Changing Decimals To Fractions Worksheet

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

Converting decimals to fractions is a fundamental skill in mathematics, crucial for a strong foundation in algebra, calculus, and beyond. This comprehensive guide will equip you with the knowledge and practice to confidently convert any decimal to its fractional equivalent. We'll cover various types of decimals, from simple terminating decimals to repeating decimals, providing clear explanations and step-by-step instructions. We'll also include downloadable worksheets to solidify your understanding and build your proficiency.

Understanding Decimals and Fractions: A Refresher

Before diving into the conversion process, let's refresh our understanding of decimals and fractions. A decimal is a way of writing a number that is not a whole number. It uses a decimal point to separate the whole number part from the fractional part. For example, in 2.5, '2' is the whole number part, and '.5' represents the fractional part.

A fraction, on the other hand, represents a part of a whole. It is expressed as a ratio of two integers, a numerator (the top number) and a denominator (the bottom number). For instance, 1/2 represents one part out of two equal parts. Understanding the relationship between these two representations is key to successful conversion.

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

Terminating decimals are decimals that have a finite number of digits after the decimal point. Converting these to fractions is a straightforward process. Here's how:

Step 1: Write the decimal as a fraction with a denominator of 1.

For example, let's convert 0.75 to a fraction:

0.75/1

Step 2: Multiply both the numerator and the denominator by a power of 10 to remove the decimal point. The power of 10 you choose depends on the number of digits after the decimal point. If there is one digit after the decimal, multiply by 10; two digits, multiply by 100; three digits, multiply by 1000, and so on.

In our example, we have two digits after the decimal point, so we multiply by 100:

(0.75 x 100) / (1 x 100) = 75/100

Step 3: Simplify the fraction to its lowest terms. Find the greatest common divisor (GCD) of the numerator and denominator and divide both by it.

The GCD of 75 and 100 is 25. Dividing both by 25, we get:

75/100 = 3/4

Therefore, 0.75 is equivalent to 3/4.

Let's try another example: Convert 0.375 to a fraction.

1. 0.375/1
2. (0.375 x 1000) / (1 x 1000) = 375/1000
3. Simplify: The GCD of 375 and 1000 is 125. 375/1000 = 3/8

Therefore, 0.375 is equivalent to 3/8.

Converting Non-Terminating, Repeating Decimals to Fractions: A More Advanced Approach

Non-terminating, repeating decimals, also known as recurring decimals, have an infinite number of digits that repeat in a pattern. Converting these to fractions requires a slightly more advanced technique.

Step 1: Set the repeating decimal equal to 'x'.

Let's convert 0.333... (where the 3s repeat infinitely) to a fraction. We set:

x = 0.333...

Step 2: Multiply both sides of the equation by a power of 10 that shifts the repeating block to the left of the decimal point. The power of 10 depends on the length of the repeating block. If the repeating block has one digit, multiply by 10; two digits, multiply by 100, and so on.

In our example, the repeating block is one digit (3), so we multiply by 10:

10x = 3.333...

Step 3: Subtract the original equation (Step 1) from the new equation (Step 2). This eliminates the repeating part.

10x - x = 3.333... - 0.333...

9x = 3

Step 4: Solve for 'x' to find the fractional representation.

x = 3/9

Step 5: Simplify the fraction to its lowest terms.

x = 1/3

Therefore, 0.333... is equivalent to 1/3.

Let's try a more complex example: Convert 0.142857142857... (where the block 142857 repeats infinitely) to a fraction.

1. x = 0.142857142857...
2. Multiply by 1,000,000 (since the repeating block has six digits): 1,000,000x = 142857.142857...
3. Subtract the original equation: 1,000,000x - x = 142857.142857... - 0.142857... This simplifies to 999,999x = 142857
4. Solve for x: x = 142857/999999
5. Simplify: The GCD of 142857 and 999999 is 142857. 142857/999999 = 1/7

Therefore, 0.142857142857... is equivalent to 1/7.

Dealing with Mixed Numbers and Negative Decimals

Mixed Numbers: If your decimal represents a mixed number (a whole number and a decimal), convert the decimal part to a fraction as described above, then add the whole number. For example, 2.75 becomes 2 + 3/4 = 11/4

Negative Decimals: Treat the negative sign separately. Convert the decimal to a fraction as usual, and then add the negative sign. For example, -0.5 becomes -1/2.

Practice Worksheets: Sharpen Your Skills

To truly master the conversion of decimals to fractions, consistent practice is key. Below are some examples of questions to practice with, categorized by difficulty. Remember to always simplify your fractions to their lowest terms. You can create your own worksheets using these examples as a starting point.

Worksheet 1: Terminating Decimals

1. 0.5
2. 0.25
3. 0.7
4. 0.625
5. 0.125
6. 0.875
7. 0.375
8. 0.6
9. 0.9
10. 0.05

Worksheet 2: Repeating Decimals

1. 0.333...
2. 0.666...
3. 0.111...
4. 0.222...
5. 0.142857142857...
6. 0.555...
7. 0.777...
8. 0.8333...
9. 0.1666...
10. 0.090909...

Worksheet 3: Mixed Practice

1. 0.12
2. 0.875
3. 0.666...
4. 1.5
5. 2.75
6. -0.25
7. 0.005
8. 3.333...
9. 0.04
10. -1.75

Frequently Asked Questions (FAQ)

Q: What if the decimal has a non-repeating, non-terminating sequence of digits? These are irrational numbers (like π or √2) and cannot be expressed exactly as a fraction. You can approximate them using a fraction, but you will never get an exact equivalent.

Q: How can I check if my fraction is simplified? Find the greatest common divisor (GCD) of the numerator and the denominator. If the GCD is 1, the fraction is already in its simplest form.

Q: Are there any online tools to check my answers? While this guide encourages working through the problems manually to build your understanding, various online calculators can verify your answers.

Conclusion

Converting decimals to fractions is a valuable skill that builds a strong foundation for advanced mathematical concepts. By mastering the techniques outlined in this guide and practicing diligently using the provided worksheets, you'll gain confidence and proficiency in this essential area of mathematics. Remember to break down complex problems into smaller, manageable steps and always check your work to ensure accuracy. Happy converting!