Express the repeating decimal as the ratio of two integers. 0.03 = 0.0333… 1. The ratio that represents the repeating decimal is (Type an integer or a simplified fraction.)
The Correct Answer and Explanation is :
To express the repeating decimal ( 0.0333\ldots ) (where the 3 repeats indefinitely) as a ratio of two integers, we need to convert the repeating decimal into a fraction.
Step-by-step process:
- Let ( x = 0.0333\ldots ). This means the number is ( 0.0 ) followed by a repeating decimal ( 333\ldots ).
- Multiply both sides of the equation by 10 to move the decimal point to the right of the first digit. This results in:
[
10x = 0.3333\ldots
] - Subtract the original equation from the new equation:
[
10x – x = 0.3333\ldots – 0.0333\ldots
]
Simplifying this:
[
9x = 0.3
] - Solve for ( x ) by dividing both sides of the equation by 9:
[
x = \frac{0.3}{9}
] - Convert the decimal 0.3 into a fraction:
[
0.3 = \frac{3}{10}
]
So now the equation becomes:
[
x = \frac{\frac{3}{10}}{9}
] - Simplify the expression:
[
x = \frac{3}{10} \times \frac{1}{9} = \frac{3}{90}
] - Simplify the fraction:
[
\frac{3}{90} = \frac{1}{30}
]
Thus, the repeating decimal ( 0.0333\ldots ) is equal to the fraction ( \frac{1}{30} ).
Conclusion:
The repeating decimal ( 0.0333\ldots ) (where the 3 repeats indefinitely) can be expressed as the ratio ( \frac{1}{30} ). This conversion shows how repeating decimals can be transformed into simple fractions using algebraic methods, which is useful for many applications in mathematics.