There are three types of decimal numbers: decimal numbers that terminate, decimal numbers that repeat forever, and decimal number that do not repeat but go on forever.
Decimal Numbers That Terminate
Example 1
$${\mathtt{0.6}}$$
1. To turn $${\mathtt{0.6}}$$ into a fraction, notice that the $${\mathtt{6}}$$ is in the tenths spot. Write the fraction as six-tenths: $${\frac{{\mathtt{6}}}{{\mathtt{10}}}}$$
2. Now take $${\frac{{\mathtt{6}}}{{\mathtt{10}}}}$$ and reduce to lowest form: $${\frac{{\mathtt{3}}}{{\mathtt{5}}}}$$
$${\mathtt{0.6}}$$ as a fraction is $${\frac{{\mathtt{3}}}{{\mathtt{5}}}}$$
Example 2
$${\mathtt{0.75}}$$
1. To turn $${\mathtt{0.75}}$$ into a fraction, notice that the $${\mathtt{7}}$$ is in the tnths place and the $${\mathtt{5}}$$ is in the hundredth place. Write the fraction as seventy-five hundredths: $${\frac{{\mathtt{75}}}{{\mathtt{100}}}}$$
2. Now take $${\frac{{\mathtt{75}}}{{\mathtt{100}}}}$$ and reduce it to lowest form: $${\frac{{\mathtt{3}}}{{\mathtt{4}}}}$$
$${\mathtt{0.75}}$$ as a fraction is $${\frac{{\mathtt{3}}}{{\mathtt{4}}}}$$
Decimal Number That Repeat Forever
$${\mathtt{0.555\: \!555\: \!555\: \!555\: \!555\: \!5}}$$...
1. To turn $${\mathtt{0.555\: \!555\: \!555\: \!555\: \!555\: \!5}}$$... into a fraction, let $${\mathtt{X}} = {\mathtt{0.555\: \!555\: \!555\: \!555\: \!555\: \!5}}$$...
2. Multiply both sides by $${\mathtt{10}}$$: $${\mathtt{10}}{\mathtt{\,\times\,}}{\mathtt{X}} = {\mathtt{5.555\: \!555\: \!555\: \!555\: \!555}}$$...
3. Subtract $${\mathtt{X}}$$ to both sides: $${\mathtt{9}}{\mathtt{\,\times\,}}{\mathtt{X}} = {\mathtt{5}}$$
4. Divide both sides by $${\mathtt{9}}$$: $${\mathtt{X}} = {\frac{{\mathtt{5}}}{{\mathtt{9}}}}$$
5. Reduce $${\frac{{\mathtt{5}}}{{\mathtt{9}}}}$$ to its lowest form if possible: $${\frac{{\mathtt{5}}}{{\mathtt{9}}}}$$
$${\mathtt{0.555\: \!555\: \!555\: \!555\: \!555\: \!5}}$$... as a fraction is $${\frac{{\mathtt{5}}}{{\mathtt{9}}}}$$
Example 2
$${\mathtt{0.121\: \!212\: \!121\: \!212\: \!121\: \!2}}$$...
1. To turn $${\mathtt{0.121\: \!212\: \!121\: \!212\: \!121\: \!2}}$$... into a fraction, let $${\mathtt{X}} = {\mathtt{0.121\: \!212\: \!121\: \!212\: \!121\: \!2}}$$...
2. Multiply both sides by $${\mathtt{100}}$$: $${\mathtt{100}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{12.121\: \!212\: \!121\: \!212\: \!12}}$$...
3. Subtract X to both sides: $${\mathtt{99}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{12}}$$
4. Divide both sides by $${\mathtt{99}}$$: $${\mathtt{X}} = {\frac{{\mathtt{12}}}{{\mathtt{99}}}}$$
5. Reduce $${\frac{{\mathtt{12}}}{{\mathtt{99}}}}$$ to its lowest form if possible: $${\frac{{\mathtt{4}}}{{\mathtt{33}}}}$$
$${\mathtt{0.121\: \!212\: \!121\: \!212\: \!121\: \!2}}$$... as a fraction is $${\frac{{\mathtt{4}}}{{\mathtt{33}}}}$$
Example 3
$${\mathtt{0.144\: \!444\: \!444\: \!444\: \!444\: \!4}}$$...
1. To turn $${\mathtt{0.144\: \!444\: \!444\: \!444\: \!444\: \!4}}$$... into a fraction let $${\mathtt{X}} = {\mathtt{0.144\: \!444\: \!444\: \!444\: \!444\: \!4}}$$...
2. Multiply both sides by $${\mathtt{100}}$$: $${\mathtt{100}}{\mathtt{\,\times\,}}{\mathtt{X}} = {\mathtt{14.444\: \!444\: \!444\: \!444\: \!444}}$$...
3. Subtract X to both sides: $${\mathtt{99}}{\mathtt{\,\times\,}}{\mathtt{X}} = {\mathtt{14.3}}$$
4. Divide both sides by $${\mathtt{99}}$$: $${\mathtt{X}} = {\frac{{\mathtt{14.3}}}{{\mathtt{99}}}}$$
5. Multiply the Numerator and Dominator by $${\mathtt{10}}$$ to get rid of the decimal number in the numerator: $${\frac{{\mathtt{143}}}{{\mathtt{990}}}}$$
6. Reduce $${\frac{{\mathtt{143}}}{{\mathtt{990}}}}$$ to its lowest form if possible: $${\frac{{\mathtt{13}}}{{\mathtt{90}}}}$$
$${\mathtt{0.144\: \!444\: \!444\: \!444\: \!444\: \!4}}$$... as a fraction is $${\frac{{\mathtt{13}}}{{\mathtt{90}}}}$$
Decimal Numbers That Do Not Repeat But Go On Forever
Example 1
$${\mathtt{\pi}}$$
$${\mathtt{\pi}}$$ is a decimal number that goes on forever but does not repeat. That decimal number is $${\mathtt{3.141\: \!592\: \!653\: \!589\: \!793\: \!2}}$$... Because $${\mathtt{\pi}}$$ does not repeat but goes on forever, there is no fraction that can represent $${\mathtt{\pi}}$$.
Example 2
$${\mathtt{e}}$$
$${\mathtt{e}}$$ is a decimal number that goes on forever but does not repeat. That decimal number is $${\mathtt{2.718\: \!281\: \!828\: \!459\: \!045\: \!2}}$$... Because $${\mathtt{e}}$$ does not repeat but goes on forever, there is no fratin that can represent $${\mathtt{e}}$$.
If the decimal is 0.9, it means 9/10. The 9 is in the tenths place. If the decimal is 0.99, it means 99/100, the second 9 is in the hundredths place. If you continue 0.999 means 999/1000, etc.
There are three types of decimal numbers: decimal numbers that terminate, decimal numbers that repeat forever, and decimal number that do not repeat but go on forever.
Decimal Numbers That Terminate
Example 1
$${\mathtt{0.6}}$$
1. To turn $${\mathtt{0.6}}$$ into a fraction, notice that the $${\mathtt{6}}$$ is in the tenths spot. Write the fraction as six-tenths: $${\frac{{\mathtt{6}}}{{\mathtt{10}}}}$$
2. Now take $${\frac{{\mathtt{6}}}{{\mathtt{10}}}}$$ and reduce to lowest form: $${\frac{{\mathtt{3}}}{{\mathtt{5}}}}$$
$${\mathtt{0.6}}$$ as a fraction is $${\frac{{\mathtt{3}}}{{\mathtt{5}}}}$$
Example 2
$${\mathtt{0.75}}$$
1. To turn $${\mathtt{0.75}}$$ into a fraction, notice that the $${\mathtt{7}}$$ is in the tnths place and the $${\mathtt{5}}$$ is in the hundredth place. Write the fraction as seventy-five hundredths: $${\frac{{\mathtt{75}}}{{\mathtt{100}}}}$$
2. Now take $${\frac{{\mathtt{75}}}{{\mathtt{100}}}}$$ and reduce it to lowest form: $${\frac{{\mathtt{3}}}{{\mathtt{4}}}}$$
$${\mathtt{0.75}}$$ as a fraction is $${\frac{{\mathtt{3}}}{{\mathtt{4}}}}$$
Decimal Number That Repeat Forever
$${\mathtt{0.555\: \!555\: \!555\: \!555\: \!555\: \!5}}$$...
1. To turn $${\mathtt{0.555\: \!555\: \!555\: \!555\: \!555\: \!5}}$$... into a fraction, let $${\mathtt{X}} = {\mathtt{0.555\: \!555\: \!555\: \!555\: \!555\: \!5}}$$...
2. Multiply both sides by $${\mathtt{10}}$$: $${\mathtt{10}}{\mathtt{\,\times\,}}{\mathtt{X}} = {\mathtt{5.555\: \!555\: \!555\: \!555\: \!555}}$$...
3. Subtract $${\mathtt{X}}$$ to both sides: $${\mathtt{9}}{\mathtt{\,\times\,}}{\mathtt{X}} = {\mathtt{5}}$$
4. Divide both sides by $${\mathtt{9}}$$: $${\mathtt{X}} = {\frac{{\mathtt{5}}}{{\mathtt{9}}}}$$
5. Reduce $${\frac{{\mathtt{5}}}{{\mathtt{9}}}}$$ to its lowest form if possible: $${\frac{{\mathtt{5}}}{{\mathtt{9}}}}$$
$${\mathtt{0.555\: \!555\: \!555\: \!555\: \!555\: \!5}}$$... as a fraction is $${\frac{{\mathtt{5}}}{{\mathtt{9}}}}$$
Example 2
$${\mathtt{0.121\: \!212\: \!121\: \!212\: \!121\: \!2}}$$...
1. To turn $${\mathtt{0.121\: \!212\: \!121\: \!212\: \!121\: \!2}}$$... into a fraction, let $${\mathtt{X}} = {\mathtt{0.121\: \!212\: \!121\: \!212\: \!121\: \!2}}$$...
2. Multiply both sides by $${\mathtt{100}}$$: $${\mathtt{100}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{12.121\: \!212\: \!121\: \!212\: \!12}}$$...
3. Subtract X to both sides: $${\mathtt{99}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{12}}$$
4. Divide both sides by $${\mathtt{99}}$$: $${\mathtt{X}} = {\frac{{\mathtt{12}}}{{\mathtt{99}}}}$$
5. Reduce $${\frac{{\mathtt{12}}}{{\mathtt{99}}}}$$ to its lowest form if possible: $${\frac{{\mathtt{4}}}{{\mathtt{33}}}}$$
$${\mathtt{0.121\: \!212\: \!121\: \!212\: \!121\: \!2}}$$... as a fraction is $${\frac{{\mathtt{4}}}{{\mathtt{33}}}}$$
Example 3
$${\mathtt{0.144\: \!444\: \!444\: \!444\: \!444\: \!4}}$$...
1. To turn $${\mathtt{0.144\: \!444\: \!444\: \!444\: \!444\: \!4}}$$... into a fraction let $${\mathtt{X}} = {\mathtt{0.144\: \!444\: \!444\: \!444\: \!444\: \!4}}$$...
2. Multiply both sides by $${\mathtt{100}}$$: $${\mathtt{100}}{\mathtt{\,\times\,}}{\mathtt{X}} = {\mathtt{14.444\: \!444\: \!444\: \!444\: \!444}}$$...
3. Subtract X to both sides: $${\mathtt{99}}{\mathtt{\,\times\,}}{\mathtt{X}} = {\mathtt{14.3}}$$
4. Divide both sides by $${\mathtt{99}}$$: $${\mathtt{X}} = {\frac{{\mathtt{14.3}}}{{\mathtt{99}}}}$$
5. Multiply the Numerator and Dominator by $${\mathtt{10}}$$ to get rid of the decimal number in the numerator: $${\frac{{\mathtt{143}}}{{\mathtt{990}}}}$$
6. Reduce $${\frac{{\mathtt{143}}}{{\mathtt{990}}}}$$ to its lowest form if possible: $${\frac{{\mathtt{13}}}{{\mathtt{90}}}}$$
$${\mathtt{0.144\: \!444\: \!444\: \!444\: \!444\: \!4}}$$... as a fraction is $${\frac{{\mathtt{13}}}{{\mathtt{90}}}}$$
Decimal Numbers That Do Not Repeat But Go On Forever
Example 1
$${\mathtt{\pi}}$$
$${\mathtt{\pi}}$$ is a decimal number that goes on forever but does not repeat. That decimal number is $${\mathtt{3.141\: \!592\: \!653\: \!589\: \!793\: \!2}}$$... Because $${\mathtt{\pi}}$$ does not repeat but goes on forever, there is no fraction that can represent $${\mathtt{\pi}}$$.
Example 2
$${\mathtt{e}}$$
$${\mathtt{e}}$$ is a decimal number that goes on forever but does not repeat. That decimal number is $${\mathtt{2.718\: \!281\: \!828\: \!459\: \!045\: \!2}}$$... Because $${\mathtt{e}}$$ does not repeat but goes on forever, there is no fratin that can represent $${\mathtt{e}}$$.
