crucial for converting decimals to fractions. By knowing the place value, we can rewrite a decimal as a fraction with the correct denominator. For example, 0.239 is rewritten as \(\frac{239}{1000}\).
greatest common divisor
To simplify a fraction, we often use the greatest common divisor (GCD). The GCD is the largest number that divides two numbers without leaving a remainder. For example, in the fraction \(\frac{239}{1000}\), we need to find the GCD of 239 and 1000.

Finding the GCD involves:
  • Listing the factors of each number.
  • Identifying the common factors.
  • Choosing the largest common factor.
However, in this specific case, 239 is a prime number, meaning it only has two factors: 1 and 239. Since 1000 is not divisible by 239, the GCD is 1. Hence, the fraction \(\frac{239}{1000}\) is already in its simplest form.
fraction simplification
Fraction simplification means reducing a fraction to its simplest form. This involves dividing both the numerator and the denominator by their GCD. Simplified fractions are easier to understand and work with in further calculations. Let's review our example:
  • Original fraction: \(\frac{239}{1000}\)
  • GCD of 239 and 1000: 1
  • Since the GCD is 1, we already have the simplified fraction: \(\frac{239}{1000}\)
In cases where the GCD is greater than 1, divide both the numerator and the denominator by the GCD to achieve simplification. Practice identifying GCD and simplifying various fractions to become proficient in this essential math skill.

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