Solution to Examples and Exercises

Oct 13, 2025 by magicalceylontours.com 50 views

Introduction

In mathematics, decimals are a way of representing fractions with a denominator that is a power of 10. When we have a decimal with a repeating pattern, we can express it in a specific form using algebraic notation. In this article, we will explore how to express decimals in the specified form, focusing on the examples $0.3\overrightarrow{6}$ and $1.2\overline{7}$ .

Expressing $0.3\overrightarrow{6}$

Let's start by examining the decimal $0.3\overrightarrow{6}$ . The bar over the 6 indicates that it is a repeating decimal. To express this decimal in a specified form, we can use the following steps:

Let $x = 0.3\overrightarrow{6}$
Multiply both sides of the equation by 10 to get $10x = 3.\overrightarrow{6}$
Subtract the original equation from the new equation to get $9x = 3$
Divide both sides of the equation by 9 to get $x = \frac{3}{9} = \frac{1}{3}$

Therefore, we can express the decimal $0.3\overrightarrow{6}$ as the fraction $\frac{1}{3}$ .

Expressing $1.2\overline{7}$

Now, let's move on to the decimal $1.2\overline{7}$ . Again, the bar over the 7 indicates that it is a repeating decimal. To express this decimal in a specified form, we can use the following steps:

Let $x = 1.2\overline{7}$
Multiply both sides of the equation by 10 to get $10x = 12.\overline{7}$
Subtract the original equation from the new equation to get $9x = 11$
Divide both sides of the equation by 9 to get $x = \frac{11}{9}$

Therefore, we can express the decimal $1.2\overline{7}$ as the fraction $\frac{11}{9}$ .

Conclusion

In conclusion, we have seen how to express decimals in the specified form using algebraic notation. By following the steps outlined above, we can convert decimals with repeating patterns into fractions. This is a useful skill to have in mathematics, as it allows us to work with decimals in a more abstract and general way.

Examples and Exercises

Here are some examples and exercises to help you practice expressing decimals in the specified form:

Express the decimal $0.4\overrightarrow{2}$ as a fraction.
Express the decimal $2.5\overline{9}$ as a fraction.
Express the decimal $0.1\overrightarrow{6}$ as a fraction.

Solution to Examples and Exercises

Let $x = 0.4\overrightarrow{2}$
Multiply both sides of the equation by 10 to get $10x = 4.\overrightarrow{2}$
Subtract the original equation from the new equation to get $9x = 4$
Divide both sides of the equation by 9 to get $x = \frac{4}{9}$
Let $x = 2.5\overline{9}$
Multiply both sides of the equation by 10 to get $10x = 25.\overline{9}$
Subtract the original equation from the new equation to get $9x = 24$
Divide both sides of the equation by 9 to get $x = \frac{24}{9} = \frac{8}{3}$
Let $x = 0.1\overrightarrow{6}$
Multiply both sides of the equation by 10 to get $10x = 1.\overrightarrow{6}$
Subtract the original equation from the new equation to get $9x = 1$
Divide both sides of the equation by 9 to get $x = \frac{1}{9}$

Conclusion

Q: What is a repeating decimal?

A: A repeating decimal is a decimal that has a repeating pattern of digits. For example, the decimal 0.3\overrightarrow{6} has a repeating pattern of 6.

Q: How do I express a repeating decimal as a fraction?

A: To express a repeating decimal as a fraction, you can use the following steps:

Let x be the repeating decimal.
Multiply both sides of the equation by 10 to get 10x.
Subtract the original equation from the new equation to get 9x.
Divide both sides of the equation by 9 to get x.

Q: What is the difference between a repeating decimal and a non-repeating decimal?

A: A non-repeating decimal is a decimal that does not have a repeating pattern of digits. For example, the decimal 0.5 is a non-repeating decimal. A repeating decimal, on the other hand, has a repeating pattern of digits.

Q: Can all decimals be expressed as fractions?

A: Yes, all decimals can be expressed as fractions. However, some decimals may have a repeating pattern of digits, which can make them more difficult to express as fractions.

Q: How do I determine if a decimal is repeating or non-repeating?

A: To determine if a decimal is repeating or non-repeating, you can look for a repeating pattern of digits. If you see a repeating pattern, the decimal is repeating. If you do not see a repeating pattern, the decimal is non-repeating.

Q: What are some examples of repeating decimals?

A: Some examples of repeating decimals include:

0.3\overrightarrow{6}
1.2\overline{7}
0.4\overrightarrow{2}
2.5\overline{9}

Q: What are some examples of non-repeating decimals?

A: Some examples of non-repeating decimals include:

0.5
0.25
0.75
0.125

Q: Can I use a calculator to express a repeating decimal as a fraction?

A: Yes, you can use a calculator to express a repeating decimal as a fraction. However, you may need to use a special function or setting on the calculator to do so.

Q: Are there any limitations to expressing decimals as fractions?

A: Yes, there are some limitations to expressing decimals as fractions. For example, some decimals may have a very large or very small value, which can make them difficult to express as fractions.

Conclusion

In conclusion, we have seen how to express decimals in the specified form using algebraic notation. We have also answered some frequently asked questions about repeating decimals and non-repeating decimals. By following the steps outlined above, you can convert decimals with repeating patterns into fractions. This is a useful skill to have in mathematics, as it allows you to work with decimals in a more abstract and general way.