Use Point 9 Repeating To Solve Fractions! – Mind Your Decisions

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!

Or keep reading.
.
.

“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.

.
.

.
.
.
.
M
I
N
D
.
Y
O
U
R
.
D
E
C
I
S
I
O
N
S
.
P
U
Z
Z
L
E
.
.
.
.
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.

MY BOOKS

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Book ratings are from January 2023.

(US and worldwide links)
https://mindyourdecisions.com/blog/my-books

Mind Your Decisions is a compilation of 5 books:

(1) The Joy of Game Theory: An Introduction to Strategic Thinking
(2) 40 Paradoxes in Logic, Probability, and Game Theory
(3) The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias
(4) The Best Mental Math Tricks
(5) Multiply Numbers By Drawing Lines

The Joy of Game Theory shows how you can use math to out-think your competition. (rated 4.3/5 stars on 290 reviews)

40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. (rated 4.2/5 stars on 54 reviews)

The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.1/5 stars on 33 reviews)

The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.3/5 stars on 116 reviews)

Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.4/5 stars on 37 reviews)

Mind Your Puzzles is a collection of the three “Math Puzzles” books, volumes 1, 2, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.

Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 112 reviews.

Math Puzzles Volume 2 is a sequel book with more great problems. (rated 4.2/5 stars on 33 reviews)

Math Puzzles Volume 3 is the third in the series. (rated 4.2/5 stars on 29 reviews)

KINDLE UNLIMITED

Teachers and students around the world often email me about the books. Since education can have such a huge impact, I try to make the ebooks available as widely as possible at as low a price as possible.

Currently you can read most of my ebooks through Amazon’s “Kindle Unlimited” program. Included in the subscription you will get access to millions of ebooks. You don’t need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. I have compiled links to programs in some countries below. Please check your local Amazon website for availability and program terms.

US, list of my books (US)
UK, list of my books (UK)
Canada, book results (CA)
Germany, list of my books (DE)
France, list of my books (FR)
India, list of my books (IN)
Australia, book results (AU)
Italy, list of my books (IT)
Spain, list of my books (ES)
Japan, list of my books (JP)
Brazil, book results (BR)
Mexico, book results (MX)