Converting Repeating Decimals to Fractional Form

Oct 13, 2025 by talynt.org 57 views

Introduction

In mathematics, repeating decimals are a type of decimal that has a repeating pattern of digits after the decimal point. These decimals can be converted into fractional form, which is a fraction of two integers. In this article, we will explore the concept of repeating decimals and learn how to convert them into fractional form.

What is a Repeating Decimal?

A repeating decimal is a decimal that has a repeating pattern of digits after the decimal point. For example, the decimal 0.333... is a repeating decimal because the digit 3 repeats indefinitely. Similarly, the decimal 0.666... is also a repeating decimal because the digit 6 repeats indefinitely.

Converting Repeating Decimals to Fractional Form

To convert a repeating decimal into fractional form, we can use a simple algebraic technique. Let's consider the repeating decimal 0.888... as an example. We can represent this decimal as x and multiply it by 10 to get 8.888... . Now, if we subtract the original decimal from this new decimal, we get:

10x - x = 8.888... - 0.888...

This simplifies to:

9x = 8

Now, we can divide both sides by 9 to get:

x = 8/9

Therefore, the fractional form of the repeating decimal 0.888... is 8/9.

Applying the Technique to the Given Problem

Now, let's apply this technique to the given problem. We are asked to find the fractional form of the repeating decimal 0.888... . Using the same technique as before, we can represent this decimal as x and multiply it by 10 to get 8.888... . Now, if we subtract the original decimal from this new decimal, we get:

10x - x = 8.888... - 0.888...

This simplifies to:

9x = 8

Now, we can divide both sides by 9 to get:

x = 8/9

Therefore, the fractional form of the repeating decimal 0.888... is 8/9.

Conclusion

In conclusion, repeating decimals can be converted into fractional form using a simple algebraic technique. By representing the repeating decimal as x and multiplying it by 10, we can subtract the original decimal from this new decimal to get a simple equation. Solving this equation, we can find the fractional form of the repeating decimal. In this article, we applied this technique to the given problem and found that the fractional form of the repeating decimal 0.888... is 8/9.

Answer

The correct answer is C. 8/9.

Discussion

This problem is a classic example of a repeating decimal and how to convert it into fractional form. The technique used in this problem is a simple and effective way to convert repeating decimals into fractional form. This technique can be applied to any repeating decimal, making it a useful tool for mathematicians and students alike.

References

Mathematics: Mathematics is the study of numbers, quantities, and shapes.
Decimal: Decimal is a way of representing a number using a point as a separator between the whole number and the fractional part.
Fraction: Fraction is a way of representing a number as a fraction of two integers.

Conclusion

Q: What is a repeating decimal?

A: A repeating decimal is a type of decimal that has a repeating pattern of digits after the decimal point. For example, the decimal 0.333... is a repeating decimal because the digit 3 repeats indefinitely.

Q: How do I convert a repeating decimal to fractional form?

A: To convert a repeating decimal to fractional form, you can use a simple algebraic technique. Let's consider the repeating decimal 0.888... as an example. We can represent this decimal as x and multiply it by 10 to get 8.888... . Now, if we subtract the original decimal from this new decimal, we get:

10x - x = 8.888... - 0.888...

This simplifies to:

9x = 8

Now, we can divide both sides by 9 to get:

x = 8/9

Therefore, the fractional form of the repeating decimal 0.888... is 8/9.

Q: What is the fractional form of the repeating decimal 0.666...?

A: To find the fractional form of the repeating decimal 0.666..., we can use the same technique as before. Let's represent this decimal as x and multiply it by 10 to get 6.666... . Now, if we subtract the original decimal from this new decimal, we get:

10x - x = 6.666... - 0.666...

This simplifies to:

9x = 6

Now, we can divide both sides by 9 to get:

x = 6/9

Therefore, the fractional form of the repeating decimal 0.666... is 6/9, which can be simplified to 2/3.

Q: How do I know if a decimal is repeating or not?

A: To determine if a decimal is repeating or not, you can look for a pattern of digits after the decimal point. If the digits repeat indefinitely, then the decimal is repeating. For example, the decimal 0.123456... is not repeating because the digits do not repeat indefinitely.

Q: Can I use this technique to convert any repeating decimal to fractional form?

A: Yes, this technique can be used to convert any repeating decimal to fractional form. The key is to represent the repeating decimal as x and multiply it by 10 to get a new decimal. Then, subtract the original decimal from this new decimal to get a simple equation. Solving this equation, you can find the fractional form of the repeating decimal.

Q: What are some examples of repeating decimals?

A: Some examples of repeating decimals include:

0.333...
0.666...
0.888...
0.999...
0.123456...

Q: Can I use this technique to convert a decimal with a finite number of digits to fractional form?

A: No, this technique is only used to convert repeating decimals to fractional form. If a decimal has a finite number of digits, it is not a repeating decimal and cannot be converted to fractional form using this technique.

Q: What are some real-world applications of repeating decimals and fractional form?

A: Repeating decimals and fractional form have many real-world applications, including:

Finance: Repeating decimals are used to calculate interest rates and investment returns.
Science: Repeating decimals are used to calculate physical constants and scientific measurements.
Engineering: Repeating decimals are used to calculate engineering measurements and designs.

Conclusion

In conclusion, repeating decimals can be converted into fractional form using a simple algebraic technique. By representing the repeating decimal as x and multiplying it by 10, we can subtract the original decimal from this new decimal to get a simple equation. Solving this equation, we can find the fractional form of the repeating decimal. In this article, we answered some common questions about repeating decimals and fractional form, and provided some examples of real-world applications.