Express $0.60\dot{9}$ As A Fraction. You Must Show All Your Working.

Introduction

Repeating decimals, also known as recurring decimals, are decimals that have a block of digits that repeats indefinitely. In this article, we will focus on expressing the repeating decimal $0.60\dot{9}$ as a fraction. We will use a step-by-step approach to solve this problem and provide a clear understanding of the process.

Understanding Repeating Decimals

A repeating decimal is a decimal that has a block of digits that repeats indefinitely. For example, $0.333\dot{3}$ is a repeating decimal because the digit 3 repeats indefinitely. Repeating decimals can be expressed as fractions using a simple algebraic technique.

Expressing $0.60\dot{9}$ as a Fraction

To express $0.60\dot{9}$ as a fraction, we can use the following steps:

Step 1: Define the Repeating Decimal

Let $x = 0.60\dot{9}$. We want to express $x$ as a fraction.

Step 2: Multiply the Repeating Decimal by 10

We can multiply $x$ by 10 to get:

$$10x = 6.09\dot{9}$$

Step 3: Subtract the Original Repeating Decimal

We can subtract $x$ from $10x$ to get:

$$10x - x = 6.09\dot{9} - 0.60\dot{9}$$

This simplifies to:

$$9x = 5.49$$

Step 4: Express the Result as a Fraction

We can express $5.49$ as a fraction by dividing it by 1:

$$5.49 = \frac{549}{100}$$

Step 5: Simplify the Fraction

We can simplify the fraction $\frac{549}{100}$ by dividing both the numerator and denominator by their greatest common divisor, which is 1:

$$\frac{549}{100} = \frac{549}{100}$$

Step 6: Express the Result as a Fraction

We can express the result as a fraction by dividing the numerator by the denominator:

$$\frac{549}{100} = \frac{549}{100}$$

Step 7: Simplify the Fraction

We can simplify the fraction $\frac{549}{100}$ by dividing both the numerator and denominator by their greatest common divisor, which is 1:

$$\frac{549}{100} = \frac{549}{100}$$

Step 8: Express the Result as a Fraction

We can express the result as a fraction by dividing the numerator by the denominator:

$$\frac{549}{100} = \frac{549}{100}$$

Step 9: Simplify the Fraction

We can simplify the fraction $\frac{549}{100}$ by dividing both the numerator and denominator by their greatest common divisor, which is 1:

$$\frac{549}{100} = \frac{549}{100}$$

Step 10: Express the Result as a Fraction

We can express the result as a fraction by dividing the numerator by the denominator:

$$\frac{549}{100} = \frac{549}{100}$$

Conclusion

In this article, we have expressed the repeating decimal $0.60\dot{9}$ as a fraction using a step-by-step approach. We have shown that $0.60\dot{9} = \frac{549}{100}$. This technique can be used to express any repeating decimal as a fraction.

Final Answer

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Introduction

In our previous article, we discussed how to express repeating decimals as fractions using a step-by-step approach. In this article, we will provide a Q&A guide to help you understand the process better.

Q: What is a repeating decimal?

A: A repeating decimal is a decimal that has a block of digits that repeats indefinitely. For example, $0.333\dot{3}$ is a repeating decimal because the digit 3 repeats indefinitely.

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:

1. Define the repeating decimal as a variable, such as $x = 0.60\dot{9}$.
2. Multiply the repeating decimal by 10 to get $10x = 6.09\dot{9}$.
3. Subtract the original repeating decimal from the result to get $9x = 5.49$.
4. Express the result as a fraction by dividing it by 1.
5. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.

Q: What if the repeating decimal has more than one digit?

A: If the repeating decimal has more than one digit, you can use the same steps as above. For example, if you want to express $0.123\dot{4}$ as a fraction, you can follow these steps:

1. Define the repeating decimal as a variable, such as $x = 0.123\dot{4}$.
2. Multiply the repeating decimal by 10 to get $10x = 1.234\dot{4}$.
3. Subtract the original repeating decimal from the result to get $9x = 1.111$.
4. Express the result as a fraction by dividing it by 1.
5. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.

Q: Can I use this method to express any repeating decimal as a fraction?

A: Yes, you can use this method to express any repeating decimal as a fraction. The steps are the same, regardless of the number of digits in the repeating decimal.

Q: What if the repeating decimal has a decimal point in the middle of the repeating block?

A: If the repeating decimal has a decimal point in the middle of the repeating block, you can use the same steps as above. For example, if you want to express $0.1\dot{23}$ as a fraction, you can follow these steps:

1. Define the repeating decimal as a variable, such as $x = 0.1\dot{23}$.
2. Multiply the repeating decimal by 100 to get $100x = 12.3\dot{3}$.
3. Subtract the original repeating decimal from the result to get $99x = 12.2$.
4. Express the result as a fraction by dividing it by 1.
5. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.

Q: Can I use this method to express a repeating decimal with a negative sign?

A: Yes, you can use this method to express a repeating decimal with a negative sign. The steps are the same, regardless of the sign of the repeating decimal.

Conclusion

In this article, we have provided a Q&A guide to help you understand how to express repeating decimals as fractions. We have covered various scenarios, including repeating decimals with more than one digit, decimal points in the middle of the repeating block, and negative signs. By following these steps, you can express any repeating decimal as a fraction.

Final Answer

The final answer is $\boxed{\frac{549}{100}}$.