Question

In the form of \(\frac{p}{2^n \times 5^m}\), 0.505 can be written as

• A. \(\frac{101}{2^1 \times 5^2}\)
• B. \(\frac{101}{2^1 \times 5^3}\)
• C. \(\frac{101}{2^2 \times 5^2}\)
• D. \(\frac{101}{2^3 \times 5^2}\)

Hint:

For terminating decimals, the number of digits after the decimal point gives the power of 10 in the denominator. For example, 0.505 has 3 digits after the decimal, so the denominator is \(10^3 = 1000\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To express a decimal number in the form of a rational number \(\frac{p}{q}\), where the denominator \(q\) is in the form of \(2^n \times 5^m\), we first convert the decimal to a fraction and then simplify it by finding the prime factorization of the numerator and the denominator.

Step 2: Detailed Explanation:
First, convert the decimal 0.505 into a fraction.
\[ 0.505 = \frac{505}{1000} \] Now, find the prime factorization of the numerator and the denominator.
The numerator is 505.
\[ 505 = 5 \times 101 \] The denominator is 1000.
\[ 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \] Now, write the fraction with its prime factors:
\[ \frac{505}{1000} = \frac{5 \times 101}{2^3 \times 5^3} \] Cancel out the common factor of 5 from the numerator and the denominator.
\[ = \frac{101}{2^3 \times 5^{(3-1)}} = \frac{101}{2^3 \times 5^2} \]

Step 3: Final Answer:
Thus, 0.505 can be written as \(\frac{101}{2^3 \times 5^2}\). This matches option (D).