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Chapter 5: Problem 26

In the following exercises, convert each decimal to a fraction or mixed number. $$0.049$$

Short Answer

Expert verified
0.049 as a fraction is \(\frac{49}{1000}\).

Step by step solution

01

- Understanding the decimal

Identify the place value of the farthest digit to the right in the decimal number 0.049. The farthest digit is 9, which is in the thousandths place.
02

- Converting the decimal to a fraction

Write the decimal 0.049 as a fraction: \ \[ 0.049 = \frac{49}{1000} \]
03

- Simplifying the fraction

Find the greatest common divisor (GCD) of 49 and 1000. The GCD of 49 and 1000 is 1. Hence, \ \[ \frac{49}{1000} \] 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.

Place Value
When dealing with decimals, understanding place value is crucial. Place value refers to the position of a digit in the number, which determines its value. For instance, in the decimal number 0.049, each digit has a specific place value:
  • 0 is in the tenths place
  • 4 is in the hundredths place
  • 9 is in the thousandths place
This means that 9 is worth nine thousandths, 4 is worth four hundredths, and so on. Recognizing these positions helps convert the decimal to a fraction accurately.
Simplifying Fractions
Simplifying fractions makes them easier to understand and work with. To simplify a fraction, you divide both the numerator and the denominator by their Greatest Common Divisor (GCD). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.
In the case of the fraction \(\frac{49}{1000}\), the GCD of 49 and 1000 is 1. Thus, \[ \frac{49}{1000} \] is already in its simplest form since no number other than 1 divides both 49 and 1000 without leaving a remainder.
Greatest Common Divisor
The Greatest Common Divisor (GCD) is a key concept in simplifying fractions. To find the GCD, list the factors of both numbers and identify the largest one shared by both. Here鈥檚 a step-by-step process:
1. List the factors: Find all numbers that divide evenly into each number.
2. Identify the common factors: Look for numbers that appear in both lists.
3. Choose the greatest one: This is the GCD.
For example:
  • Factors of 49 are 1 and 49 (since 49 is a prime number).
  • Factors of 1000 are 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, and 1000.
Since the only common factor between 49 and 1000 is 1, the GCD is 1. Hence, \[ \frac{49}{1000} \] is already simplified.

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