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Chapter 11: Problem 51

Find fraction notation. $$0.131313 \ldots, \text { or } 0 . \overline{13}$$

Short Answer

Expert verified
The fraction notation is \( \frac{13}{99} \)

Step by step solution

01

Assign a variable

Let x be the repeating decimal: \( x = 0.131313\text{...} \text{ or } 0.\bar{13} \)
02

Multiply by a power of 10

To eliminate the repeating decimals, multiply both sides of the equation by 100 (since 13 is two digits): \( 100x = 13.131313\text{...} \)
03

Subtract the original variable

To isolate the repeating part, subtract the original equation (Step 1) from this new equation (Step 2): \( 100x - x = 13.131313\text{...} - 0.131313\text{...} \)
04

Simplify the equation

Simplify the left side and right side:\( 99x = 13 \)
05

Solve for x

Divide both sides by 99 to solve for x: \( x = \frac{13}{99} \)

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Key Concepts

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

fraction notation
Fraction notation is a way to express numbers that are not whole numbers.
It helps to represent decimals in a fraction form.
A fraction consists of a numerator (top part) and a denominator (bottom part).
For example, in the fraction \( \frac{1}{2} \), 1 is the numerator and 2 is the denominator.
When converting repeating decimals to fractions, we use fraction notation to represent the repeating decimal precisely.
In the exercise, the repeating decimal \( 0.131313... \) is converted to the fraction \( \frac{13}{99} \).
The fraction notation makes it easier to understand and work with numbers in mathematical problems.
repeating decimals
Repeating decimals are numbers that have one or more digits that repeat infinitely.
For example, \( 0.131313...\) has the digits '13' repeating forever.
These are different from terminating decimals, which come to an end, like \( 0.25 \).
To work with repeating decimals, we can use a bar notation.
So, \( 0.131313... \) can also be written as \( 0.\overline{13} \).
When converting repeating decimals to fractions, identifying the repeating sequence is crucial.
In this exercise, recognizing that '13' repeats helps you to set up the equations needed for conversion.
algebraic manipulation
Algebraic manipulation involves using algebraic methods to solve equations and simplify expressions.
When converting repeating decimals to fractions, algebraic manipulation is essential.
Here鈥檚 how it works:
  • Assign a variable: Set \( x \) to the repeating decimal, like \( x = 0.131313... \).

  • Multiply by a power of 10: Move the decimal point by multiplying both sides by 100 (since 13 is two digits), giving \( 100x = 13.131313... \).

  • Subtract the original variable: Isolate the repeating part by subtracting \( x \) from \( 100x \): \( 100x - x = 13.131313... - 0.131313... \).

  • Simplify the equation: Simplifying \( 100x - x \) gives \( 99x = 13 \).

  • Solve for \( x \): Finally, divide both sides by 99 to find \( x \), so \( x = \frac{13}{99} \).

Through algebraic manipulation, we convert the repeating decimal into a simplified fraction.

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