To convert the decimal 0.735 to a fraction in simplest form, follow these steps:
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Write the decimal as a fraction: \[ 0.735 = \frac{735}{1000} \]
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Simplify the fraction by finding the greatest common divisor (GCD) of 735 and 1000. First, we can find the prime factorization of both numbers.
- For 735:
- 735 is divisible by 5 (since it ends with a 5): \[ 735 \div 5 = 147 \]
- Now, factor 147. The sum of the digits \(1 + 4 + 7 = 12\) is divisible by 3, so: \[ 147 \div 3 = 49 \]
- Factor 49, which is \(7 \times 7 = 7^2\).
Thus, the prime factorization of 735 is: \[ 735 = 5 \times 3 \times 7^2 \]
- For 1000: \[ 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \]
- For 735:
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Now we need to find the GCD. The common factors are \(5\), and the highest power of 5 that divides both is \(5^1\).
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Divide both the numerator and the denominator by the GCD to simplify the fraction: \[ \frac{735 \div 5}{1000 \div 5} = \frac{147}{200} \]
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Check if \(147\) and \(200\) can be simplified further. The prime factorization of \(147\) is \(3 \times 7^2\) and for \(200\) it is \(2^3 \times 5^2\). They share no common factors, which means \(\frac{147}{200}\) is in simplest form.
Therefore, the decimal \(0.735\) as a fraction in simplest form is: \[ \frac{147}{200} \]