Fractions
By the end of this lesson, you should be able to:
4. Identify fractions as proper and improper fractions.
6. Correctly add, subtract, multiply and divide fractions.
7. Raise a fraction to a whole number power, and raise a number to a fractional power.
Fractions are simple math that sometimes confuses each of us. Some people opt to use decimals, but fractions are often more precise. For example, "How much of a yard is one foot?" The answer is not .3, or .3333, or even .333333333333. They are all a little off. But if you answered 1/3 you would be correct, not off by even a little. (The symbol for "approximately equal to" is like an equals sign with wavy lines.)
Therefore, many people find it helpful to be able to deal with fractions, but also to convert them to decimals when appropriate. In certain technical fields, some people have found it very useful to have gained the skill of quickly finding the decimal equivalent of common fractions in their heads, without a calculator or even a pencil.
To use a calculator to determine the decimal equivalent of a fraction, type in the fraction and press Enter. Most calculators will return the decimal equivalent. Some may even be able to convert decimals back to fractions.Without a calculator, if you needed to determine the decimal equivalent of , say, 2/7, you would divide 7 into 2.000, to find it quotient of approximately 0.286.
Although a calculator is ideal for converting a fraction to a decimal, there are many equivalents that it pays to have memorized. This depends on the requirements of your job and the type of work you normally do. In general, you are advised to be able to immediately convert between fractions and decimals for the following (in your head without a pencil or calculator):
Here are some tricks:half thirds fourths fifths sixths sevenths eighths ninths tenths elevenths twelfths sixteenths twentieths
Half
1/2 = .5
Easy
Thirds
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1/3 = .3
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2/3 = .6
That bar on top of the numbers usually means a repeating unit of a non-terminating decimal. 1/3 is not equal to .3, it is equal to .333333... which goes on for ever.
Fourths
1/4 = .25
2/4 = 1/2 = .5
3/4 = .75
Using quarter dollars is a good way to teach this equivalent.
Fifths
1/5 = .2
2/5 = .4
3/5 = .6
4/5 = .8
Easy: 2 4 6 8
Sixths
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1/6 = .16
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2/6 = 1/3 = .3
3/6 = 1/2 = .5
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4/6 = 2/3 = .6
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5/6 = .83
The trick to the sixths is getting the equivalent of 1/6, which is half of 1/3.
Half of .3 is .15
Half of .33 is .15 + .015, or .165
Thus half of .333333.... is .166666....
This is 1/6, and if we subtract it from 1.0, we get 5/6, or .8333333.....
The 2/6, 3/6, and 3/6 have reduced forms that have already been covered.
Sevenths
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1/7 = .142857
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2/7 = .285714
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3/7 = .428571
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4/7 = .571428
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5/7 = .714285
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6/7 = .857142
This is my favorite. Here's the trick:
What is twice 7? 14
What is twice 14? 28
What is twice 28, but because this is a trick, add 1? 56 + 1 = 57
Put these three together and you get "142857"
1/7 is a non-terminating decimal beginning
with the 1, or the lowest of these numerals:
1/7 = .142857142857142....
2/7 begins with the next higher numeral the two, and so on.
Eighths
1/8 = .125
2/8 = 1/4 = .25
3/8 = .375
4/8 = 1/2 = .5
5/8 = .625
6/8 = 3/4 = .75
7/8 = .875
An eighth is half of a quarter; half of .250 is .125.
To get 3/8, we can add .125 to .250 in our head to get .375.
5/8 is .500 plus .125, or .625.
7/8 is 1.000 minus .125 or .875.
Ninths
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1/9 = .1
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2/9 = .2
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3/9 = 1/3 = .3
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4/9 = .4
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5/9 = .5
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6/9 = 2/3 = .6
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7/9 = .7
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8/9 = .8
1/9 is .1111111.....
This pattern continues for the other numbers.
(But does that mean that 9/9 is really only .99999....?)
Tenths
1/10 = .1
2/10 = 1/5 = .2
3/10 = .3
4/10 = .2/5 = 4
5/10 = .5
6/10 = 3/5 = .6
7/10 = .7
8/10 = 4/5 = .8
9/10 = .9
Tenths are easy, but thinking of dimes helps some people.
Elevenths
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1/11 = .09
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2/11 = .18
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3/11 = .27
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4/11 = .36
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5/11 = .45
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6/11 = .54
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7/11 = .63
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8/11 = .72
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9/11 = .81
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10/11 = .90
"If you know your 9 times table, you know your elevenths." The nine times table begins, 9, 18, 27, 36, 45.
Look at the non-terminating decimal equivalents of the elevenths.
1/11 = .0909090909...
2/11 = .1818181818...
Twelfths
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1/12 = .083
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2/12 = 1/6 = .16
3/12 = 1/4 = .25
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4/12 = 1/3 = .3
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5/12 = .416
6/12 = 1/2 = .5
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7/12 = .583
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8/12 = 2/3 = .6
9/12 = 3/4 = .75
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10/12 = 5/6 = .83
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11/12 = .916
The trick to twelfths is to determine the amount of 1/12 by dividing 1/6 in half.
1/6 is .16666...
Half of .16 is .08,
and half of .00666... is .003333...
Thus, 1/12 is .0833333....
5/12 can be quickly found by subtracting .083333... from .5000 in your head.
7/12 can be quickly found by adding .083333... to .50000 in your head.
11/12 can be quickly found by subtracting
.083333... from 1.000 in your head.
Sixteenths
1/16 = .0625
2/16 = 1/8 = .125
3/16 = .1875
4/16 = 1/4 = .25
5/16 = .3125
6/16 = 3/8 = .375
7/16 = .4375
8/16 = 1/2 = .5
9/16 = .5625
10/16 = 5/8 = .625
11/16 = .6875
12/16 = 3/4 = .75
13/16 = .8125
14/16 = 7/8 = .875
15/16 = .9375
Most technicians and engineers would probably say that the sixteenths are the most used fractions. You should be able to quickly arrive at the decimal equivalents in your head, without a pencil or calculator, in about three seconds
1/16 is half of 1/8,
so half of .125 is half of .12 (or .06)
plus half of .0050 (or .0025).
Memorize that 1/16 is .0625.
3/16 and 5/16 can be found by either subtracting .0625 from .2500, or adding .0625 to .2500.
Similarly, 7/16 and 9/16 surround 1/2, so they can be found by subtracting .0625 from .5000, or adding .0625 to .5000.
Again, 11/16 and 13/16 surround 3/4, so they can be found by subtracting .0625 from .7500, or adding .0625 to .7500.
Finally, 15/16 can be found by subtracting .0625 from 1.0000.
Twentieths
1/20 = .05
2/20 = 1/10 = .1
3/20 = .15
4/20 = 1/5 = .2
5/20 = 1/4 = .25
6/20 = 3/10 = .3
7/20 = .35
8/20 = 2/5 = .4
9/20 = .45
10/20 = 1/2 = .5
11/20 = .55
12/20 = 3/5 = .6
13/20 = .65
14/20 = 7/10 = .7
15/20 = .75
16/20 = 4/5 = .8
17/20 = .85
18/20 = 9/10 = .9
19/20 = .95
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Twentieths are easy, especially in light of the five times table, and the use of nickels.
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Simplifying Fractions
Sometimes a fraction is not in its lowest terms. 4/8 is an example. 4/8 can be simplified to 2/4, but that is still not the lowest term. By dividing both the numerator and denominator by 2, we can reduce it to 1/2. Any fraction that has a common factor (other than 1) in the numerator and denominator can be reduced to simpler terms.
Proper and Improper Fractions
A proper fraction is defined as one where the (absolute value of the) denominator is greater than the (absolute value of the) numerator. 1/8, 6/7, 3/6, and 342/982 are all proper fractions.An improper fraction has a bigger number for the numerator than for the denominator, such as 5/4 and 89/31. These can be simplified by creating a mixed number, which has a whole number component and a fractional component. For example, five fourths is equal to one and one fourth, or 1.25 in decimal.
Reciprocals
The reciprocal of a fraction flips the numerator and denominator.
- The reciprocal of 2/7 is 7/2, which reduces to 3.5.
- The reciprocal of 5 is 1/5.
Addition and subtraction:Be sure to find a common denominator.
- 1/3 + 1/9 = 3/9 + 1/9 = 4/9
- 2/5 + 1/3 = 6/15 + 5/15 = 11/15
Multiplication:Multiply numerators and denominators.The word, "of," can be interpreted as "times."
- 2/9 * 3/5 = (2*3) / (9*5) = 6/45 = 2/15
- For example, what is one-third of two-fifths?
Answer: 1/3 * 2/5 = 2/15
Division:To divide by a fraction, multiply by its reciprocal.
- (3/7) / (4/5) = (3/7) * (5/4) = (3*5) / (7*4) = 15/28
Exponents:When you raise a fraction to a power, raise both the numerator and denominator to that power.When a number is raised to a fractional power, raise it to the power of the numerator, and find the root indicated in the denominator.
- (2/5)3 = 23 / 53 = 8 / 125
- 82/3 = (82)1/3 [or the cubed root of 82]
= 641/3 [or the cubed root of 64]
= 4
© Jim Flowers
Department of Technology, Ball State University