Write 6.75 As A Mixed Number.

Writing 6.75 as a Mixed Number: A Comprehensive Guide

Converting decimals to fractions, and specifically, expressing decimals as mixed numbers, is a fundamental skill in mathematics. This comprehensive guide will walk you through the process of converting the decimal 6.75 into a mixed number, explaining the underlying concepts and providing various approaches for solving similar problems. We'll also explore the broader context of decimal-fraction conversions and their applications.

Understanding Decimals and Mixed Numbers

Before diving into the conversion, let's review the definitions:

• Decimal: A decimal is a number that uses a decimal point to separate the whole number part from the fractional part. For example, 6.75 has a whole number part of 6 and a fractional part of 0.75.

• Mixed Number: A mixed number combines a whole number and a fraction. It represents a value greater than one. For instance, 2 ¾ is a mixed number where 2 is the whole number and ¾ is the fraction.

• Improper Fraction: An improper fraction is a fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). For example, 11/4 is an improper fraction.

Converting 6.75 to a Mixed Number: Step-by-Step

The conversion of 6.75 to a mixed number involves two main steps:

Step 1: Convert the decimal part to a fraction.

The decimal part of 6.75 is 0.75. To convert this to a fraction, we consider the place value of the last digit. In 0.75, the last digit, 5, is in the hundredths place. Therefore, we can write 0.75 as a fraction with a denominator of 100:

0.75 = 75/100

Step 2: Simplify the fraction and combine with the whole number.

The fraction 75/100 can be simplified by finding the greatest common divisor (GCD) of 75 and 100. The GCD of 75 and 100 is 25. Dividing both the numerator and the denominator by 25, we get:

75/100 = (75 ÷ 25) / (100 ÷ 25) = 3/4

Now, combine this simplified fraction with the whole number part of 6.75, which is 6:

6.75 = 6 + 3/4 = 6 ¾

Therefore, 6.75 expressed as a mixed number is 6 ¾.

Alternative Methods for Conversion

While the above method is straightforward, other approaches can be used, particularly for more complex decimal numbers:

Method 1: Using the Place Value Directly

Instead of converting 0.75 to 75/100 and then simplifying, you can directly identify the fraction by observing the place value. 0.75 represents seventy-five hundredths, which is 75/100. Then, simplify this fraction as shown above.

Method 2: Converting to an Improper Fraction First

You can first convert the entire decimal (6.75) to an improper fraction. This involves writing the decimal as a fraction with a denominator of 1 (6.75/1). Then, multiply both the numerator and denominator by 100 (or 1000, 10000, etc., depending on the number of decimal places) to remove the decimal point:

(6.75/1) * (100/100) = 675/100

Now, simplify the improper fraction by finding the GCD of 675 and 100 (which is 25):

675/100 = (675 ÷ 25) / (100 ÷ 25) = 27/4

Finally, convert the improper fraction (27/4) to a mixed number by dividing the numerator by the denominator:

27 ÷ 4 = 6 with a remainder of 3.

This gives us the mixed number 6 ¾.

Practical Applications of Decimal-to-Fraction Conversions

The ability to convert decimals to fractions, and vice versa, is crucial in various fields:

• Engineering and Construction: Precise measurements and calculations often require working with fractions and decimals interchangeably.

• Baking and Cooking: Recipes frequently use both fractional and decimal measurements for ingredients.

• Finance: Calculating interest rates, discounts, and proportions often involves converting between decimals and fractions.

• Science: Many scientific calculations and measurements utilize both decimal and fractional representations of data.

• Software Development: Understanding decimal-fraction conversions is fundamental for working with numerical data in programming.

• Everyday Life: From splitting bills to measuring ingredients, the ability to work with fractions and decimals is a valuable life skill.

Troubleshooting Common Mistakes

Several common errors can occur during decimal-to-fraction conversions:

• Incorrect simplification of fractions: Always ensure that the fraction is simplified to its lowest terms by dividing both the numerator and denominator by their greatest common divisor.

• Errors in place value: Accurately identifying the place value of the last digit in the decimal is essential for setting the denominator of the initial fraction.

• Mistakes in division: When converting an improper fraction to a mixed number, be careful when performing the division to obtain the whole number and remainder.

Expanding on Decimal-Fraction Conversions: Advanced Concepts

Beyond the basic conversion of 6.75, let's explore more complex scenarios:

Converting Repeating Decimals: Repeating decimals (like 0.333...) require a different approach. They are converted to fractions using algebraic techniques.

Converting Decimals with Multiple Decimal Places: The process remains similar for decimals with more decimal places. The denominator of the initial fraction will be a power of 10 corresponding to the number of decimal places. For instance, 0.1234 would become 1234/10000.

Conclusion

Converting 6.75 to a mixed number, which results in 6 ¾, is a straightforward process involving converting the decimal part to a fraction and then combining it with the whole number. Understanding this fundamental concept, along with its practical applications and potential pitfalls, is crucial for mathematical proficiency and its application in various real-world contexts. Mastering this skill equips you with a strong foundation for more advanced mathematical concepts and problem-solving. Remember to practice regularly to improve your fluency in converting between decimals and fractions.