| DECIMALS TO FRACTIONS |
Fractions Disguised as Whole Numbers
Decimals are fractions. Their denominators are exclusively powers of 10. They were developed so we could express parts of a whole (fractions) with the same place value system we use for whole numbers. It wasn't always so.
In ancient days, mathematicians and scientists used the sexagesimal notation for fractions with 60 as its base. We still use it today on clocks and to measure angles. On the clock, an hour (whole) is divided into 60 minutes, and each minute is divided into 60 seconds. With angles, whole numbers are called degrees not hours. The table of equivalences is:
1° (degree) = 60' (minutes), and 1' (minute) = 60" (seconds).
As always in math, there are symbols ( ° , ' , and " ) to represent degrees, minutes and seconds. The number 3.5 was written 3° 30' back then because 30 minutes is one half of a degree, and 9.75 was written 9 ° 45', since 45 minutes is ¾ of 60 = 0.75 of a degree.
Today, what with calculators, we can use our regular decimal system to measure angles but we still use the sexagesimal system to measure time.
Three Kinds of Decimals
There are 3 distinct types of decimals:
terminating, repeating and infinite or never ending decimals.
Terminating Decimals: have a finite (countable) number of digits. Like the name says, these decimals end at a specific place value.
0.175, 0.00093, 0.5478931 are terminating decimals.
Repeating Decimals: are decimals in which one or more digits keep repeating without end. They represent fractions with denominators that are prime numbers other than 2 or 5.
0.3333..., 0.11111..., and 0.090909... are repeating decimals.
We indicate repeating decimals with either a dot or (better) a "bar" over the repeating digits. The repeating decimals above are written this way:
Infinite or Non-repeating Decimals: are numbers like pi and the square roots of 2 or 3. The decimal digits never repeat and never end. These are called irrational numbers because they can't be expressed as a fraction (rational number). The closest fraction to pi is 22/7. The actual value of pi to 5 decimal places is 3.14159. The value of 22 ÷ 7 = 3.14285. They are exact only to the hundredths place. Infinite decimals cannot be expressed as fractions.
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Converting Finite Decimals to Fractions
A finite decimal is easily converted to a fraction. We just write the decimal's digits in the numerator of a fraction with the appropriate power of ten in the denominator.
Then, if we can, we reduce the fraction to lowest terms.
Examples:
and
Note: The number of zeros in the denominator = the number of decimal places.
Converting Repeating Decimals to Fractions
To convert a repeating decimal to a fraction, we use multiplication, subtraction,and division.
Let's do 2 examples before we discuss the procedure.
Example 1: convert 0.33333... to a fraction.
Solution:
We let x = 0.33333..
so 10x = 3. 33333....
subtract the 1st equation from the 2nd
10x - x = 9x
and 3.33333... - 0.33333... = 3.0
If 9x = 3, then x = 1/3
The fraction equivalent of 0.33333... = 1/3
Here, we multiplied by 10 to move the decimal one place to the right. We did this because there was only one repeating digit in this decimal.
Example 2: convert 0.090909... to a fraction.
Solution:
We let x = 0.090909...
so 100x = 09.090909....
subtract the 1st equation from the 2nd
100x - x = 99x.
and 09.090909.... - 0.090909... = 9.0
If 99 x = 9, then x = 9/99 = 1/11
The fraction equivalent of 0.090909... = 1/11
Here, we multiplied by 100 to move the decimal 2 places to the right. We did this because there were 2 repeating digits in the decimal.
What Did We Do?
First, we named the repeating decimal x to create an equation.
Then we multiplied by the correct power of 10 to move the decimal point to the right of one complete repeat pattern.
Next, we subtracted the first equation from the 2nd.
This gave us a whole number multiple of x on the left side of the equation.
On the right side, we get a whole number since the repeating digits (after the decimal point) cancel each other out.
Finally, we divided by the coefficient (multiple) of x on the left to get the fraction equivalent to the repeating decimal.
In both these examples, the repeating pattern started immediately after the decimal place in the tenths column. This isn't always the case.
Let's do another example with what I call problem child repeating decimals to see what to do when the repeating pattern doesn't start immediately right of the decimal point.
Example: Find the fraction equivalent to 0.2787878...
notice that the "2" doesn't repeat -- just the 787878...
Solution:
To set it up so the repeating digits start immediately right of the decimal point,
we start with 10x instead of x like this:
We let 10x = 2.787878...
now we multiply by 100 to move the decimal 2 more places, past a complete repeat pattern.
1000x = 278.787878....
Now we subtract the 1st equation from the 2nd
1000x - 10x = 990x.
and 278.787878.... - 2.787878... = 276.0
If 990 x = 276, then x = 276/990 = 138/495
The fraction equivalent of 0.2787878.... = 138/495
If we have to convert 0.27565656 into a fraction, we first multiply by 100 to move the decimal point 2 places right past the 27 since those digits don't repeat. Then we multiply by 100 again to move an entire repeat pattern left of the decimal. Finally we subtract and divide to find the fraction form of that repeating decimal.
If we have a whole number with a repeating decimal like 3.717171... , we first convert the decimal part to a fraction, then write the answer as a mixed number.
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Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.
Practice Exercises
1) Write these repeating decimals using bar notation:
| a) 0.343434... | b) 12.727272... | c) 0.2571571571... |
2) Express these decimals as a fraction in lowest terms:
| a) 0.75 | b) 0.213 | c) 0.3462 | d) 0.0045 |
3) Express these repeating decimals as a fraction in lowest terms:
| a) 0.777... | b) 0.222... | c) 0.343434... | d) 0.10171717... |
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Solutions
1) Write these repeating decimals using bar notation:
a) 0.343434...=  |
b) 12.727272...=  |
c) 0.2571571571...=  |
2) Express these decimals as a fraction in lowest terms:
a) 0.75 = |
b) 0.213 = |
c) 0.3462 =  |
d) 0.0045 =  |
3) Express these repeating decimals as a fraction in lowest terms:
| a) 0.777... x = 0.777, 10x = 7.777
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b) 0.222... x = 0.222, 10x = 2.222
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| c) 0.343434... x = 0.3434..., 100x = 34.3434...
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d) 0.10171717... 100x = 10.171717..., 10 000x = 1017.1717...
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