Compare − 3. 5 ‾ -3.\overline{5} − 3. 5 And − 10 3 -\frac{10}{3} − 3 10 ​ Using Symbols $\ \textless \ $, $\ \textgreater \ $, Or = = = .A. − 3. 5 ‾ \textless − 10 3 -3.\overline{5} \ \textless \ -\frac{10}{3} − 3. 5 \textless − 3 10 ​ B. $-3.\overline{5} \

Introduction

In mathematics, decimals and fractions are two fundamental ways to represent numbers. While decimals are often used in everyday calculations, fractions provide a more precise and elegant way to express ratios. In this article, we will compare two numbers, $-3.\overline{5}$ and $-\frac{10}{3}$, using the symbols $\ \textless \ $, $\ \textgreater \ $, or $=$. We will delve into the world of repeating decimals and fractions, exploring their properties and relationships.

Repeating Decimals

A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. For example, $0.\overline{3}$ is a repeating decimal where the digit $3$ repeats indefinitely. Repeating decimals can be represented as fractions using the concept of infinite geometric series.

Converting Repeating Decimals to Fractions

To convert a repeating decimal to a fraction, we can use the following steps:

1. Let $x = 0.\overline{a_1a_2...a_n}$ be a repeating decimal.
2. Multiply both sides of the equation by $10^n$ to shift the decimal point $n$ places to the right.
3. Subtract the original equation from the new equation to eliminate the repeating part.
4. Simplify the resulting equation to obtain the fraction.

Example: Converting $-3.\overline{5}$ to a Fraction

Let's apply the above steps to convert $-3.\overline{5}$ to a fraction.

1. Let $x = -3.\overline{5}$.
2. Multiply both sides of the equation by $10^1$ to shift the decimal point $1$ place to the right.
3. Subtract the original equation from the new equation:

$$\begin{aligned} 10x &= -35.\overline{5} \\ -x &= -3.\overline{5} \\ \hline 9x &= -32 \end{aligned}$$

4. Simplify the resulting equation to obtain the fraction:

$$\begin{aligned} x &= -\frac{32}{9} \end{aligned}$$

Therefore, $-3.\overline{5} = -\frac{32}{9}$.

Comparing $-3.\overline{5}$ and $-\frac{10}{3}$

Now that we have converted $-3.\overline{5}$ to a fraction, we can compare it with $-\frac{10}{3}$.

$$\begin{aligned} -3.\overline{5} &= -\frac{32}{9} \\ -\frac{10}{3} &= -\frac{10}{3} \end{aligned}$$

To compare these two fractions, we can convert them to equivalent fractions with a common denominator.

$$\begin{aligned} -\frac{32}{9} &= -\frac{32 \times 3}{9 \times 3} \\ &= -\frac{96}{27} \\ -\frac{10}{3} &= -\frac{10 \times 9}{3 \times 9} \\ &= -\frac{90}{27} \end{aligned}$$

Now we can compare the two fractions:

$$\begin{aligned} -\frac{96}{27} &\ \textless \ -\frac{90}{27} \end{aligned}$$

Therefore, $-3.\overline{5} \ \textless \ -\frac{10}{3}$.

Conclusion

In this article, we compared two numbers, $-3.\overline{5}$ and $-\frac{10}{3}$, using the symbols $\ \textless \ $, $\ \textgreater \ $, or $=$. We converted $-3.\overline{5}$ to a fraction using the concept of infinite geometric series and compared it with $-\frac{10}{3}$. Our analysis showed that $-3.\overline{5} \ \textless \ -\frac{10}{3}$. This comparison highlights the importance of understanding the properties of repeating decimals and fractions in mathematics.

References

• [1] "Repeating Decimals" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Converting Repeating Decimals to Fractions" by Purplemath. Retrieved 2023-02-20.

Further Reading

• "Fractions and Decimals" by Khan Academy. Retrieved 2023-02-20.
• "Repeating Decimals" by Wolfram MathWorld. Retrieved 2023-02-20.
  Frequently Asked Questions: Repeating Decimals and Fractions ===========================================================

Q: What is a repeating decimal?

A: A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. For example, $0.\overline{3}$ is a repeating decimal where the digit $3$ repeats indefinitely.

Q: How do I convert a repeating decimal to a fraction?

A: To convert a repeating decimal to a fraction, you can use the following steps:

1. Let $x = 0.\overline{a_1a_2...a_n}$ be a repeating decimal.
2. Multiply both sides of the equation by $10^n$ to shift the decimal point $n$ places to the right.
3. Subtract the original equation from the new equation to eliminate the repeating part.
4. Simplify the resulting equation to obtain the fraction.

Q: Can you give an example of converting a repeating decimal to a fraction?

A: Let's convert $-3.\overline{5}$ to a fraction.

1. Let $x = -3.\overline{5}$.
2. Multiply both sides of the equation by $10^1$ to shift the decimal point $1$ place to the right.
3. Subtract the original equation from the new equation:

$$\begin{aligned} 10x &= -35.\overline{5} \\ -x &= -3.\overline{5} \\ \hline 9x &= -32 \end{aligned}$$

4. Simplify the resulting equation to obtain the fraction:

$$\begin{aligned} x &= -\frac{32}{9} \end{aligned}$$

Q: How do I compare two fractions?

A: To compare two fractions, you can convert them to equivalent fractions with a common denominator. Then, you can compare the numerators.

Q: Can you give an example of comparing two fractions?

A: Let's compare $-\frac{32}{9}$ and $-\frac{10}{3}$.

First, we need to convert them to equivalent fractions with a common denominator.

$$\begin{aligned} -\frac{32}{9} &= -\frac{32 \times 3}{9 \times 3} \\ &= -\frac{96}{27} \\ -\frac{10}{3} &= -\frac{10 \times 9}{3 \times 9} \\ &= -\frac{90}{27} \end{aligned}$$

Now we can compare the two fractions:

$$\begin{aligned} -\frac{96}{27} &\ \textless \ -\frac{90}{27} \end{aligned}$$

Therefore, $-3.\overline{5} \ \textless \ -\frac{10}{3}$.

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

A: A non-repeating decimal is a decimal number that does not have a block of digits that repeats indefinitely. For example, $0.5$ is a non-repeating decimal.

Q: Can you give an example of converting a non-repeating decimal to a fraction?

A: Let's convert $0.5$ to a fraction.

$$\begin{aligned} x &= 0.5 \\ 10x &= 5 \\ \hline 9x &= 5 \end{aligned}$$

$$\begin{aligned} x &= \frac{5}{9} \end{aligned}$$

Q: Why is it important to understand the properties of repeating decimals and fractions?

A: Understanding the properties of repeating decimals and fractions is important because it helps you to:

• Convert between decimals and fractions
• Compare fractions
• Solve equations involving decimals and fractions
• Understand the concept of infinite geometric series

Conclusion

In this article, we answered some frequently asked questions about repeating decimals and fractions. We covered topics such as converting repeating decimals to fractions, comparing fractions, and the difference between repeating and non-repeating decimals. We hope that this article has helped you to better understand the properties of repeating decimals and fractions.

References

• [1] "Repeating Decimals" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Converting Repeating Decimals to Fractions" by Purplemath. Retrieved 2023-02-20.
• [3] "Fractions and Decimals" by Khan Academy. Retrieved 2023-02-20.
• [4] "Repeating Decimals" by Wolfram MathWorld. Retrieved 2023-02-20.