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Chapter 1: Problem 615

Write each decimal as a fraction. $$ 0.095 $$

Short Answer

Expert verified
The decimal 0.095 as a fraction is \( \frac{19}{200} \).

Step by step solution

01

Understand the decimal

Recognize that the decimal 0.095 represents a number in the tenths, hundredths, and thousandths place.
02

Write the decimal as a fraction with a denominator of 1000

Since 0.095 reaches the thousandths place, write it as \( \frac{95}{1000} \).
03

Simplify the fraction

Find the greatest common divisor (GCD) of 95 and 1000, which is 5. Divide both the numerator and the denominator by 5: \( \frac{95 \div 5}{1000 \div 5} = \frac{19}{200} \).

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Key Concepts

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

decimal representation
Decimals are a way to represent fractions or parts of whole numbers. The value of each digit in a decimal depends on its position. For example, in the decimal number 0.095, the digit '9' is in the hundredths place, and the digit '5' is in the thousandths place. Understanding how to read a decimal is crucial:
  • The first digit after the decimal point represents tenths (1/10).
  • The second digit represents hundredths (1/100).
  • The third digit represents thousandths (1/1,000).
By recognizing this, you know that 0.095 is the same as 95 thousandths, which helps in converting it to a fraction with 1000 as the denominator.
fraction simplification
Simplifying fractions is an essential skill in mathematics. Once you have converted a decimal to a fraction, the next step is to simplify it. Simplification involves reducing the fraction to its simplest form. To do that, you divide both the numerator (top number) and the denominator (bottom number) by a common factor. Here is a simple process to follow:
  • Identify the numerator and the denominator. For example, in the fraction \(\frac{95}{1000}\), 95 is the numerator and 1000 is the denominator.
  • Find the greatest common divisor (GCD), which is the largest number that divides both the numerator and the denominator without leaving a remainder.
  • Divide both the numerator and the denominator by the GCD.
  • Write down the resulting fraction. Example: Dividing both 95 and 1000 by their GCD of 5, you get \(\frac{19}{200}\).
Simplification makes fractions easier to understand and work with.
greatest common divisor
The greatest common divisor (GCD) is the largest number that can divide two or more numbers without leaving a remainder. Finding the GCD is a key step in simplifying fractions. Here's a simple method to find the GCD:
  • List all the factors of both the numerator and the denominator.
  • Identify the largest factor that appears in both lists. This is the GCD.
Example: To find the GCD of 95 and 1000:
  • Factors of 95: 1, 5, 19, 95
  • Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
  • The common factors are 1 and 5. The greatest of these is 5, so the GCD is 5.
Using the GCD, you can divide both terms of the fraction to simplify it efficiently. Understanding how to find and use the GCD helps you in various mathematical processes.

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