Fractions


Objectives:
By the end of this lesson, you should be able to: 
1. Explain how fractions
are often more precise than decimals.
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. 


The
Accuracy of Fractions
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.) 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. 

Half 1/2 = .5 
Easy 
Thirds _ 1/3 = .3 _ 2/3 = .6 
That bar on top of the numbers usually means a repeating unit of a nonterminating 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 _ 1/6 = .16 _ 2/6 = 1/3 = .3 3/6 = 1/2 = .5

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 ______ 1/7 = .142857 ______ 2/7 = .285714 ______ 3/7 = .428571 ______ 4/7 = .571428 ______ 5/7 = .714285 ______ 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 nonterminating decimal beginning
with the 1, or the lowest of these numerals:
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 _ 1/9 = .1 _ 2/9 = .2 _ 3/9 = 1/3 = .3 _ 4/9 = .4 _ 5/9 = .5 _ 6/9 = 2/3 = .6 _ 7/9 = .7 _ 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 __ 1/11 = .09 __ 2/11 = .18 __ 3/11 = .27 __ 4/11 = .36 __ 5/11 = .45 __ 6/11 = .54 __ 7/11 = .63 __ 8/11 = .72 __ 9/11 = .81 __ 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 nonterminating decimal equivalents of the elevenths. 1/11 = .0909090909... 2/11 = .1818181818... 
Twelfths _ 1/12 = .083 _ 2/12 = 1/6 = .16 3/12 = 1/4 = .25
6/12 = 1/2 = .5
9/12 = 3/4 = .75

The trick to twelfths is to determine the amount of 1/12 by dividing 1/6 in half. 1/6 is .16666...
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,
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

Twentieths are easy, especially in light of the five times table, and the use of nickels. ................................... 
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. Reciprocals The reciprocal of a fraction flips the numerator and denominator. Addition and subtraction:Be sure to find a common denominator. 



All information
is subject to change without notification.
© Jim Flowers Department of Technology, Ball State University 