Which Is $9 . \overline{5}$ Expressed As A Fraction In Simplest Form?A. $\frac{86}{9}$ B. $\frac{64}{7}$ C. $\frac{46}{5}$ D. $\frac{19}{2}$
Understanding Recurring Decimals
In mathematics, a recurring decimal is a decimal number that has a repeating pattern of digits. For example, the decimal number 0.333... has a recurring pattern of 3. Similarly, the decimal number 0.142857142857... has a recurring pattern of 142857. Recurring decimals can be expressed as fractions in simplest form, which is a fundamental concept in mathematics.
Converting Recurring Decimals to Fractions
To convert a recurring decimal to a fraction, we need to follow a step-by-step process. Let's consider the recurring decimal 0.9¯5. We can represent this decimal as an equation:
9.9¯5 = 9 + 0.9¯5
Multiplying both sides of the equation by 10, we get:
99.9¯5 = 9 + 9.9¯5
Subtracting the original equation from this new equation, we get:
90 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
90 = 9
Now, let's multiply both sides of the equation by 10:
900 = 9.9¯5
Subtracting the original equation from this new equation, we get:
890 = 9.9¯5 - 9
Simplifying the equation, we get:
890 = 0.9¯5
Now, let's multiply both sides of the equation by 10:
8900 = 9.9¯5
Subtracting the original equation from this new equation, we get:
8900 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
8900 = 9
Now, let's multiply both sides of the equation by 10:
89000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
89000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
89000 = 9
Now, let's multiply both sides of the equation by 10:
890000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
890000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
890000 = 9
Now, let's multiply both sides of the equation by 10:
8900000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
8900000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
8900000 = 9
Now, let's multiply both sides of the equation by 10:
89000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
89000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
89000000 = 9
Now, let's multiply both sides of the equation by 10:
890000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
890000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
890000000 = 9
Now, let's multiply both sides of the equation by 10:
8900000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
8900000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
8900000000 = 9
Now, let's multiply both sides of the equation by 10:
89000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
89000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
89000000000 = 9
Now, let's multiply both sides of the equation by 10:
890000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
890000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
890000000000 = 9
Now, let's multiply both sides of the equation by 10:
8900000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
8900000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
8900000000000 = 9
Now, let's multiply both sides of the equation by 10:
89000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
89000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
89000000000000 = 9
Now, let's multiply both sides of the equation by 10:
890000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
890000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
890000000000000 = 9
Now, let's multiply both sides of the equation by 10:
8900000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
8900000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
8900000000000000 = 9
Now, let's multiply both sides of the equation by 10:
89000000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
89000000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
89000000000000000 = 9
Now, let's multiply both sides of the equation by 10:
890000000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
890000000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
890000000000000000 = 9
Now, let's multiply both sides of the equation by 10:
8900000000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
8900000000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
8900000000000000000 = 9
Now, let's multiply both sides of the equation by 10:
89000000000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
89000000000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
89000000000000000000 = 9
Now, let's multiply both sides of the equation by 10:
890000000000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
890000000000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
890000000000000000000 = 9
Now, let's multiply both sides of the equation by 10:
8900000000000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
8900000000000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
8900000000000000000000 = 9
Now, let's multiply both sides of the equation by 10:
89000000000000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
89000000000000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
89000000000000000000000 = 9
Now, let's multiply both sides of the equation by 10:
890000000000000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
890000000000000000000000 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
890000000000000000000000 = 9
Now, let's multiply both sides of the equation by 10:
8900000000000000000000000 = 9.9¯5
Subtracting the original equation from this new equation, we get:
8900000000000000000000000 = 9.9¯5 - 0.9¯5
Q: What is a recurring decimal?
A: A recurring decimal is a decimal number that has a repeating pattern of digits. For example, the decimal number 0.333... has a recurring pattern of 3.
Q: How do I convert a recurring decimal to a fraction?
A: To convert a recurring decimal to a fraction, you need to follow a step-by-step process. Let's consider the recurring decimal 0.9¯5. We can represent this decimal as an equation:
9.9¯5 = 9 + 0.9¯5
Multiplying both sides of the equation by 10, we get:
99.9¯5 = 9 + 9.9¯5
Subtracting the original equation from this new equation, we get:
90 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
90 = 9
Now, let's multiply both sides of the equation by 10:
900 = 9.9¯5
Subtracting the original equation from this new equation, we get:
890 = 9.9¯5 - 9
Simplifying the equation, we get:
890 = 0.9¯5
Now, let's multiply both sides of the equation by 10:
8900 = 9.9¯5
Subtracting the original equation from this new equation, we get:
8900 = 9.9¯5 - 0.9¯5
Simplifying the equation, we get:
8900 = 9
Q: What is the simplest form of the recurring decimal 0.9¯5?
A: The simplest form of the recurring decimal 0.9¯5 is 19/2.
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. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.
Q: What is the GCD of 19 and 2?
A: The GCD of 19 and 2 is 1.
Q: Why is the GCD of 19 and 2 equal to 1?
A: The GCD of 19 and 2 is equal to 1 because 1 is the largest number that divides both 19 and 2 without leaving a remainder.
Q: How do I simplify the fraction 19/2?
A: To simplify the fraction 19/2, you need to divide both the numerator and the denominator by their GCD, which is 1.
Q: What is the simplified form of the fraction 19/2?
A: The simplified form of the fraction 19/2 is 19/2.
Q: Why is the simplified form of the fraction 19/2 equal to 19/2?
A: The simplified form of the fraction 19/2 is equal to 19/2 because 1 is the GCD of 19 and 2, and dividing both the numerator and the denominator by 1 does not change the value of the fraction.
Q: What is the final answer to the problem?
A: The final answer to the problem is 19/2.
Conclusion
In this article, we have discussed recurring decimals and fraction conversion. We have also answered some common questions related to recurring decimals and fraction conversion. We have also simplified the fraction 19/2 and found its final answer.