Question

Express the recurring decimal 0.034 as a fraction in its simplest form but only the 3 and 4 recur.

Options:
a) 17/500
b) 34/999
c) 34/1001
d) 17/499

Answer 1

The recurring decimal 0.034, with only the 3 and 4 recurring, is expressed as the fraction 34/999 when converting it into its simplest form. The correct option is b) 34/999.

Step-by-step explanation:

To express the recurring decimal 0.034 where only the 3 and 4 recur as a fraction in its simplest form, we'll denote the recurring part of the number, which is 34 repeating, and separate it from the non-repeating part. Let the recurring decimal be represented by 'x':

x = 0.0343434...

Now, multiply 'x' by 1000 to shift the decimal point three places to the right:

1000x = 34.343434...

Since the digits 34 are repeating, we can set up the following equation:

1000x - x = 34.343434... - 0.0343434...

999x = 34.309

Now divide both sides of the equation by 999:

x = 34.309 / 999

We can simplify this fraction by noticing that 34.309 is the same as 34309 divided by 1000, so:

x = 34309 / (999 * 1000)

x = 34309 / 999000

Finally, you can simplify this fraction either manually or using a calculator to find that:

x = 34/999

Therefore, the correct answer is option b) 34/999.