To convert the decimal \(0.315\) to a fraction, we can express it as:
\[ 0.315 = \frac{315}{1000} \]
Next, we will simplify this fraction. We need to find the greatest common divisor (GCD) of \(315\) and \(1000\).
First, we find the prime factorization of both numbers:
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For 315:
- \(315 \div 3 = 105\)
- \(105 \div 3 = 35\)
- \(35 \div 5 = 7\)
- \(7\) is prime.
Thus, the prime factorization of \(315\) is: \[ 315 = 3^2 \times 5^1 \times 7^1 \]
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For 1000:
- \(1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3\)
Thus, the prime factorization of \(1000\) is: \[ 1000 = 2^3 \times 5^3 \]
Next, we identify the common prime factors:
- The common prime factor is \(5\).
Now we calculate the GCD: \[ \text{GCD}(315, 1000) = 5^1 = 5 \]
Now, we divide both the numerator and the denominator by the GCD:
\[ \frac{315 \div 5}{1000 \div 5} = \frac{63}{200} \]
Now, \(63\) and \(200\) have no common factors other than \(1\). Therefore, the fraction \(\frac{63}{200}\) is in its lowest terms.
Thus, the decimal \(0.315\) as a fraction in its lowest terms is:
\[ \frac{63}{200} \]