# Decimal Representation of Fractions

Fractions can be represented in decimal form as either exact decimal equivalents, or as rough approximations.

In mental calculations, especially division, it is frequently helpful to know how to convert fractions with denominators up to 11 into decimal form. It’s a simple matter to memorize them.

## Commonly Known Decimal Representations

In this section are

^{1}⁄_{2}= 0.5^{1}⁄_{3}= 0.3̅ (a bar over a number or numbers denotes that they repeat infinitely, in this case: 0.33333...)^{2}⁄_{3}= 0.6̅^{1}⁄_{4}= 0.25^{2}⁄_{4}= 0.5^{3}⁄_{4}= 0.75

## Lesser Known Decimal Representations Up To 11ths

Many of the below aren’t well known. However, many have patterns that, once learned, make them easy to recall.

Pattern for 5ths: Double the numerator and place a decimal in front of it.

^{1}⁄_{5}= 0.2^{2}⁄_{5}= 0.4^{3}⁄_{5}= 0.6^{4}⁄_{5}= 0.8

Pattern: Most of the 6ths you already know, as they reduce to other fractions. You simply need to learn ^{1}⁄_{6} and ^{5}⁄_{6}.

^{1}⁄_{6}= 0.16̅^{2}⁄_{6}=^{1}⁄_{3}= 0.3̅^{3}⁄_{6}=^{1}⁄_{2}= 0.5^{4}⁄_{6}=^{2}⁄_{3}= 0.6̅^{5}⁄_{6}= 0.83̅

Pattern: 7ths infinitely repeat the sequence .142857 starting at different points. Multiply the numerator by 14 to get a rough idea of the starting point. For example, to figure out ^{4}⁄_{7}, multiply 4×14=56, which starts with a 5, which tells you that ^{4}⁄_{7} starts with 0.5.

^{1}⁄_{7}= 0.1̅4̅2̅8̅5̅7̅^{2}⁄_{7}= 0.2̅8̅5̅7̅1̅2̅^{3}⁄_{7}= 0.4̅2̅8̅5̅7̅1̅^{4}⁄_{7}= 0.5̅7̅1̅4̅2̅8̅^{5}⁄_{7}= 0.7̅1̅4̅2̅8̅5̅^{6}⁄_{7}= 0.8̅5̅7̅1̅4̅2̅

Pattern: Multiply the numerator by 125, and place a decimal in front of it.

^{1}⁄_{8}= 0.125^{2}⁄_{8}=^{1}⁄_{4}= 0.25^{3}⁄_{8}= 0.375^{4}⁄_{8}=^{1}⁄_{2}= 0.5^{5}⁄_{8}= 0.625^{6}⁄_{8}=^{3}⁄_{4}= 0.75^{7}⁄_{8}= 0.875

Pattern: Place a decimal in front of the numerator, and repeat the numerator endlessly.

^{1}⁄_{9}= 0.1̅^{2}⁄_{9}= 0.2̅^{3}⁄_{9}=^{1}⁄_{3}= 0.3̅^{4}⁄_{9}= 0.4̅^{5}⁄_{9}= 0.5̅^{6}⁄_{9}=^{2}⁄_{3}= 0.6̅^{7}⁄_{9}= 0.7̅^{8}⁄_{9}= 0.8̅

Pattern: Place a decimal in front of the numerator.

^{1}⁄_{10}= 0.1^{2}⁄_{10}=^{1}⁄_{5}= 0.2^{3}⁄_{10}= 0.3^{4}⁄_{10}=^{2}⁄_{5}= 0.4^{5}⁄_{10}=^{1}⁄_{2}= 0.5^{6}⁄_{10}=^{3}⁄_{5}= 0.6^{7}⁄_{10}= 0.7^{8}⁄_{10}=^{4}⁄_{5}= 0.8^{9}⁄_{10}= 0.9

Pattern: Multiply the numerator by 9, repeat that answer endlessly, and place a decimal in front of it.

^{1}⁄_{11}= 0.0̅9̅^{2}⁄_{11}= 0.1̅8̅^{3}⁄_{11}= 0.2̅7̅^{4}⁄_{11}= 0.3̅6̅^{5}⁄_{11}= 0.4̅5̅^{6}⁄_{11}= 0.5̅4̅^{7}⁄_{11}= 0.6̅3̅^{8}⁄_{11}= 0.7̅2̅^{9}⁄_{11}= 0.8̅1̅^{10}⁄_{11}= 0.9̅0̅