Question

Express the following as a rational number in its simplest form: 

`0.bar(365)`

Express the following as a fraction in simplest form.

`0.bar(365)`

Answer 1

1. Set up the initial equation:

Let x equal the repeating decimal:

x = 0.365365365...   ...(Equation 1)

2. Multiply to shift the decimal point:

Since there are 3 repeating digits under the bar, multiply both sides by 10³ = 1000:

1000x = 365.365365365...   ...(Equation 2)

3. Subtract the first equation from the second:

1000x – x = (365.365365...) – (0.365365...)

999x = 365

4. Solve for x:

`x = 365/999`

5. Simplify the fraction:

The prime factorization of the numerator and denominator are:

365 = 5 × 73

999 = 3³ × 37

Since they share no common factors, the greatest common divisor (GCD) is 1. The fraction is already in its simplest form.