The Infinite Geometric Series

Oct 13, 2025 by talynt.org 148 views

Solving the Mystery of the Repeating Decimal: Which Simplified Fraction is Equal to 0.1¯7?

In mathematics, decimals can be either terminating or repeating. A repeating decimal is a decimal number that goes on forever in a predictable pattern. In this article, we will explore the concept of repeating decimals and how to convert them into simplified fractions. We will use the given decimal 0.1¯7 as an example to find the equivalent simplified fraction.

A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. In the case of 0.1¯7, the block of digits "7" repeats indefinitely. To convert a repeating decimal into a simplified fraction, we need to use a mathematical technique called the "infinite geometric series."

The Infinite Geometric Series

The infinite geometric series is a mathematical concept that describes a series of numbers that form a geometric progression. In the case of a repeating decimal, the series can be represented as:

a + ar + ar^2 + ar^3 + ...

where a is the first term and r is the common ratio.

Converting 0.1¯7 to a Simplified Fraction

To convert 0.1¯7 to a simplified fraction, we can use the following steps:

Let x = 0.1¯7
Multiply both sides of the equation by 10 to get 10x = 1.7¯7
Subtract the original equation from the new equation to get 9x = 1.6
Divide both sides of the equation by 9 to get x = 16/90

Simplifying the Fraction

The fraction 16/90 can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 16 and 90 is 2. Therefore, we can simplify the fraction as follows:

16 ÷ 2 = 8 90 ÷ 2 = 45

So, the simplified fraction is 8/45.

In conclusion, we have successfully converted the repeating decimal 0.1¯7 to a simplified fraction using the infinite geometric series technique. The equivalent simplified fraction is 8/45. This example demonstrates the importance of understanding repeating decimals and how to convert them into simplified fractions.

Now that we have found the equivalent simplified fraction, let's compare it with the given options:

A. 9/17 B. 8/45 C. 17/9 D. 16/90

Based on our calculation, the correct answer is:

B. 8/45

The final answer is B. 8/45.
Frequently Asked Questions (FAQs) About Repeating Decimals and Simplified Fractions

A: A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. For example, 0.1¯7 is a repeating decimal because the block of digits "7" repeats indefinitely.

A: To convert a repeating decimal to a simplified fraction, you can use the infinite geometric series technique. This involves multiplying both sides of the equation by a power of 10, subtracting the original equation, and then dividing both sides by the resulting coefficient.

A: The infinite geometric series is a mathematical concept that describes a series of numbers that form a geometric progression. In the case of a repeating decimal, the series can be represented as:

a + ar + ar^2 + ar^3 + ...

where a is the first term and r is the common ratio.

A: To find the GCD of two numbers, you can use the Euclidean algorithm. This involves repeatedly dividing the larger number by the smaller number and taking the remainder until the remainder is zero. The last non-zero remainder is the GCD.

A: Simplifying fractions is important because it makes them easier to work with and understand. Simplified fractions have a smaller numerator and denominator, which makes them easier to compare and manipulate.

A: Yes, you can use a calculator to convert a repeating decimal to a simplified fraction. However, it's also important to understand the underlying mathematical concepts so that you can verify the result and make sure it's correct.

A: Some common mistakes to avoid when converting repeating decimals to simplified fractions include:

Not using the correct technique (e.g. using the wrong power of 10)
Not simplifying the fraction correctly (e.g. not dividing both the numerator and denominator by their GCD)
Not verifying the result (e.g. not checking that the simplified fraction is equivalent to the original repeating decimal)

A: Yes, you can use this technique to convert any repeating decimal to a simplified fraction. However, it's worth noting that some repeating decimals may not have a finite simplified fraction (e.g. 0.¯1 = 1/9).

A: To determine if a repeating decimal has a finite simplified fraction, you can use the following criteria:

The repeating block must have a length of 1 (e.g. 0.1¯7)
The repeating block must not have a length of 2 (e.g. 0.1¯27)
The repeating block must not have a length of 3 (e.g. 0.1¯372)

If the repeating block meets these criteria, then the repeating decimal has a finite simplified fraction.