How do you convert 0.219 (19 repeating) to a fraction?

Question

HIX Tutor · Accepted Answer

undefined Great question! Let's break it down step by step.

1. **Define the number**: Let $ x = 0.219191919...$ (where the "19" repeats).

2. **Multiply by 1000**: Since "19" has 2 digits, multiply by 1000 to move to the right:
   $$ 1000x = 219.191919...$$

3. **Multiply by 10**: Now, multiply $ x $ by 10:
   $$ 10x = 2.191919...$$

4. **Set up an equation**: Now you have two equations:
   $$ 1000x = 219.191919...$$
   $$ 10x = 2.191919...$$

5. **Subtract the second from the first**: 
   $$
   1000x - 10x = 219.191919... - 2.191919...
   $$

This simplifies to:
   $$
   990x = 217
   $$

6. **Solve for $ x $**:
   $$
   x = \frac{217}{990}
   $$

7. **Final fraction**: Thus, $ 0.219191919...$ as a fraction is:
   $$
   \frac{217}{990}
   $$

Do you want to try converting another number to a fraction?

Eva Abramson · Answer

#217/990#

There is a technique for changing recurring decimals to fractions.

Let #x = 0.21919191919....... # to infinity

#10x = 2.19191919191919...#to infinity.

#1000x = 219.1919191919 # to infinity

Do you see what will happen if we subtract #1000x -10x?#

#1000x = 219.1919191919 # to infinity #ul(-10x)" " ul(-2.19191919191919...)#to infinity. #990x = 217.0000000000000...# to infinity

Solve for #x#

#x = 217/990#

Notice the following:

Not all the decimals recur. The recurring starts after 3 decimal places. This was the reason for multiplying by #1000#. The digits must line up so that when you subtract there will be zero's to infinity.

This method can be summarised as follows:

#x = (219-2)990 = 217/990#

#291: " " # write down all the digits until they start to recur. #-2: " "#subtract the digits that do NOT recur

#div 990:" "# divide by a 9 for each recurring digit followed by a 0 for each non-recurring digit.

Avery Alexander · Answer

undefined To convert the repeating decimal 0.219 (19 repeating) to a fraction, we can represent it as $ \frac{219}{1000} $ and simplify it to $ \frac{219}{999} $.