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The decimal form of fractions
Based on what you have learned about decimal numbers, it will be easy to convert fractions with denominators of \latex{ 10; 100 } and \latex{ 1,000 }.
Example 1
Convert the following fractions to decimals.
a) \latex{\frac{3}{10}}
b) \latex{\frac{7}{100}}
c) \latex{\frac{13}{1000}}
Solution
a) \latex{\frac{3}{10} = 0.3}
b) \latex{\frac{7}{100}=0.07}
c) \latex{\frac{13}{1000}=0.013}
Some fractions are easy to convert into decimal form, even though their denominators are not \latex{ 10; 100 } or \latex{ 1,000 }. You just need to expand or simplify their denominators.
Example 2
Convert the following fractions to decimals.
a) \latex{\frac{1}{2}}
b) \latex{\frac{5}{4}}
c) \latex{\frac{12}{30}}
Solution 1
(by expansion or simplification)
a) \latex{\frac{1}{2}= \frac{5}{10} = 0.5}
b) \latex{\frac{5}{4}= \frac{125}{100} = 1.25}
b) \latex{\frac{12}{30}= \frac{4}{10} = 0.4}
Solution 2
(division)
A fraction is a quotient where the nominator is the dividend, and the denominator is the divisor.
a) \latex{\frac{1}{2} = 1\div2}
b) \latex{\frac{5}{4} = 5\div4}
c)\latex{\frac{12}{30} = 12\div30}
\latex{ 1 }
\latex{ \div }
\latex{ 2 }
\latex{= }
\latex{0 }
\latex{. }
\latex{5}
\latex{ 1 }
\latex{ 0 }
\latex{ 0 }
\latex{ 5 }
\latex{ \div }
\latex{ 4 }
\latex{ = }
\latex{ 1 }
\latex{ . }
\latex{ 2 }
\latex{ 5 }
\latex{ 1 }
\latex{ 0 }
\latex{ 2 }
\latex{ 0 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ \div }
\latex{ 3}
\latex{ =}
\latex{ 0}
\latex{ 0}
\latex{ .}
\latex{ 4}
\latex{ 1 }
\latex{ 2 }
\latex{ 0 }
\latex{ 0 }
\latex{\frac{1}{2}= 0.5}
\latex{\frac{5}{4}= 1.25}
\latex{\frac{12}{30}= 0.4}
A finite decimal (also known as a terminating decimal) has no remainder after completing the division.
If you can expand or simplify a fraction to a denominator of \latex{ 10; 100 } or \latex{ 1,000 } (or any other power of ten), then it is a finite decimal.
Repeating decimals
How can you convert fractions into decimals that cannot be expanded or simplified to denominators of \latex{ 10; 100 } or \latex{ 1,000 }?
Example 3
Convert the following fractions to decimals.
a) \latex{\frac{1}{3}}
b) \latex{\frac{2}{11}}
c) \latex{\frac{7}{27}}
d) \latex{\frac{5}{6}}
Solution
Divide the numerator by the denominator, as in Example 2.
a) \latex{\frac{1}{3} = 1\div3}
\latex{\frac{1}{3} = 0.333 \; ... = 0.\dot{3}}
\latex{ 1 }
\latex{ \div }
\latex{ 3 }
\latex{ = }
\latex{ 0 }
\latex{ . }
\latex{ 3 }
\latex{ 3 }
\latex{ 3 }
\latex{ 1 }
\latex{ 0 }
\latex{ 1 }
\latex{ 0 }
\latex{ 1 }
\latex{ 0 }
\latex{ 1 }
b) \latex{\frac{2}{11} = 2\div11}
\latex{\frac{2}{11} = 0.1818 \; ... = 0.\dot{1}\dot{8}}
\latex{ 2 }
\latex{ \div }
\latex{ 1 }
\latex{ 1 }
\latex{ = }
\latex{ 0 }
\latex{ . }
\latex{ 1 }
\latex{ 8 }
\latex{ 1 }
\latex{ 8 }
\latex{ 2 }
\latex{ 0 }
\latex{ 9 }
\latex{ 0 }
\latex{ 2 }
\latex{ 2 }
\latex{ 0 }
\latex{ 9 }
\latex{ 0 }
\latex{ 2 }
c) \latex{\frac{7}{27} = 7\div27}
\latex{\frac{7}{27} = 0.259259 \; ... = 0.\dot{2}5\dot{9}}
\latex{ 7 }
\latex{ \div }
\latex{ 2 }
\latex{ 7 }
\latex{ = }
\latex{ 0 }
\latex{ . }
\latex{ 2 }
\latex{ 5 }
\latex{ 9 }
\latex{ 2 }
\latex{ 7 }
\latex{0 }
\latex{1}
\latex{6}
\latex{0}
\latex{2}
\latex{5}
\latex{0}
\latex{7}
\latex{0}
\latex{1}
\latex{6}
d) \latex{\frac{5}{6} = 5\div6}
\latex{\frac{5}{6} = 0.833 \; ... = 0.8\dot{3}}
\latex{ 5 }
\latex{ \div }
\latex{ 6 }
\latex{ = }
\latex{ 0 }
\latex{ . }
\latex{ . }
\latex{ 8 }
\latex{ 3 }
\latex{ 3 }
\latex{ 5 }
\latex{ 0 }
\latex{ 2 }
\latex{ 0 }
\latex{ 2 }
\latex{ 0 }
\latex{ 2 }
After completing the divisions, the remainder was not \latex{ 0 } because the digits started repeating after a certain point. Repeating decimals have a set of digits that repeat uniformly without ending. The repeating term is as long as the number of digits that repeat and is marked by a dot above the first and the last digits of the term.
If you cannot expand or simplify a fraction to a denominator of \latex{ 10; 100 } or \latex{ 1,000 } (or any other power of ten), then it is an infinite (or non-terminating) decimal.
Exercises
{{exercise_number}}. Determine whether the following fractions are finite or infinite.
a) \latex{\frac{1}{2}}
b) \latex{\frac{1}{4}}
c) \latex{\frac{7}{20}}
d) \latex{\frac{13}{50}}
e) \latex{\frac{9}{25}}
f) \latex{\frac{1}{6}}
g) \latex{\frac{26}{39}}
h) \latex{\frac{1}{5}}
i) \latex{\frac{7}{10}}
j) \latex{\frac{2}{100}}
k) \latex{\frac{7}{9}}
l) \latex{\frac{4}{3}}
m) \latex{\frac{11}{6}}
n) \latex{\frac{15}{24}}
{{exercise_number}}. Convert the following fractions into decimals, then round the decimals to one, two or three digits.
a) \latex{\frac{7}{16}}
b) \latex{\frac{5}{3}}
c) \latex{\frac{3}{32}}
d) \latex{1\frac{4}{9}}
e) \latex{\frac{22}{7}}
f) \latex{\frac{9}{16}}
g) \latex{\frac{11}{6}}
h) \latex{\frac{3}{4}}
i) \latex{\frac{72}{10}}
j) \latex{\frac{13}{8}}
k) \latex{\frac{2}{3}}
l) \latex{\frac{5}{8}}
m) \latex{\frac{4}{2}}
n) \latex{\frac{2}{4}}
{{exercise_number}}. This highly specialised marvel of machinery is capable of converting fractions into decimals. What are the products of its conversions?
\latex{\frac{5}{6}}
a)
b)
\latex{\frac{9}{8}}
c)
\latex{\frac{7}{3}}
d)
\latex{\frac{12}{18}}
e)
\latex{\frac{65}{52}}
f)
\latex{\frac{31}{7}}
{{exercise_number}}. Convert the following decimals to simple fractions.
a) \latex{ 0.25 };
b) \latex{ 3.2 }
c) \latex{ 0.17 }
d) \latex{ 12.12 }
e) \latex{ 20.1 }
f) \latex{ 5.005 }
{{exercise_number}}. You have a set of dominoes that you can only place next to each other if the fraction on one piece is the same as the decimal on the other piece. At the end of the game, you're left with one domino unplaced. Which one is it?
\latex{\textbf{0.4}}
\latex{\frac{\textbf{3}}{\textbf{4}}}
\latex{\textbf{0.48}}
\latex{\textbf{0.}\dot{\textbf{6}}}
\latex{\textbf{0.55}}
\latex{\textbf{0.75}}
\latex{\textbf{1.5}}
\latex{\textbf{1.}\dot{\textbf{6}}}
\latex{\textbf{0.9}}
\latex{\textbf{0.}\dot{\textbf{3}}}
\latex{\textbf{0.8}}
\latex{\textbf{0.25}}
\latex{\textbf{0.35}}
\latex{\frac{\textbf{2}}{\textbf{5}}}
\latex{\frac{\textbf{1}}{\textbf{4}}}
\latex{\frac{\textbf{3}}{\textbf{5}}}
\latex{\frac{\textbf{7}}{\textbf{20}}}
\latex{\frac{\textbf{1}}{\textbf{3}}}
\latex{\frac{\textbf{36}}{\textbf{40}}}
\latex{\frac{\textbf{2}}{\textbf{3}}}
\latex{\frac{\textbf{4}}{\textbf{5}}}
\latex{\frac{\textbf{5}}{\textbf{3}}}
\latex{\frac{\textbf{12}}{\textbf{25}}}
\latex{\frac{\textbf{3}}{\textbf{2}}}
{{exercise_number}}. Which of the following fractions have finite decimal forms?
a) \latex{\frac{16}{25}} ; \latex{\frac{25}{16}}
b) \latex{\frac{9}{8}} ; \latex{\frac{8}{9}}
c) \latex{\frac{12}{7}} ; \latex{\frac{7}{12}}
d) \latex{\frac{35}{28}} ; \latex{\frac{28}{35}}
{{exercise_number}}. After converting the following fractions to decimals, what number stands at the tenth place after the decimal point?
a) \latex{\frac{9}{12}}
b) \latex{\frac{10}{16}}
c) \latex{\frac{12}{15}}
e) \latex{\frac{35}{28}}
f) \latex{1\frac{2}{3}}
g) \latex{2\frac{3}{4}}
{{exercise_number}}. After converting the following fractions to decimals, what number stands at the \latex{ 2,006 }th place after the decimal point?
a) \latex{\frac{5}{9}}
b) \latex{\frac{6}{11}}
c) \latex{\frac{7}{110}}
d) \latex{\frac{1}{7}}
e) \latex{\frac{3}{13}}
f) \latex{1\frac{9}{11}}
g) \latex{1\frac{9}{11}}
