Problem 6 - Calculus Single Variable

Question

Convert the decimal to a rational fraction. (Ellipses are included in some exercises to indicate repetition.) \(15.7231231231 \ldots\)

Step-by-Step Solution

Answer

The decimal 15.7231231231... converts to the rational fraction \(\frac{15707399}{999000}\).

Step 1: Identify the Repeating Part

First, we need to identify the non-repeating and repeating parts of the decimal. Here, the integer part is 15, the non-repeating part is 723, and "123" is the repeating sequence.

Step 2: Set Up Equation for the Decimal

Let's denote the decimal as \( x = 15.723123123... \). Since "123" repeats, we'll need a strategy to handle both the non-repeating and repeating parts.

Step 3: Eliminate the Decimal

Multiply \( x \) by an appropriate power of 10 to shift decimal places. First, multiply by 1000 to eliminate non-repeating decimals: \( 1000x = 15723.123123... \). Then, multiply by 1000000 to target the repeating part: \( 1000000x = 15723123.123123... \).

Step 4: Subtract to Remove Repeating Sequence

By subtracting the first equation from the second, the repeating decimal cancels out: \[1000000x - 1000x = 15723123.123123... - 15723.123123... \]Simplifies to: \[999000x = 15707399 \].

Step 5: Solve for x

Divide both sides of the equation by 999000 to solve for \( x \): \[ x = \frac{15707399}{999000} \].

Step 6: Simplify the Fraction

Finally, simplify \( \frac{15707399}{999000} \) by finding the greatest common divisor (GCD) of the numerator and the denominator, if possible.

Key Concepts

Repeating Decimals

Some decimals go on forever and have a pattern that repeats. They are called repeating decimals. In the decimal given, the part after the decimal point is "123," which repeats continuously. That's why it is noted as "123123123..." Many students often wonder how to handle such decimals. The key is to identify the pattern that repeats. Once you find this pattern, like identifying the word in a song chorus that keeps coming back, managing it becomes easier. Let's consider that identifying these repeating sequences is the first step towards converting them into fractions.

Rational Numbers

When we talk about converting decimals to fractions, we're essentially discussing transforming decimals into rational numbers. A rational number is any number that can be represented as the quotient or fraction of two integers. It's like translating the decimal into a universally understood form, a little like saying the same thing in different languages. For example, when we convert our repeating decimal to a fraction, we express it as a ratio between two numbers. This shows that the decimal indeed was a rational number all along. Understanding that every repeating decimal can be turned into a fraction is key to mastering this concept.

Greatest Common Divisor

A crucial part of converting a decimal to a simplified fraction is using the greatest common divisor (GCD). Think of the GCD as the largest puzzle piece that fits perfectly into both numbers involved. When you find the GCD, you can simplify the fraction to its most basic form, making it easier to understand and use. For instance, if you have a fraction like \( \frac{6}{8} \), the GCD of 6 and 8 is 2. Dividing both the numerator and denominator by 2 gives you the simplified fraction \( \frac{3}{4} \). Simplifying fractions with the GCD is essential in reducing your answer to its neatest form.

Factorization

Factorization is about breaking down numbers into their basic building blocks. For fractions, this process helps when you need to simplify. When you look at a number, think of it like a large LEGO house. Factorization will give you the small LEGO bricks that made it up, making your understanding crisp and clear. By looking at both the numerator and the denominator this way, you spot common factors, which you can use to simplify your fraction. In the case of large numbers, breaking them down to prime numbers, or factorizing, can greatly help in finding how they relate to each other, and ultimately simplify your results.