Express the recurring decimal 0.004 as a fraction.
The Correct Answer and Explanation is :
To express the recurring decimal (0.004) as a fraction, we first need to clarify what is meant by “recurring decimal.” In this case, it seems there is some confusion, as (0.004) does not have a repeating part; it is a terminating decimal. However, I’ll explain how to convert both a terminating decimal and a recurring decimal to a fraction.
Converting a Terminating Decimal to a Fraction
- Understanding the Decimal: The decimal (0.004) can be expressed as:
[
0.004 = \frac{4}{1000}
] - Simplifying the Fraction: Next, we simplify (\frac{4}{1000}):
- The greatest common divisor (GCD) of 4 and 1000 is 4.
- Divide both the numerator and the denominator by their GCD:
[
\frac{4 \div 4}{1000 \div 4} = \frac{1}{250}
]
Thus, the decimal (0.004) as a fraction is (\frac{1}{250}).
Converting a Recurring Decimal to a Fraction
If we were to consider a true recurring decimal, such as (0.004444…) (where the digit 4 repeats), the process would be different:
- Let (x = 0.004444…).
- Multiply by a power of 10 that moves the decimal point to the right of the repeating part:
[
1000x = 4.444…
] - Subtract the original equation from this new equation:
[
1000x – x = 4.444… – 0.004444…
]
This simplifies to:
[
999x = 4.44
] - Convert (4.44) to a fraction: (4.44 = \frac{444}{100}).
- Now solve for (x):
[
999x = \frac{444}{100} \implies x = \frac{444}{99900}
] - Simplify: Divide both the numerator and the denominator by the GCD (which is 111):
[
x = \frac{4}{900} = \frac{1}{225}
]
In conclusion, (0.004) is (\frac{1}{250}) as a fraction, while a repeating decimal such as (0.004444…) would convert to (\frac{1}{225}).