Question

Convert the decimal below to a fraction in simplest form.
0.465

Answer 1

what about 0.786

Answer 2

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:
\[
\frac{93}{200}
\]

Answer 3

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:
\[
\frac{393}{500}
\]