Converting Fractions to Recurring Decimals

Oct 13, 2025 by miodeya.com 77 views

Introduction

In mathematics, fractions and decimals are two ways to represent numbers. A fraction is a way to express a part of a whole, while a decimal is a way to express a number in a base-10 system. In this article, we will explore how to convert fractions to recurring decimals and use this knowledge to solve a division problem.

Converting Fractions to Recurring Decimals

A recurring decimal is a decimal that has a repeating pattern of digits. To convert a fraction to a recurring decimal, we can use long division. Let's start with the fraction $\frac{5}{9}$ .

Step 1: Set up the long division

To convert $\frac{5}{9}$ to a recurring decimal, we need to set up a long division problem. We will divide 5 by 9.

  ____________________
9 | 5.000000000000000

Step 2: Perform the long division

Now, we will perform the long division. We will divide 5 by 9 and get a quotient of 0 with a remainder of 5.

  0.5555555555555556
  ____________________
9 | 5.000000000000000
  - 45.00000000000000
  ____________________
  - 40.00000000000000
  - 36.00000000000000
  ____________________
  - 4.000000000000000
  - 3.600000000000000
  ____________________
  - 0.400000000000000

Step 3: Identify the recurring pattern

Now, we will identify the recurring pattern in the decimal. We can see that the pattern 5 repeats indefinitely.

  0.5555555555555556

Therefore, the recurring decimal representation of $\frac{5}{9}$ is 0.5555555555555556.

Solving Division Problems

Now that we have learned how to convert fractions to recurring decimals, let's use this knowledge to solve a division problem. Mumtaz chose two two-digit numbers and divided one by the other. His answer was 2.12121212. We need to find the two numbers used by Mumtaz.

Step 1: Identify the recurring pattern

We can see that the recurring pattern in the decimal is 1212.

  2.121212121212121

Step 2: Convert the recurring decimal to a fraction

Now, we will convert the recurring decimal 2.121212121212121 to a fraction. We can see that the pattern 1212 repeats indefinitely.

  2.121212121212121 = 2 + 0.121212121212121
  0.121212121212121 = 1212/9999
  2.121212121212121 = 2 + 1212/9999

Step 3: Simplify the fraction

Now, we will simplify the fraction 1212/9999. We can see that both the numerator and the denominator have a common factor of 3.

  1212/9999 = (1212/3)/(9999/3)
  1212/9999 = 404/3333

Step 4: Find the two numbers used by Mumtaz

Now, we will find the two numbers used by Mumtaz. We can see that the fraction 404/3333 can be written as 1212/9999.

  404/3333 = 1212/9999

Therefore, the two numbers used by Mumtaz are 1212 and 9999.

Conclusion

In this article, we have learned how to convert fractions to recurring decimals and use this knowledge to solve a division problem. We have seen that the recurring decimal representation of $\frac{5}{9}$ is 0.5555555555555556 and that the two numbers used by Mumtaz are 1212 and 9999. We hope that this article has provided you with a better understanding of fractions and decimals and how to convert them to recurring decimals.

References

[1] "Converting Fractions to Decimals" by Math Open Reference.
Frequently Asked Questions: Converting Fractions to Recurring Decimals and Solving Division Problems =============================================================================================

Q: What is a recurring decimal?

A: A recurring decimal is a decimal that has a repeating pattern of digits. For example, the decimal 0.5555555555555556 is a recurring decimal because the pattern 5 repeats indefinitely.

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

A: To convert a fraction to a recurring decimal, you can use long division. For example, to convert the fraction $\frac{5}{9}$ to a recurring decimal, you would divide 5 by 9 and get a quotient of 0 with a remainder of 5.

Q: What is the recurring decimal representation of $\frac{5}{9}$ ?

A: The recurring decimal representation of $\frac{5}{9}$ is 0.5555555555555556.

Q: How do I solve a division problem using recurring decimals?

A: To solve a division problem using recurring decimals, you need to identify the recurring pattern in the decimal and convert it to a fraction. For example, if the recurring pattern is 1212, you would convert it to the fraction 1212/9999.

Q: What is the fraction equivalent of the recurring decimal 2.121212121212121?

A: The fraction equivalent of the recurring decimal 2.121212121212121 is 2 + 1212/9999.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both numbers by the GCD. For example, to simplify the fraction 404/3333, you would find the GCD of 404 and 3333, which is 1, and divide both numbers by 1.

Q: What are the two numbers used by Mumtaz in the division problem?

A: The two numbers used by Mumtaz in the division problem are 1212 and 9999.

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

A: To know if a decimal is recurring or not, you need to look for a repeating pattern of digits. If you see a repeating pattern, then the decimal is recurring.

Q: Can I convert a recurring decimal to a fraction?

A: Yes, you can convert a recurring decimal to a fraction. To do this, you need to identify the recurring pattern in the decimal and convert it to a fraction.

Q: What is the importance of converting fractions to recurring decimals?

A: Converting fractions to recurring decimals is important because it helps us to solve division problems and understand the relationship between fractions and decimals.

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

A: Yes, you can use a calculator to convert fractions to recurring decimals. However, it's also important to understand the underlying math concepts so that you can solve problems on your own.

Q: How do I practice converting fractions to recurring decimals?

A: To practice converting fractions to recurring decimals, you can try converting different fractions to recurring decimals and then check your answers using a calculator or by simplifying the fraction. You can also try solving division problems using recurring decimals and see if you can get the correct answer.