What will be the decimal expansion of the rational number `(27)/(1250)` ?

Options

A. 0.0125
B. 0.0021
C. 0.0315
D. 0.0216

Answer

To find the decimal expansion of the rational number \( \frac{27}{1250} \), we can follow these steps:

### Step 1: Prime Factorization of the Denominator
First, we need to perform the prime factorization of the denominator, which is 1250.

- Divide 1250 by 2:
\[ 1250 \div 2 = 625 \]
- Now, divide 625 by 5:
\[ 625 \div 5 = 125 \]
- Divide 125 by 5:
\[ 125 \div 5 = 25 \]
- Divide 25 by 5:
\[ 25 \div 5 = 5 \]
- Finally, divide 5 by 5:
\[ 5 \div 5 = 1 \]

Thus, the prime factorization of 1250 is:
\[ 1250 = 2^1 \times 5^4 \]

### Step 2: Adjusting the Denominator
To convert the fraction into a form that can be easily expressed as a decimal, we want the denominator to be a power of 10. Since \( 10 = 2 \times 5 \), we need to balance the powers of 2 and 5.

- The current powers are \( 2^1 \) and \( 5^4 \). To make them equal, we can multiply the numerator and denominator by \( 2^3 \) (which is 8) to make the denominator \( 2^4 \times 5^4 \).

### Step 3: Multiplying by the Necessary Factor
Now we multiply both the numerator and the denominator by \( 2^3 = 8 \):
\[ \frac{27 \times 8}{1250 \times 8} = \frac{216}{10000} \]

### Step 4: Converting to Decimal
Now, we can convert \( \frac{216}{10000} \) into decimal form:
\[ \frac{216}{10000} = 0.0216 \]

### Final Answer
Thus, the decimal expansion of the rational number \( \frac{27}{1250} \) is:
\[ \boxed{0.0216} \]

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