What is 0.1 repeating. Convert to a FRACTION
The Correct Answer and Explanation is :
To convert the repeating decimal ( 0.1\overline{3} ) (which means ( 0.13333…)) into a fraction, we can use a simple algebraic method. Here’s a step-by-step explanation:
- Define the repeating decimal: Let ( x = 0.1\overline{3} ).
- Multiply by a power of 10: To eliminate the repeating part, we multiply both sides by 10 (since the decimal repeats after one digit):
[
10x = 1.3\overline{3}
] - Set up an equation: Now, we have two equations:
[
x = 0.1\overline{3} \quad \text{(1)}
]
[
10x = 1.3\overline{3} \quad \text{(2)}
] - Subtract equation (1) from equation (2): This eliminates the repeating part:
[
10x – x = 1.3\overline{3} – 0.1\overline{3}
]
Simplifying gives:
[
9x = 1.2
] - Solve for ( x ): Divide both sides by 9:
[
x = \frac{1.2}{9}
] - Convert ( 1.2 ) to a fraction: We can express ( 1.2 ) as ( \frac{12}{10} ):
[
x = \frac{12/10}{9} = \frac{12}{90}
] - Simplify the fraction: Both the numerator and denominator can be divided by 6:
[
x = \frac{2}{15}
]
Thus, ( 0.1\overline{3} = \frac{2}{15} ).
Summary
The decimal ( 0.1\overline{3} ) is equivalent to the fraction ( \frac{2}{15} ). This conversion involves defining the repeating decimal as a variable, manipulating it algebraically, and simplifying to find the fractional representation. This process is a common method for converting repeating decimals into fractions, providing a clear, systematic approach to handling such numbers.