Which is greater, \begin{align*}\frac{18}{99}\end{align*} or \begin{align*}\frac{15}{80}\end{align*}? How can you use decimals to help you with this problem?
Watch This
Khan Academy Converting Fractions to Decimals
Guidance
To change a fraction to a number in decimal form, the numerator must be divided by the denominator. You can use long division or a calculator for this calculation. If, after the division, the numbers after the decimal point end, it is a terminating decimal. If, after the division, the numbers after the decimal point repeat in a pattern forever, it is a periodic decimal (also known as a repeating decimal).
- \begin{align*}\frac{3}{4}=3\div 4=0.75\end{align*}. This is a terminating decimal.
- \begin{align*}\frac{3}{13}=3\div 13=0.230769230769...\end{align*}. This is a periodic or repeating decimal. The period is 230769 because this is the set of numbers that repeats.
Keep in mind that a rational number is any number that can be written in the form \begin{align*}\frac{a}{b}\end{align*} where \begin{align*}b \ne 0\end{align*}. Therefore, periodic decimals and terminating decimals are both rational numbers.
Example A
What fraction is equal to 0.45454545... ?
Solution: This is a periodic or repeating decimal. The period has a length of two because the pattern that is repeating consists of 2 digits. To express the number as a fraction, follow these steps:
Step 1: Let \begin{align*}x=0.45454545\end{align*}
Step 2: The repeating digit is 45. Place the repeating digit to the left of the decimal point by moving the decimal point 2 places to the right.
45.454545
To move the decimal point two places to the right, the decimal number was multiplied by 100. If you go back to step 1, you will see the equation \begin{align*}x=0.45454545\end{align*}. If you multiply one side of the equation by 100, you must multiply the other side by 100. You now have \begin{align*}100x=45.454545\end{align*}
Step 3: Subtract the two equations and solve for \begin{align*}x\end{align*}.
\begin{align*}&100x=45.454545\\ & \underline{\;\; -x=0.45454545}\\ & \frac{99x}{99}=\frac{45}{99}\\ & \quad \ x=\frac{45}{99}=\frac{5}{11}\end{align*}
Example B
What fraction is equal to 0.125?
Solution: This number appears to be a terminating decimal number. The steps to follow to express 0.125 as a fraction are:
Step 1: Express the number as a whole number by moving the decimal point to the right. In this case, the decimal must be moved three places to the right.
Step 2: \begin{align*}0.125=125\end{align*}
Express 125 as a fraction with a denominator of 1 and three zeros. The three zeros represent the number of places that the decimal point was moved.
\begin{align*}\frac{125}{1000}\end{align*}
Step 3: If possible, simplify the fraction. If you are not sure of the simplified form, your graphing calculator can do the calculations.
Therefore, the decimal number of 0.125 is equivalent to the fraction \begin{align*}\frac{1}{8}\end{align*}.
The method shown above is one that can be used if you can’t remember the place value associated with the decimal numbers. If you remember the place values, you can simply write the decimal as a fraction and simplify that fraction.
Example C
Are the following decimal numbers terminating or periodic? If they are periodic, what is the period and what is its length?
i) 0.318181818...
ii) 0.375
iii) 0.3125
iv) 0.1211221112...
Solution:
i) A periodic decimal with a period of 18. The length of the period is 2.
ii) 0.375 A terminating decimal.
iii) 0.3125 A terminating decimal.
iv) 0.1211221112 This decimal is not a terminating decimal nor is it a periodic decimal. Therefore, the decimal is not a rational number. Decimals that are non-periodic belong to the set of irrational numbers.
Concept Problem Revisited
You can convert both fractions to decimals in order to figure out which is greater.
\begin{align*}\frac{18}{99}=.1818...\end{align*}
\begin{align*}\frac{15}{80}=.1875\end{align*}
You can see that \begin{align*}\frac{15}{80}\end{align*} is greater.
Vocabulary
- Irrational Numbers
- An irrational number is the set of non-periodic decimal numbers. Some examples of irrational numbers are \begin{align*}\sqrt{3},\sqrt{2}\end{align*} and \begin{align*}\pi\end{align*}.
- Periodic Decimal
- A periodic decimal is a decimal number that has a pattern of digits that repeat. The decimal number 0.1465325325..., is a periodic decimal.
- Rational Numbers
- A rational number is any number that be written in the form \begin{align*}\frac{a}{b}\end{align*} where \begin{align*}b \ne 0\end{align*}. Therefore, periodic decimal numbers and terminating decimal numbers are rational numbers.
- Terminating Decimal
- A terminating decimal is a decimal number that ends. The process of dividing the fraction ends when the remainder is zero. The decimal number 0.25 is a terminating decimal.
Guided Practice
1. Express 2.018181818 in the form \begin{align*}\frac{a}{b}\end{align*}.
2. Express \begin{align*}\frac{15}{11}\end{align*} in decimal form.
3. If one tablet of micro K contains 0.5 grams of potassium, how much is contained in \begin{align*}2\frac{3}{4}\end{align*} tablets?
Answers:
1. Let \begin{align*}x=2.018181818\end{align*} The period is 18.
\begin{align*}1000x=2018.181818\end{align*}
\begin{align*}10x=20.18181818\end{align*}
\begin{align*}& 1000 x=2018.181818 \qquad \text{These are the two equations that must be subtracted.}\\ & \underline{-10x=20.18181818}\\ & \frac{990x}{990}=\frac{1998}{990} \qquad \qquad \quad \ \ \text{Solve for} \ x.\\ & \quad \ \ x=\frac{1998}{990}\end{align*}
Use your calculator to simplify the fraction.
\begin{align*}x=\frac{1998}{990}\end{align*}
\begin{align*}x=\frac{111}{55}\end{align*}
2. \begin{align*}\frac{15}{11}=1.3636...\end{align*}
3. The number of tablets is given as a mixed number. \begin{align*} 2 \frac{3}{4}=2.75\end{align*}. \begin{align*}2.75 \times 0.5=1.375 \ grams\end{align*}.
Practice
Express the following fractions in decimal form.
- \begin{align*}\frac{1}{12}\end{align*}
- \begin{align*}\frac{6}{11}\end{align*}
- \begin{align*}\frac{3}{20}\end{align*}
- \begin{align*}\frac{1}{13}\end{align*}
- \begin{align*}\frac{3}{8}\end{align*}
Express the following numbers in the form \begin{align*}\frac{a}{b}\end{align*}.
- 0.325
- 3.72727272...
- 0.245454545...
- 0.618
- 0.36363636...
Complete the following table.
Fraction | Decimal | |
---|---|---|
11. | \begin{align*}\frac{5}{64}\end{align*} | |
12. | \begin{align*}\frac{11}{32}\end{align*} | |
13. | \begin{align*}\frac{1}{20}\end{align*} | |
14. | \begin{align*}0.0703125\end{align*} | |
15. | \begin{align*}0.1875\end{align*} |