 Let’s say you need to convert the following fractions into decimal form.

1/3 =
1/11 =
5/11 =
1/33 =
1/101 =
1/271 =
1/6 =
1/7 =
1/17 =

There’s a little trick to use point 9 repeating to make the calculations easier! As usual, watch the video for an explanation.

As usual, watch the video for a solution.

Trick To Convert Fractions To Decimals!

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“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

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Use Point 9 Repeating To Solve Fractions!

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

What is 1/3 = ? You probably know it is 0.3 repeating. But how do we know this? I learned to do the calculation in school using the long division algorithm, illustrated below. That’s a lot of work. Is there an easier way? The trick is that 1 = 0.9 repeating!

1/3
= (0.9999…)/3
= 0.3333…

That’s it! Let’s solve the other problems.

1/11
= (0.9999…)/11
= 0.0909…

5/11 =
= 5×(0.9999…)/11
= 5×(0.0909…)
= 0.4545…

1/33
= (0.9999…)/33
= 0.0303…

To do 1/101, note that 101×99 = 9999. Since we have 4 digits of 9, we need to repeat 4 decimal places. So we need to have 2 leading 0s to have 0099 repeat.

1/101
= (0.9999…)/101
= 0.00990099…

To do 1/271, note that 271×369 = 99999. Since we have 5 digits of 9, we need to repeat 5 decimal places. So we need to have 2 leading 0s to have 00369 repeat.

1/271
= (0.99999…)/271
= 0.000369…

To do 1/6, there is no simple trick. However, using 1 = 0.999… can simplify the long division calculation. To work out 1/7, notice 7×142857 = 999999. The decimal needs 6 digits to repeat, so there is no leading 0.

1/7
= (0.999999…)/7
= 0.142857…

Working out 1/17 is a bit of a joke. But notice 17×588235294117647 = 9999999999999999. The decimal needs 6 digits to repeat, so there is no leading 0.

0.0588235294117647

1/17
= (0.9999999999999999…)/17
= 0.0588235294117647…

Justification

The decimal 0.9 repeating can be defined as the limit of a sum of a geometric series as n goes to infinity. Start with the finite sum:

0.9 repeated to n spots
= 9/10 + 9/100 + … + 9/10n

If we were to add 1/10n to this finite sum we would get 1. So this sum is equal to

= 1 – 1/10n

Taking the limit as n goes to infinity, the second term will vanish. So we have:

= 1 – 0
= 1

Thus 0.9 repeating is equal to 1.

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