Which Fraction Represents The Decimal $0.\overline{12}$?A. $\frac{1}{12}$ B. $ 3 25 \frac{3}{25} 25 3 [/tex] C. $\frac{4}{33}$ D. $\frac{33}{4}$
Understanding the Problem
When dealing with repeating decimals, it's essential to find a way to express them as fractions. This is because fractions are more manageable and easier to work with in mathematical operations. In this problem, we're given the decimal and asked to find the fraction that represents it.
The Concept of Repeating Decimals
A repeating decimal is a decimal number that goes on forever in a repeating pattern. In this case, the decimal repeats the pattern indefinitely. To express this as a fraction, we need to find a way to capture the repeating pattern.
Converting Repeating Decimals to Fractions
One way to convert a repeating decimal to a fraction is to use algebraic manipulation. Let's assume that the repeating decimal is equal to some fraction . We can then set up an equation to solve for .
Setting Up the Equation
Let . Since the decimal repeats every two digits, we can multiply by to shift the decimal two places to the right. This gives us .
Subtracting the Original Equation
Now, let's subtract the original equation from the equation . This will help us eliminate the repeating decimal.
Simplifying the Equation
When we subtract the original equation from the equation , we get:
This simplifies to:
Solving for x
Now, we can solve for by dividing both sides of the equation by .
Reducing the Fraction
The fraction can be reduced by dividing both the numerator and the denominator by their greatest common divisor, which is .
Conclusion
Therefore, the fraction that represents the decimal is .
Answer
The correct answer is:
C.
Explanation
The other options are incorrect because they do not accurately represent the decimal . Option A, , is a different fraction that does not have the same decimal representation. Option B, , is also incorrect because it does not match the decimal . Option D, , is the reciprocal of the correct answer, but it is not the correct representation of the decimal .
Real-World Applications
Understanding how to convert repeating decimals to fractions is essential in various real-world applications, such as finance, engineering, and science. For example, in finance, repeating decimals can be used to calculate interest rates and investments. In engineering, repeating decimals can be used to calculate the dimensions of structures and machines. In science, repeating decimals can be used to calculate the values of physical constants and properties.
Tips and Tricks
When dealing with repeating decimals, it's essential to remember the following tips and tricks:
- Use algebraic manipulation to convert repeating decimals to fractions.
- Set up an equation to solve for the fraction.
- Subtract the original equation from the equation to eliminate the repeating decimal.
- Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor.
- Check your answer by converting the fraction back to a decimal.
By following these tips and tricks, you can confidently convert repeating decimals to fractions and solve problems in various real-world applications.
Q: What is a repeating decimal?
A: A repeating decimal is a decimal number that goes on forever in a repeating pattern. For example, the decimal repeats the pattern indefinitely.
Q: Why is it important to convert repeating decimals to fractions?
A: Converting repeating decimals to fractions is essential in various real-world applications, such as finance, engineering, and science. Fractions are more manageable and easier to work with in mathematical operations.
Q: How do I convert a repeating decimal to a fraction?
A: To convert a repeating decimal to a fraction, you can use algebraic manipulation. Let's assume that the repeating decimal is equal to some fraction . You can then set up an equation to solve for .
Q: What is the step-by-step process for converting a repeating decimal to a fraction?
A: The step-by-step process for converting a repeating decimal to a fraction is as follows:
- Let .
- Multiply by to shift the decimal two places to the right. This gives us .
- Subtract the original equation from the equation .
- Simplify the equation to get .
- Solve for by dividing both sides of the equation by .
- Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor.
Q: What are some common mistakes to avoid when converting repeating decimals to fractions?
A: Some common mistakes to avoid when converting repeating decimals to fractions include:
- Not using algebraic manipulation to convert the repeating decimal to a fraction.
- Not setting up an equation to solve for the fraction.
- Not subtracting the original equation from the equation to eliminate the repeating decimal.
- Not reducing the fraction by dividing both the numerator and the denominator by their greatest common divisor.
Q: How do I check my answer when converting a repeating decimal to a fraction?
A: To check your answer when converting a repeating decimal to a fraction, you can convert the fraction back to a decimal. If the decimal representation matches the original repeating decimal, then your answer is correct.
Q: What are some real-world applications of converting repeating decimals to fractions?
A: Some real-world applications of converting repeating decimals to fractions include:
- Finance: Converting repeating decimals to fractions can be used to calculate interest rates and investments.
- Engineering: Converting repeating decimals to fractions can be used to calculate the dimensions of structures and machines.
- Science: Converting repeating decimals to fractions can be used to calculate the values of physical constants and properties.
Q: What are some tips and tricks for converting repeating decimals to fractions?
A: Some tips and tricks for converting repeating decimals to fractions include:
- Use algebraic manipulation to convert repeating decimals to fractions.
- Set up an equation to solve for the fraction.
- Subtract the original equation from the equation to eliminate the repeating decimal.
- Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor.
- Check your answer by converting the fraction back to a decimal.
By following these tips and tricks, you can confidently convert repeating decimals to fractions and solve problems in various real-world applications.