Express the repeating decimal as the ratio of two integers. 0.3 = 0.3333… The ratio of two integers is . (Type an integer or a simplified fraction.)
The Correct Answer and Explanation is :
To express the repeating decimal ( 0.3333\ldots ) (also written as ( 0.\overline{3} )) as a ratio of two integers, we need to convert the decimal into a fraction.
Step-by-step conversion process:
- Define the repeating decimal:
Let ( x = 0.3333\ldots ), where the digit 3 repeats infinitely. - Multiply both sides by 10:
To get rid of the repeating decimal, multiply both sides of the equation by 10. This gives us:
[
10x = 3.3333\ldots
]
Now, the decimal part on the right-hand side is the same as the original ( x ), so we can write the equation as:
[
10x = 3 + 0.3333\ldots
]
Therefore, we have:
[
10x = 3 + x
] - Solve for ( x ):
Now, subtract ( x ) from both sides to isolate ( x ) on one side of the equation:
[
10x – x = 3
]
Simplifying this:
[
9x = 3
]
Finally, solve for ( x ) by dividing both sides by 9:
[
x = \frac{3}{9}
] - Simplify the fraction:
The fraction ( \frac{3}{9} ) can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3:
[
x = \frac{3 \div 3}{9 \div 3} = \frac{1}{3}
]
Final result:
Thus, the repeating decimal ( 0.3333\ldots ) is equivalent to the fraction ( \frac{1}{3} ).
Explanation:
A repeating decimal occurs when a digit or group of digits repeats infinitely. In the case of ( 0.\overline{3} ), the digit 3 repeats indefinitely. By using algebraic manipulation, we can convert repeating decimals into fractions, which can be expressed as the ratio of two integers. This is a common technique used in mathematics to simplify and work with repeating decimals. In this case, the repeating decimal ( 0.\overline{3} ) simplifies to the fraction ( \frac{1}{3} ), which is the ratio of the integers 1 and 3.