Download 47 ←Numerator←Denominator

JH WEEKLIES ISSUE #22
2012-2013
Mathematics—Fractions
Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3…etc.). The
preferred way to represent these partial numbers (rational numbers) is with the fraction.
I.
The Fraction
Before we begin our discussion of fractions, there is some terminology that you should be
familiar with:
47 ←Numerator←Denominator
The center bar in a vertical fraction such as this is formally called the vinculum
(informally, the fraction bar).
Common fractions include numbers such as 1/2, 3/4, and 2/5. Simply put, fractions
represent division: we divide the numerator into the number of parts specified by the
denominator. This means that the denominator in a fraction cannot ever be zero because
it is prohibited by mathematics to divide something into zero parts (there is always at
least one part to every whole).
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II.
Comparison of Fractions
When you have two fractions that do not have the same denominator, it can occasionally
be difficult to compare them—to figure out which is larger or smaller. One easy trick
involves multiplication.
Example II.1: Comparison of Fractions
Which fraction is larger: 35 or 1564?
While we could go ahead and find the decimal representation (see §VI) to
compare the fractions, or we could the multiply comparison method.
By multiplying the numerator of the first fraction by the denominator of the
second, and the numerator of the second by the denominator of the first, we can
compare the fractions.
3515643∙6415∙519275
Because we know that 192 is greater than 75, we also know that the first fraction
35 is larger than the second fraction 1564. The dot that we use between numbers
above is another way of representing multiplication, and will be used throughout
this document.
We could also simplify the fractions or find a common denominator.
III.
Simplification and Equivalence of Fractions
You can have an infinite number of fractions that all represent the same decimal number.
Example III.1: Equivalence of Fractions
Represent the number 0.5 with a fraction.
Of course, the simplest fraction that we could use would be 1/2. We could also use
2/4, or 3/6, 4/8, 5/10, 70/140, 1959437/3918874 or an infinite number of other fractions.
This multiplicity of fractions can get kind of confusing; what if you have two fractions,
but they both look like that last example (1959437/3918874)? How could you tell that they are
equivalent?
We can simplify fractions by factoring the numerator and denominators and looking for
common factors. For most problems, you will be asked to give your answer in
“simplified form” or in “lowest terms.”
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Example III.2: Simplifying Fractions by Factoring
Simplify the fraction 6084.
We can start simplifying this fraction by factoring the numerator into simpler
numbers. First, we recognize that 60 is an even number—this means that we can
factor a 2 out of it.
60=30∙2
Again, 30 is an even number, so we can factor a 2 out of it.
60=30∙2=15∙2∙2
We know that 15 is equal to 3 ∙ 5
60=30∙2=15∙2∙2=2∙2∙3∙5
All four of these factors are prime numbers, which means that we cannot factor
them further. We now turn to the denominator, recognizing similarly that 84 is an
even number, so we can factor out a 2.
84=2∙42
Similarly:
84=2∙42=2∙2∙21=2∙2∙3∙7
Reassembling our fraction from these factored components…
6084=2∙2∙3∙52∙2∙3∙7
We know from multiplication that we can factor out those three common factors
as separate fractions
6084=2∙2∙3∙52∙2∙3∙7=22∙22∙33∙57
For more on fraction multiplication, see §V.3. We know that any number divided
by itself is equal to 1, so we have
6084=2∙2∙3∙52∙2∙3∙7=22∙22∙33∙57=1∙1∙1∙57=57
Thus, we have simplified the fraction 6084 to 57.
Another way to look at this problem is repeated division of the numerator and
denominator to remove common factors:
6084=2∙302∙42=3042=2∙152∙21=1521=3∙53∙7=57
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IV.
Proper and Improper Fractions, Mixed Numbers
Mathematics states that fractions are “proper” when the numerator is less than the
denominator.
Example IV.1: Proper Fractions
Proper fractions include fractions like 3/8, 15/35, and 2/5.
When the numerator of a fraction is greater than its denominator, then the fraction is
called “improper.”
Example IV.2: Improper Fractions
Improper fractions include fractions like 4/3, 7/2, and 19/4.
Even though they are called “improper,” improper fractions still represent numbers—they
simply represent numbers that are greater than 1.
In order to make improper fractions acceptable to mathematics, we need to separate the
whole number part from the fractional part.
Example IV.3: Separating Improper Fractions
Separate 133 into a whole number and a fraction.
By separating the numerator into added factors, we can re-write the fraction
133=3+3+3+3+13=33+33+33+33+13
For more information on adding fractions, please see §V.1. Now, because any
number divided by itself is 1, we can rewrite again…
133=1+1+1+1+13=4+13
Notice now that we have a whole number and a proper fraction: 4+13.
We can now rewrite our separated improper fraction as a mixed number, which is a whole
number and fraction pair. They are often used for cooking (i.e. 21/2 cups of flour).
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Example IV.4: Building a Mixed Number
Using the result from IV.3, create a mixed number.
This is very simple; we just remove the plus sign and back the fraction and the
whole number up next to each-other.
4+13=413
Note that our resulting mixed number symbolizes adding 4 to 1/3—not multiplying
them.
Alternatively, we could use the division method with remainders to create our mixed
number.
Example IV.5: Building a Mixed Number by Division
Rewrite 13/3 as a mixed number.
We divide 13 by 3 with remainders:
04r13)13
-12
1
Thus, we see that 3 goes into 13 four times with 1 remaining. Therefore, our
mixed number is 41/3, which agrees with our result of example IV.4.
V.
Adding, Subtracting, Multiplying, and Dividing Fractions
For adding and subtracting, you must have a common denominator. It is
not necessary to have a common denominator for multiplying and dividing.
1.
Adding Fractions
If two fractions have a common denominator, then it is very easy to add them
together—you simply add the numerators and leave the denominator the same.
Example V.1.i: Adding Fractions with a Common Denominator
Add the fractions 4/9 and 3/9.
We proceed as detailed above:
49+39=4+39=79
4/9
+ 3/9 equals 7/9.
However, if two fractions do not have the same denominator, we have to find
equivalent fractions so the denominators are the same. We can go about this by
three different methods:
a.
If one denominator is a factor of another (such as 3 is of 9), then
you can easily convert one fraction to the other’s denominator.
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Example V.1.ii: Adding Fractions with Similar Denominators
Add the fractions 4/9 and 1/3.
We recognize that 3 is a factor of 9 (3 ∙ 3 = 9), so we can
multiply 1/3 by 3/3 to get a denominator of 9.
13∙33=39
1/3
is equivalent to 3/9.
We can perform the multiplication above due to the
multiplicative identity property (any number multiplied by
1 is itself), and the fact that any number divided by itself is
equal to 1.
Alternatively, the least common multiple of 3 and 9 is
9—please see §V.1.c for more information.
The problem then becomes adding 4/9 and 3/9, which we
have already done in example V.1.i.
b.
We can multiply each fraction by an equivalent of 1 based on the
other’s denominator (see example) and then simplify; this trick is
typically good when the denominators are less than 15.
Example V.1.iii: Adding Fractions with Different Denominators
Add the fractions 6/7 and 2/3, giving your answer as a
simplified improper fraction.
We proceed by multiplying 6/7 by 3/3, and 2/3 by 7/7:
67∙33=1821; 23∙77=1421
Both fractions now have the denominator of 21, which is
the least common multiple (§V.1.c). We can proceed:
1821+1421=3221
As it turns out, there are no common factors between 32
and 21 (besides 1), and we are asked to leave our answer as
an improper fraction. We are done: 6/7 + 2/3 = 32/21.
c.
We can find the least common multiple (LCM) of the
denominators, which is the smallest whole number of which both
denominators are divisors. This can be done in two different ways:
the simpler is presented here, the more complex in the appendix
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because it requires more explanation. The LCM methods take the
longest, but are more useful when you are adding more than two
fractions or when the denominators are larger numbers.
To find the LCM, list multiples of the denominators, looking for
the smallest number that is common between the two lists. That
will be your common denominator. Start by listing a few of the
larger denominator’s multiples.
Be careful not to confuse multiples with factors; factors are
component parts of a number where as multiples are where you
take the number and add it to itself (see example). It may help you
to remember that “multiple” sounds like “multiply.”
Example V.1.iv: Adding Fractions by LCM
Add 1/2 and 4/23.
We start by listing multiples of each of the denominators,
looking for the LCM:
Multiples of 2:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22,
24, 26, 28, 30, 32, 34, 36, 38, 40, 42,
44, 46, 48, 50
Multiples of 23:
23, 46, 69
The lowest common multiple is 46. Thus, we want to
convert our fractions so they have a common denominator
of 46. Making the conversions, we have
12+423=2346+846=3146
Coincidently, we would have come to this result directly by
the multiplication method listed in (b) above.
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2.
Subtracting Fractions
When subtracting fractions, all of the logic applied above in the adding fractions
section still applies: we can only subtract fractions that have a common
denominator. The only wrinkle in subtracting fractions is that we can have
negative answers.
Example V.2.i: Subtracting Fractions
Perform the operation 6/7 – 3/5, giving your answer in lowest terms.
First, we use the techniques listed in V.1 to convert our fractions so they
have common denominators.
67=3035; 35=2135
Now we perform the operation
67-35=3035-2135=30-2135=935
Our answer (9/35) is as simplified as we can make it—it is in lowest
terms—so we are done.
Example V.2.ii: Subtracting Fractions
Perform the operation 3/4 – 8/9, giving your answer in lowest terms.
Again, we use the techniques in V.1 to convert our fractions so they have
common denominators:
34=2736; 89=3236
We now perform the operation
44-89=2736-3236=27-3236=-536
Our answer (-5/36) is in lowest terms, so we are done.
Example V.2.iii: Subtracting Fractions
Perform the operation 11/15 – 2/5, giving your answer in simplified form.
We begin by finding equivalent fractions with the same denominator for
both fractions:
1115=1115; 25=615
We can now perform the operation
1115-615=515
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Now we simplify our result to put it in lowest terms.
515=13
The result is now in simplified form: 11/15 – 2/5 = 1/3
3.
Multiplying Fractions
Multiplying is one of the easiest operations to perform on fractions because you
can do as much or as little simplification as you would like at the beginning to
reduce the numbers you need to carry. This is accomplished by factoring and
canceling numbers out of the numerators and denominators of your participating
fractions. Otherwise, you simply multiply numerators and multiply denominators,
then simplify.
Example V.3.i: Multiplying Fractions
Perform the operation 3/4 ∙ 2/3, giving your answer in lowest terms.
We proceed with straight multiplication.
34∙23=3∙24∙3=612
Simplifying by pulling out a factor of 6 from both the numerator and the
denominator, we get
612=66∙12=1∙12=12
Our answer is 1/2.
Alternatively, we could have performed the simplification ahead of time
by canceling a 2 from the denominator of 3/4 with the numerator of 2/3, and
a 3 from the numerator of 3/4 and the denominator of 2/3:
34∙23=32∙2∙23=3∙22∙2∙3
1∙1∙12∙1∙1=12
Again, our answer is 1/2.
Example V.3.ii: Multiplying Fractions
Perform the operation 168/448 ∙ -24/216, giving your answer in lowest terms.
We begin by simplifying each of the two fractions:
168448=2∙2∙2∙3∙72∙2∙2∙7∙8=3∙78∙7=38
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-24216=-2∙2∙2∙32∙2∙2∙3∙3∙3=-33∙9=-19
We can further simplify the fractions when we multiply them together:
38∙-19=-38∙3∙3
-1∙18∙3=-124
Our answer, in lowest terms, is -1/24.
4.
Dividing Fractions
It is similarly easy to divide fractions because of a very handy trick. Division of
fraction A by fraction B is directly equivalent to the multiplication of fraction A
and the reciprocal of fraction B. Once you have your new multiplication pair,
simply proceed as with multiplication.
To get the reciprocal of a fraction, you simply swap the numerator and the
denominator. Therefore, the reciprocal of 2/5 is 5/2. Please note that if you are
dividing by a fraction where the numerator is 0, then you cannot perform the
operation because the reciprocal of that fraction would be dividing by zero.
Example V.4.i: Dividing Fractions
Perform the operation 3/7 ÷ 30/21, giving your answer in lowest terms.
Formally, the operation calls for this:
373021
Such a fraction is in the form of a complex fraction, and is directly
equivalent to 3/7 ÷ 30/21.
But because we have our reciprocal trick, we know the following:
37÷3021=37∙2130
By simplifying, we find that our problem reduces to
3∙3∙73∙7∙10=11∙11∙310=310
Our answer 3/10 is in lowest terms, so we are finished.
VI.
Converting Fractions to Decimals
The most common method for converting the fraction into a decimal is to go through
long division, dividing the numerator by the denominator. We can do this because a
fraction is nothing more than a division equation:
Example VI.1: Fractions to Decimals by Long Division
Convert 35 to decimal form.
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Because 35=3÷5, we begin by setting up a division form.
5)3
5 does not go evenly into 3, so we add a decimal zero.
0.5)3.0
5 does evenly go into “30,” so we proceed with our normal division:
0.65)3.0-3.0
0
The decimal representation of 35 is 0.6
This works fine; it can be slightly tedious and eat up a lot of time, however, especially for
more complex fractions. Luckily, there are several tricks that can be used to speed up
your computation if the denominator can be easily manipulated.
If you can manipulate the fraction so the denominator is a power of 10, then you can
easily compute the decimal from the fraction by simply moving around the decimal point
in the numerator.
Example VI.2: Fractions to Decimals by Manipulation
Convert 35 to decimal form.
This time, we recognize that the denominator of the fraction (5) is exactly half of
10. This means that if we multiply both the denominator and the numerator by 2,
we can get a fraction where the denominator is 10.
35∙22=3∙25∙2=610
We are allowed to do this because of the multiplicative property—any number
times 1 is equal to that number—and because 2/2 is equal to 1. For more on
fraction multiplication, please see §V.3.
Now that we have our fraction with a denominator of 10, we know from division
that we can simply move the decimal point in the numerator one place to the left
to represent division by 10. Therefore, our answer is 0.6, which agrees with our
answer by long division.
Similarly, if we were to get our fraction in terms of 100, we would move the decimal
point in the numerator twice over to the left.
Further, if you can manipulate your fraction into a fraction that is recognizable, the
decimals are easy to compute. For this purpose, we recommend memorizing a table of
common decimal values (included in the appendix)—after all, if you can reduce a
fraction to 5/8 but do not remember the decimal equivalent, you will be stuck computing
the decimal by long division (for reference, 5/8 = 0.625).
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VII.
Converting Decimals to Fractions
1.
Terminating Decimals to Fractions
When a decimal value has a specific number of decimal places (i.e. 0.625, 0.5),
then it is very easy to convert the decimal value into a fraction.
Simply count up the number of decimal places (i.e. 0.265 has 3, 0.25 has 2),
multiply your decimal by a one followed by that many zeros (for 3 decimal
places, use 1000; for 2 decimal places, use 100); this is your numerator. Your
denominator is the power of ten that you just multiplied with. Simplify your
fraction, and you are done.
Example VII.1.i: Converting Terminating Decimals to Fractions
Convert 0.525 to a fraction in lowest terms.
The decimal value 0.525 has three decimal places, so we multiply by
1000.
0.525∙1000=525
Therefore, 525 is our numerator. Our denominator is simply the number
we just multiplied with, 1000. Ergo, our unsimplified fraction is:
5251000
Simplifying using techniques in this document, we find that our final
fraction is 2140; 0.525=5251000=2140.
Terminating decimals are known as rational numbers; they can be represented in
a fractional form.
2.
Repeating Decimals to Fractions
When a decimal value has an infinite number of decimal places—typically
represented by an ellipses (…)—it is possible to find a fractional representation if
the decimals repeat (like 0.23232323232… or 0.1010101010…). Such repeating
decimals are occasionally represented by a vinculum over repeating decimals,
such as 0.23=0.2323232323… or 0.193193193…=0.193.
The technique for finding the fractional representation of repeating decimals
formally goes through some algebra; here, we present the technique in a similar
manner to finding the representation of a terminating decimal.
Cut your repeating decimal off when the digits begin to repeat. Then count up the
number of repeating decimal values and multiply the repeating value by a one
followed by that many zeros; this is your numerator. Your denominator is the
power of ten that you just multiplied with minus one. Simplify your fraction, and
you are done. This technique takes advantage of the repeating property of nines.
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Example VII.2.i: Converting Repeating Decimals to Fractions
Convert 0.3 to a fraction in lowest terms.
We begin by identifying the repeating element, which in this case is 3.
This one value repeats an infinite number of times. Therefore, we cut off
our decimal at 0.3 and multiply 0.3 by 10.
0.3∙10=3
This 3 is our numerator. Our denominator, therefore, is 10 – 1, or 9. We
assemble our fraction and then simplify:
39=13
Therefore, 1/3 is equal to 0.333333…
Example VII.2.ii: Converting Repeating Decimals to Fractions
Convert 0.265265265265… to a fraction in lowest terms.
We begin by identifying the repeating element—265. There are three
repeating values in this decimal. Therefore, we cut off our decimal at
0.265, and multiply 0.265 by 1000:
0.265∙1000=265
This 265 is our numerator. Our denominator, therefore, is 1000 – 1, or
999. Then, we simplify.
265999
Turns out that
265=265999.
265/999
is a fraction in lowest terms. Our work is done: 0.
Like terminating decimals, repeating decimals are known as rational numbers;
they can be represented in a fractional form.
3.
Non-Repeating Non-Terminating Decimal Values
When a decimal value never repeats and has an infinite number of places, then
that number is called irrational—it cannot be represented as a fraction. Common
irrational numbers include 2, π, and e.
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EXERCISES
Exercise I: Comparison of Fractions
Given each set of fractions, rank them from greatest to least.
1.
3.
27, 35, 1123
13, 34, 49, 52
2.
46, 13, 12
4.
154, 38, 1112, 37, 33,
910
Exercise II: Simplification of Fractions
Simplify each of the following fractions to lowest terms.
1.
4.
3060
927
2.
93
5.
1845
3.
8402500
6.
2025
Exercise III: Proper and Improper Fractions, Mixed Numbers
For the following set of fractions, identify those that are proper and those that are
improper. If the fraction is improper, convert it to a mixed number.
1.
4.
1345
1928
2.
1210
5.
2010
3.
4514
6.
13025
Exercise IV: Operations on Fractions
Perform the following operations, giving your answer in improper reduced form.
1.
5.
27+36=?
23∙46=?
2.
716+48=?
6.
25∙1218=?
3.
-1820-34=?
7.
1618÷46=?
4.
710-45=?
8.
-25÷49=?
Exercise V: Fractions to Decimals
Convert the following fractions into decimal form.
1.
3.
58
2.
1824
4.
712
825
Exercise VI: Decimals to Fractions
Convert the following decimal numbers into fractions.
1.
0.184
2.
0.5163516351635163…
ANSWERS TO EXERCISES
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Exercise I
1.
35,1123, 27
2.
46, 12, 13
Exercise II
1.
12
3.
52, 34, 49, 13
4.
154, 33, 1112, 910, 38
4.
13
2.
3
5.
25
3.
42125
6.
45
Exercise III
1.
proper
4.
proper
2.
115
5.
2
3.
3314
6.
515
Exercise IV
1.
1114
5.
49
2.
1516
6.
415
3.
-3320
7.
43
4.
-110
8.
-910
3.
0.583
4.
0.32
2.
17213333
Exercise V
1.
0.625
2.
0.75
Exercise VI
1.
23125
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APPENDIX (1/2)
Least Common Multiple Method 2
This method for finding the LCM is very useful when numbers start to get large or when you are
dealing with more than two fractions.
1. Factor the denominators of the subject fractions down to prime factors.
2. Count the number of times each prime factor appears in the factorizations.
a. If a factor appears in more than one denominator’s factorization, the count should
be for the denominator where the factor appears the most.
b. For example, if the factor 2 appears three times in the factorization of A and
twice in the factorization of B, then the count for the factor 2 is three.
3. Multiply all of the unique factors that appeared in the factorizations together; each unique
factor should be multiplied in the number of the count for that factor.
4. This resulting number is your least common multiple. Proceed with your operation.
It is helpful to look at an example.
Example Appendix: LCM by Prime Factors
Find the least common multiple between 16, 17, and 18.
We need to find the prime factorization of the denominators 6, 7, and 8.
6=2∙3
7=7
8=2∙2∙2
The counts for each of the prime factors are 2: three times, 3: one time, 7: one time.
Therefore, the LCM is 2∙2∙2∙3∙7=168.
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APPENDIX CONTINUED (2/2)
Table of Fractions, Decimals, and Percentages*†
Fraction
1/2
1/3
2/3
1/4
3/4
1/5
2/5
3/5
4/5
1/6
5/6
1/8
3/8
5/8
7/8
1/9
2/9
4/9
5/9
7/9
8/9
1/10
3/10
7/10
9/10
1/50
1/100
Decimal
0.5
0.333…
0.666…
0.25
0.75
0.2
0.4
0.6
0.8
0.1666…
0.8333…
0.125
0.375
0.625
0.875
0.111…
0.222…
0.444…
0.555…
0.777…
0.888…
0.1
0.3
0.7
0.9
0.02
0.01
Percent
50%
33.333…%
66.666…%
25%
75%
20%
40%
60%
80%
16.666…%
83.333…%
12.5%
37.5%
62.5%
87.5%
11.111…%
22.222…%
44.444…%
55.555%
77.777%
88.888%
10%
30%
70%
90%
2%
1%
*We know that percentages were not covered in this issue of Weeklies, but felt it prudent to include them
for reference. You can convert a decimal to a percentage by moving the decimal point twice to the left.
†Further, this table is incomplete—these are the fraction-decimal conversions that we recommend that
you memorize. It is always helpful to have others as well.
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