Show That The Following Recurring Decimals Can Be Expressed As Fractions:(a) 0. 4 ˙ 0.\dot{4} 0. 4 ˙ (b) 0. 2 ˙ I 0.\dot{2}i 0. 2 ˙ I (c) 19.45 19.45 19.45 (d) 0.12 4 ˙ 0.12\dot{4} 0.12 4 ˙

Introduction

Recurring decimals, also known as repeating decimals, are decimals that have a block of digits that repeats indefinitely. In this article, we will explore how to express recurring decimals as fractions, which is a fundamental concept in mathematics. We will examine four different recurring decimals and show that they can be expressed as fractions.

Expressing $0.\dot{4}$ as a Fraction

The recurring decimal $0.\dot{4}$ can be expressed as a fraction by using the following steps:

1. Let $x = 0.\dot{4}$
2. Multiply both sides of the equation by 10 to get $10x = 4.\dot{4}$
3. Subtract the original equation from the new equation to get $9x = 4$
4. Divide both sides of the equation by 9 to get $x = \frac{4}{9}$

Therefore, the recurring decimal $0.\dot{4}$ can be expressed as the fraction $\frac{4}{9}$.

Expressing $0.\dot{2}i$ as a Fraction

The recurring decimal $0.\dot{2}i$ can be expressed as a fraction by using the following steps:

1. Let $x = 0.\dot{2}i$
2. Multiply both sides of the equation by 10 to get $10x = 2.\dot{2}i$
3. Subtract the original equation from the new equation to get $9x = 2i$
4. Divide both sides of the equation by 9 to get $x = \frac{2i}{9}$

Therefore, the recurring decimal $0.\dot{2}i$ can be expressed as the fraction $\frac{2i}{9}$.

Expressing $19.45$ as a Fraction

The decimal $19.45$ can be expressed as a fraction by using the following steps:

1. Let $x = 19.45$
2. Multiply both sides of the equation by 100 to get $100x = 1945$
3. Subtract the original equation from the new equation to get $99x = 1926$
4. Divide both sides of the equation by 99 to get $x = \frac{1926}{99}$

Therefore, the decimal $19.45$ can be expressed as the fraction $\frac{1926}{99}$.

Expressing $0.12\dot{4}$ as a Fraction

The recurring decimal $0.12\dot{4}$ can be expressed as a fraction by using the following steps:

1. Let $x = 0.12\dot{4}$
2. Multiply both sides of the equation by 100 to get $100x = 12.14$
3. Subtract the original equation from the new equation to get $99x = 12.02$
4. Multiply both sides of the equation by 100 to get $999x = 1202$
5. Subtract the original equation from the new equation to get $900x = 1190$
6. Divide both sides of the equation by 900 to get $x = \frac{1190}{900}$

Therefore, the recurring decimal $0.12\dot{4}$ can be expressed as the fraction $\frac{1190}{900}$.

Conclusion

In this article, we have shown that four different recurring decimals can be expressed as fractions. We have used the method of multiplying both sides of the equation by a power of 10, subtracting the original equation from the new equation, and then dividing both sides of the equation by the resulting number. This method can be used to express any recurring decimal as a fraction.

References

• [1] "Recurring Decimals" by Math Open Reference
• [2] "Fractions" by Khan Academy

Further Reading

• "Recurring Decimals and Fractions" by Purplemath
• "Converting Recurring Decimals to Fractions" by Mathway

Glossary

• Recurring Decimal: A decimal that has a block of digits that repeats indefinitely.
• Fraction: A way of expressing a number as the ratio of two integers.
• Method of Multiplication: A method used to express recurring decimals as fractions by multiplying both sides of the equation by a power of 10.
  Frequently Asked Questions: Recurring Decimals and Fractions ===========================================================

Q: What is a recurring decimal?

A: A recurring decimal is a decimal that has a block of digits that repeats indefinitely. For example, the decimal 0.333... is a recurring decimal because the digit 3 repeats indefinitely.

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

A: To convert a recurring decimal to a fraction, you can use the method of multiplication. This involves multiplying both sides of the equation by a power of 10, subtracting the original equation from the new equation, and then dividing both sides of the equation by the resulting number.

Q: What is the method of multiplication?

A: The method of multiplication is a technique used to convert recurring decimals to fractions. It involves multiplying both sides of the equation by a power of 10, subtracting the original equation from the new equation, and then dividing both sides of the equation by the resulting number.

Q: How do I use the method of multiplication to convert a recurring decimal to a fraction?

A: To use the method of multiplication, follow these steps:

1. Let x be the recurring decimal.
2. Multiply both sides of the equation by a power of 10 (e.g. 10, 100, 1000, etc.).
3. Subtract the original equation from the new equation.
4. Divide both sides of the equation by the resulting number.

Q: What are some examples of recurring decimals that can be converted to fractions using the method of multiplication?

A: Some examples of recurring decimals that can be converted to fractions using the method of multiplication include:

• 0.333...
• 0.444...
• 0.666...
• 0.999...

Q: Can I use the method of multiplication to convert any recurring decimal to a fraction?

A: Yes, the method of multiplication can be used to convert any recurring decimal to a fraction. However, the method may become more complex for recurring decimals with multiple repeating blocks of digits.

Q: What are some common mistakes to avoid when converting recurring decimals to fractions using the method of multiplication?

A: Some common mistakes to avoid when converting recurring decimals to fractions using the method of multiplication include:

• Not multiplying both sides of the equation by a power of 10.
• Not subtracting the original equation from the new equation.
• Not dividing both sides of the equation by the resulting number.

Q: Can I use a calculator to convert recurring decimals to fractions?

A: Yes, you can use a calculator to convert recurring decimals to fractions. However, the method of multiplication is a useful technique to learn for converting recurring decimals to fractions.

Q: What are some real-world applications of converting recurring decimals to fractions?

A: Some real-world applications of converting recurring decimals to fractions include:

• Calculating interest rates on loans.
• Converting between different units of measurement (e.g. inches to feet).
• Calculating the area and perimeter of shapes.

Q: Can I use the method of multiplication to convert decimals with multiple repeating blocks of digits to fractions?

A: Yes, the method of multiplication can be used to convert decimals with multiple repeating blocks of digits to fractions. However, the method may become more complex and require multiple steps.

Q: What are some tips for converting recurring decimals to fractions using the method of multiplication?

A: Some tips for converting recurring decimals to fractions using the method of multiplication include:

• Use a power of 10 that is a multiple of the number of repeating digits.
• Make sure to subtract the original equation from the new equation.
• Divide both sides of the equation by the resulting number.

Conclusion

In this article, we have answered some frequently asked questions about recurring decimals and fractions. We have discussed the method of multiplication, which is a technique used to convert recurring decimals to fractions. We have also provided some examples of recurring decimals that can be converted to fractions using the method of multiplication, as well as some tips and common mistakes to avoid.