Changing Decimals To Fractions Worksheet

Mastering the Art of Converting Decimals to Fractions: A Comprehensive Worksheet Guide

Converting decimals to fractions is a fundamental skill in mathematics, crucial for various applications from basic arithmetic to advanced calculus. This comprehensive guide provides a detailed explanation of the process, accompanied by practice exercises and answers to solidify your understanding. We'll cover everything from simple decimal conversions to those involving repeating decimals, ensuring you develop a strong grasp of this essential concept. This worksheet-style guide is designed to help you master this skill, whether you're a student, a teacher, or simply someone looking to brush up on their math skills.

Understanding the Basics: Decimals and Fractions

Before diving into the conversion process, let's refresh our understanding of decimals and fractions. A decimal is a way of representing a number using base-10, where the digits to the right of the decimal point represent fractions with denominators of powers of 10 (10, 100, 1000, etc.). A fraction, on the other hand, represents a part of a whole, expressed as a ratio of two numbers (numerator/denominator).

For example, the decimal 0.5 is equivalent to the fraction ½ (one-half), 0.25 is equivalent to ¼ (one-quarter), and 0.75 is equivalent to ¾ (three-quarters). Understanding this fundamental relationship is the key to converting between these two representations.

Converting Simple Decimals to Fractions: A Step-by-Step Approach

Converting simple decimals (those with a finite number of digits after the decimal point) to fractions is a relatively straightforward process. Follow these steps:

1. Identify the place value of the last digit: Determine the place value of the last digit in the decimal. For example, in 0.35, the last digit (5) is in the hundredths place.

2. Write the decimal as a fraction: Write the decimal as a fraction with the digits after the decimal point as the numerator and the place value as the denominator. So, 0.35 becomes 35/100.

3. Simplify the fraction: Simplify the fraction to its lowest terms by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. In the example, the GCD of 35 and 100 is 5. Dividing both by 5 gives us 7/20.

Example 1: Convert 0.6 to a fraction.

• The last digit (6) is in the tenths place.
• The fraction is 6/10.
• Simplifying, we get 3/5.

Example 2: Convert 0.125 to a fraction.

• The last digit (5) is in the thousandths place.
• The fraction is 125/1000.
• Simplifying (GCD = 125), we get 1/8.

Worksheet Exercises (Simple Decimals):

Convert the following decimals to fractions in their simplest form:

1. 0.8
2. 0.45
3. 0.025
4. 0.75
5. 0.15

Answers:

1. 4/5
2. 9/20
3. 1/40
4. 3/4
5. 3/20

Converting Decimals with Repeating Digits to Fractions

Converting decimals with repeating digits (like 0.333... or 0.142857142857...) to fractions requires a slightly different approach. These are also known as recurring decimals. Here's the method:

1. Represent the repeating decimal with an algebraic equation: Let x equal the repeating decimal.

2. Multiply the equation by a power of 10: Multiply the equation by a power of 10 that shifts the repeating part to the left of the decimal point. The power of 10 will be equal to the number of digits in the repeating block. For example, if the repeating block is '333...', you'd multiply by 10; if it's '142857', you'd multiply by 1,000,000.

3. Subtract the original equation from the multiplied equation: This will eliminate the repeating part, leaving you with a simple equation.

4. Solve the equation for x: Solve the equation to find the value of x, which will be the fractional representation of the repeating decimal.

5. Simplify the fraction: Simplify the resulting fraction to its lowest terms.

Example 3: Convert 0.333... to a fraction.

1. Let x = 0.333...
2. Multiply by 10: 10x = 3.333...
3. Subtract the original equation: 10x - x = 3.333... - 0.333... This simplifies to 9x = 3.
4. Solve for x: x = 3/9 = 1/3.

Example 4: Convert 0.142857142857... to a fraction.

1. Let x = 0.142857142857...
2. Multiply by 1,000,000: 1,000,000x = 142857.142857...
3. Subtract the original equation: 999,999x = 142857
4. Solve for x: x = 142857/999999 = 1/7.

Worksheet Exercises (Repeating Decimals):

Convert the following repeating decimals to fractions in their simplest form:

1. 0.666...
2. 0.272727...
3. 0.121212...
4. 0.555...
5. 0.818181...

Answers:

1. 2/3
2. 3/11
3. 4/33
4. 5/9
5. 9/11

Converting Mixed Decimals to Fractions

A mixed decimal is a number that has both a whole number part and a decimal part (e.g., 2.5, 7.333...). To convert a mixed decimal to a fraction, follow these steps:

1. Separate the whole number and the decimal part: Write the mixed decimal as the sum of its whole number and decimal parts. For example, 2.5 = 2 + 0.5.

2. Convert the decimal part to a fraction: Convert the decimal part to a fraction using the methods described earlier.

3. Convert the whole number to a fraction: Convert the whole number to a fraction with the same denominator as the fraction from step 2.

4. Add the fractions: Add the two fractions together.

Example 5: Convert 2.75 to a fraction.

1. 2.75 = 2 + 0.75
2. Convert 0.75 to a fraction: 75/100 = 3/4
3. Convert 2 to a fraction with a denominator of 4: 8/4
4. Add the fractions: 8/4 + 3/4 = 11/4

Worksheet Exercises (Mixed Decimals):

Convert the following mixed decimals to fractions in their simplest form:

1. 3.2
2. 1.75
3. 4.666...
4. 2.125
5. 5.333...

Answers:

1. 16/5
2. 7/4
3. 14/3
4. 17/8
5. 16/3

Frequently Asked Questions (FAQ)

Q1: What is the easiest way to convert decimals to fractions?

A1: For simple, terminating decimals, the easiest method is to write the decimal as a fraction with a denominator that is a power of 10 (10, 100, 1000, etc.), then simplify the fraction.

Q2: How do I convert a repeating decimal to a fraction if the repeating block is long?

A2: The algebraic method described above remains the most efficient, even with long repeating blocks. While it may seem tedious at first, the process is consistent and reliable.

Q3: Are there any shortcuts for converting certain types of decimals?

A3: Recognizing common decimal-fraction equivalents (like 0.5 = ½, 0.25 = ¼, 0.75 = ¾) can save time.

Q4: What if the decimal has both a repeating and a non-repeating part?

A4: This requires a combination of the techniques we've discussed. First separate out the non-repeating part. Convert the repeating part using the algebraic method, then add the two resulting fractions.

Conclusion: Mastering Decimal-to-Fraction Conversion

Converting decimals to fractions is a crucial skill that builds a solid foundation in mathematics. By understanding the underlying principles and practicing the methods outlined in this worksheet, you can confidently tackle various types of decimal-to-fraction conversions. Remember to practice regularly and utilize the different techniques presented here to master this essential skill. Consistent practice will transform this initially challenging concept into a readily applicable skill. Keep practicing, and you'll soon find yourself effortlessly converting decimals to fractions.