Problem 277 - Intermediate Algebra

Question

In the following exercises, write each decimal as a fraction. $$ 0.095 $$

Step-by-Step Solution

Answer

\( \frac{19}{200} \)

Step 1: Understand the decimal

Recognize that 0.095 is a decimal number. When converting a decimal to a fraction, determine the place value of the last digit. Here, the last digit '5' is in the thousandths place.

Step 2: Write the decimal as a fraction

Since the digit '5' is in the thousandths place, 0.095 can be written as: \[ \frac{95}{1000} \]

Step 3: Simplify the fraction

To simplify \( \frac{95}{1000} \), find the greatest common divisor (GCD) of 95 and 1000. The GCD of 95 and 1000 is 5. Then, divide both the numerator and the denominator by their GCD: \[ \frac{95 \div 5}{1000 \div 5} = \frac{19}{200} \]

Step 4: Final simplified fraction

The simplified fraction of 0.095 is: \[ \frac{19}{200} \]

Key Concepts

Place Value

Understanding place value is key when converting a decimal to a fraction. Each digit in a decimal number has a specific place value depending on its position relative to the decimal point. For the decimal 0.095, the last digit '5' is in the thousandths place. This is because it is three places to the right of the decimal point. Recognizing this place value helps you convert the decimal to a fraction correctly. When you see '5' in the thousandths place, it indicates that the decimal can be written as \( \frac{95}{1000} \). Breaking down the decimal step-by-step ensures accuracy in your conversion process.

Simplifying Fractions

To simplify a fraction, you need to find common factors between the numerator (top number) and the denominator (bottom number) and divide both by the same number. This reduces the fraction to its simplest form. For instance, after converting 0.095 to \( \frac{95}{1000} \), look for any common factors. By identifying and dividing both the numerator and denominator by their greatest common factor (GCF), you reduce the fraction's complexity. For example, we can simplify \( \frac{95}{1000} \) by dividing both numbers by 5: \[ \frac{95 \div 5}{1000 \div 5} = \frac{19}{200} \] Simplifying makes the fraction easier to understand and use in further calculations.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) of two numbers is the largest number that evenly divides both of them. Finding the GCD is crucial for simplifying fractions efficiently. To find the GCD of 95 and 1000: \begin{itemize} \item List the factors of 95: 1, 5, 19, 95 \item List the factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000 \end{itemize} The common factors are 1 and 5. The largest of these is 5, making it the GCD. Dividing the original fraction by the GCD gives you: \[ \frac{95 \div 5}{1000 \div 5} = \frac{19}{200} \] This method , finding the highest number that divides both terms equally, ensures your fraction is in the simplest form.