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Convert the decimal below to a fraction in simplest form.

0.465

3 answers

To convert the decimal 0.465 to a fraction, follow these steps:

  1. Express the decimal as a fraction: \[ 0.465 = \frac{465}{1000} \]

  2. Simplify the fraction: We need to find the greatest common divisor (GCD) of 465 and 1000.

    • First, find the prime factorization:

      • 465 can be factored into \(3 \times 155 = 3 \times 5 \times 31\)
      • 1000 can be factored into \(10^3 = (2 \times 5)^3 = 2^3 \times 5^3\)

    • The factors are:

      • For 465: \(3, 5, 31\)
      • For 1000: \(2, 5\)

    The common factor is \(5\).

  3. Divide both the numerator and denominator by their GCD: \[ \frac{465 \div 5}{1000 \div 5} = \frac{93}{200} \]

  4. Check if the fraction is in simplest form: The factors of 93 are \(3 \times 31\) and the factors of 200 are \(2^3 \times 5^2\). There are no common factors between 93 and 200, confirming that \(\frac{93}{200}\) is in simplest form.

Thus, the decimal 0.465 as a fraction in simplest form is: \[ \frac{93}{200} \]

what about 0.786

To convert the decimal 0.786 to a fraction in simplest form, follow the same steps as before:

  1. Express the decimal as a fraction: \[ 0.786 = \frac{786}{1000} \]

  2. Simplify the fraction: We need to find the greatest common divisor (GCD) of 786 and 1000.

    • First, let's find the prime factorization:

      • For 786:
        • 786 is even, so divide by 2: \(786 \div 2 = 393\)
        • 393 can be factored into \(3 \times 131\) (since 393 is also divisible by 3).
        • Therefore, the prime factorization is \(2 \times 3 \times 131\).
      • For 1000:
        • As noted earlier, \(1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3\).

    • The factors are:

      • For 786: \(2, 3, 131\)
      • For 1000: \(2, 5\)

    The common factor is \(2\).

  3. Divide both the numerator and denominator by their GCD: \[ \frac{786 \div 2}{1000 \div 2} = \frac{393}{500} \]

  4. Check if the fraction is in simplest form:

    • The factors of 393 are \(3 \times 131\),
    • The factors of 500 are \(2^2 \times 5^3\). Since there are no common factors between 393 and 500, the fraction \(\frac{393}{500}\) is in simplest form.

Thus, the decimal 0.786 as a fraction in simplest form is: \[ \frac{393}{500} \]