Problem 6 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$6 \frac{3}{4}$$

Step-by-Step Solution

Answer

The mixed number \(6 \frac{3}{4}\) as a decimal is 6.75.

Step 1: Identify the Fraction and Whole Number

The mixed number given is \(6 \frac{3}{4}\), which means there is a whole number part 6 and a fractional part \(\frac{3}{4}\).

Step 2: Convert the Fraction to Decimal

To convert the fraction \(\frac{3}{4}\) into a decimal, divide the numerator by the denominator: \(3 \div 4 = 0.75\). So, \(\frac{3}{4}\) is equal to 0.75.

Step 3: Combine the Whole Number and Decimal Parts

Now, combine the whole number 6 with the decimal 0.75 obtained from the fraction. This results in: \[6 + 0.75 = 6.75\].

Key Concepts

Mixed Numbers

Mixed numbers are numbers that combine a whole number and a fraction. In everyday situations, we often encounter these numbers because they provide an easy way to represent quantities that are more than a whole but not quite another whole. For example, if you have 6 whole pizzas and three-fourths of another pizza, you could express this as 6 ¾ pizzas.

Understanding mixed numbers is important for converting them into other numerical forms, such as decimals or improper fractions. A mixed number is typically written as a whole number followed by a fraction, like in the example: 6 ¾. This expression signals that the total value is slightly more than 6 but not quite 7.

• Makes calculations involving whole parts and fraction parts more straightforward.
• Often used in measurements, recipes, and time-telling.
• Helpful for visualizing portions of a total amount.

Fraction to Decimal Conversion

Converting fractions to decimals involves understanding the relationship between the numerator and the denominator. A fraction represents a division of the numerator (the top part) by the denominator (the bottom part). For instance, in the fraction \(\frac{3}{4}\), you would divide 3 by 4 to get 0.75.

The process is straightforward and allows for an easy transition from parts of a whole to a decimal format, which can be easier to work with in most calculations. When converting, you may use long division or a calculator. The result will often be a decimal that makes it easier to comprehend the exact proportion the fraction represents.

• Decide whether the decimal is terminating (like 0.75) or repeating (like 0.333...).
• Helps in simplifying mathematical operations like addition and subtraction.
• Useful in financial calculations, technical applications, and data analysis.

Repeating Decimals

Repeating decimals are decimals that have one or more digits that repeat infinitely. They occur when converting certain fractions to decimals. A repeating decimal is expressed by placing a bar over the repeating digits.

For example, when we convert \(\frac{1}{3}\) into a decimal, the result is 0.333... with the digit '3' repeating indefinitely. Thus, we write it as \(0.\overline{3}\).

Understanding repeating decimals is critical because they frequently occur in divisions that do not result in a clean break. Recognizing and expressing these can help ensure accuracy in mathematical calculations and avoid rounding errors.

• Used to express non-terminating decimal values.
• Helps in understanding infinite series and calculus concepts.
• Necessary for precise calculations, especially in scientific contexts.