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Chapter 2: Problem 42

Express the number as a ratio of integers. \(0.46=0.46464646 \ldots\)

Short Answer

Expert verified
The ratio of integers is \( \frac{46}{99} \).

Step by step solution

01

Set up the equation

Let \( x = 0.464646\ldots \). This notation represents a repeating decimal.
02

Express the repeating decimal

Since the decimal repeats every two digits, multiply \( x \) by 100 to shift the decimal point two places to the right, so we have \( 100x = 46.464646\ldots \).
03

Subtract to eliminate the repeating part

Subtract the equation from Step 1 from the equation in Step 2 to get:\[100x - x = 46.464646\ldots - 0.464646\ldots\]This simplifies to:\[99x = 46\]
04

Solve for x

Divide both sides of the equation by 99 to solve for \( x \):\[x = \frac{46}{99}\]
05

Simplify the fraction

Check if \( \frac{46}{99} \) can be simplified. The greatest common divisor (GCD) of 46 and 99 is 1, so the fraction is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ratios
A ratio is a way to compare two quantities by using division. It shows how many times one value contains or is contained within the other. For example, if we have a repeating decimal like \(0.464646...\), expressed as a ratio, it can compare this repeating part with a whole number.
  • A ratio can be written in different forms such as \(a:b\) or \(\frac{a}{b}\).
  • Converting a repeating decimal to a fraction involves creating a ratio of integers.
To convert the repeating decimal \(0.464646...\) into a ratio of integers, we use algebraic methods. We multiply by powers of 10 to align the repeating digits and make them subtractable. This helps us formulate an equation that isolates the decimal part, allowing us to eventually express it as a fraction, or a ratio between two integers.
Integers
Integers are whole numbers that can be positive, negative, or zero. When expressing a repeating decimal as a ratio of integers, both the numerator and the denominator in the fraction must be integers.
  • The numerator represents the repeating part of the decimal converted into an integer form.
  • The denominator represents the subtractive and divisor factors used to eliminate the decimal part, making it a multiple of the repeating part.
In solving the exercise, we set \(x = 0.464646...\) and multiply by \(100\) to shift the decimal places. This provides us integers to work with, like \(46\) and \(99\), eventually leading us to the fraction \(\frac{46}{99}\), which consists entirely of integers.
Fraction Simplification
Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1.
  • To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator.
  • If the GCD is greater than 1, divide both the numerator and the denominator by this number to simplify.
For \(\frac{46}{99}\), checking for simplification involves finding the GCD of \(46\) and \(99\). Since their GCD is 1, it indicates that the fraction is already in its simplest form. Hence, \(\frac{46}{99}\) cannot be simplified further. This concept emphasizes how important it is to recognize when fractions can or cannot be reduced, ensuring the ratio is as straightforward as possible.

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