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About Fractions
TABLE OF CONTENTS
About Fractions................................................................................................................... 1
What is a FRACTION?................................................................................................... 1
Introduction......................................................................................................................... 1
Introduction..................................................................................................................... 1
Forms of Fractions .............................................................................................................. 1
Different Forms of Fractions .......................................................................................... 1
Proper Fractions .......................................................................................................... 2
Improper Fractions...................................................................................................... 2
Mixed Fractions .......................................................................................................... 2
Converting Fractions........................................................................................................... 2
Converting Fractions....................................................................................................... 2
Converting an improper fraction to a mixed fraction ................................................. 2
Converting a mixed fraction to an improper fraction ................................................. 3
Converting a fraction to a percent............................................................................... 3
Converting a percent to a fraction............................................................................... 3
Reciprocals.................................................................................................................. 3
Simplifying Fractions.......................................................................................................... 3
Simplifying Fractions...................................................................................................... 3
Equivalent Fractions ....................................................................................................... 4
Comparing Fractions........................................................................................................... 4
Comparing Fractions....................................................................................................... 4
Comparing fractions with the same denominator ....................................................... 4
Comparing fractions with different denominators...................................................... 5
Comparing fractions with decimals ............................................................................ 5
Operations Involving Fractions........................................................................................... 5
Operations Involving Fractions....................................................................................... 5
Adding or Subtracting Fractions................................................................................. 5
Multiply Fractions....................................................................................................... 6
Dividing Fractions ...................................................................................................... 6
Glossary .............................................................................................................................. 7
References........................................................................................................................... 8
About Fractions
What is a FRACTION?
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A rational number.
A form to express a portion of a whole number.
Introduction
Introduction
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In mathematics, a fraction – formally called a rational number –
expresses a portion of the whole number. It is also used to illustrate a
ratio between two numbers, as well as an expression that represents
the quotient of two numbers.
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A fraction is expressed as , where x can be any integer , and y can be
any integer except for zero. In cases where y equals 0, the fraction is
undefined.
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Consider the fraction below:
This fraction above can be read as “one sixth”, or “one over six.”
Forms of Fractions
Different Forms of Fractions
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A fraction is a generic terminology, and it can be expressed in
different forms.
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Proper Fractions
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A proper fraction is where the numerator is less than the denominator
(i.e., p/q where p < q).
Improper Fractions
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An improper fraction is where the numerator is greater than or equal
to the denominator (i.e., p/q where p >= q).
Mixed Fractions
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A mixed fraction consists of a whole number associated with a proper
fraction.
Converting Fractions
Converting Fractions
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Some forms of fractions can be converted into different forms.
Knowing the proper techniques will be critical for performing
calculations that involve fractions.
Converting an improper fraction to a mixed fraction
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Divide the numerator by the denominator:
a. The resulting quotient becomes the whole number of the new
fraction.
b. The remainder of the quotient becomes the numerator of the new
fraction.
c. The denominator remains the same as before.
2
Converting a mixed fraction to an improper fraction
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1) Multiply the whole number by the denominator, and add the
numerator of the fraction.
2) The answer obtained in step 1 becomes the numerator of the new
fraction; the denominator of the new fraction remains the same as
before.
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Note: the conversion to and from mixed numbers is only applicable to
improper fractions (and not proper fractions). Can you figure out
why?
Converting a fraction to a percent
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A percent is a special type of fraction where the denominator is 100:
1) Convert the fraction such that the denominator is equal to 100.
2) The numerator of the new fraction becomes the percent.
Converting a percent to a fraction
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1) The percent value is the numerator of the fraction.
2) The denominator of the fraction is 100.
Reciprocals
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Two numbers are reciprocals of one another when their product is
equal to 1.
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Understanding the concept of reciprocals is essential for performing
calculations involving the division of two fractions.
Simplifying Fractions
Simplifying Fractions
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A fraction can be simplified when the numerator and the denominator
are composite numbers. It is important to recognize fractions that can
be simplified to their lowest terms.
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Note: you should always simplify fractions to its lowest terms!
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To simplify a fraction:
1)
Determine a common factor between the numerator and the
denominator. A common factor is a number that is divisible by
both numbers. (i.e., 4 is a common factor of 8 and 12).
2)
Divide the numerator and the denominator by the common
factor.
3)
Repeat step 1 and 2 until there are no more common factors.
4)
A fraction is fully simplified when no more common factor
exist between the numerator and the denominator.
Equivalent Fractions
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Equivalent fractions are two or more fractions are equivalent if and
only if the fractions can be simplified to the same fraction. They
express the same amount with one another.
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Note: equivalent fractions are also multiples of one another.
Comparing Fractions
Comparing Fractions
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Knowing how to compare fractions properly is a very important skill
required for many disciplines. You will often be given two fractions,
and will be asked to determine which of the two fractions is larger or
smaller.
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A rule of thumb for comparing any form of fraction is to convert the
given fractions so that the denominators are the same.
Comparing fractions with the same denominator
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Given two fractions with the same denominator, the fraction with the
larger numerator is greater than the fraction with the smaller
numerator.
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Comparing fractions with different denominators
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Fractions with different denominators cannot be compared directly. In
these scenarios, the given fractions must be converted such that the
denominators are the same. With the same denominator, you can then
compare the fractions using the same method described previously.
Comparing fractions with decimals
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A fraction can be regarded as the quotient of two numbers (the
numerator and the denominator). By performing a straight-forward
division between the numerator and the denominator, the fraction is
converted into a decimal. Decimals can then be easily compared.
Operations Involving Fractions
Operations Involving Fractions
Adding or Subtracting Fractions
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To add or subtract fractions, the denominators of the fractions must be
the same:
1) Find the least common denominator.
2) Using the least common denominator, write its equivalent fractions.
3) Add or subtract the numerators.
4) The least common denominator is the denominator of the resulting
fraction.
5) If the operands are mixed fractions, add or subtract the whole
number accordingly.
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Multiply Fractions
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To multiply fractions, the rules are more straightforward:
1) If possible, simplify the fractions to its lowest terms.
2) Multiply the two numerators, yielding the numerator of the new
fraction.
3) Multiply the two denominators, yielding the denominator of the
new fraction.
Dividing Fractions
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The most important step in dividing fractions is using the reciprocal of
the divisor:
1) Convert the divisor into its reciprocal form.
2) Change the division sign to a multiplication sign.
3) Apply the rules of multiplying fractions as described previously.
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Glossary
Composite Number:
a number that is divisible by 1, itself, and
another integer.
Denominator:
in a fraction A/B, the variable B is the
denominator.
Equivalent Fractions:
two or more fractions are equivalent if and
only if the fractions can be simplified to the
same fraction.
Fraction:
expresses a portion of the whole number.
Improper Fraction:
a fraction where the numerator is greater
than or equal to the denominator
(i.e., p/q where p >= q).
Integer:
a set of positive or negative whole numbers,
such as -2, -1, 0, 1, 2.
Mixed Fraction:
a fraction that consists of a whole number
associated with a proper fraction.
Numerator:
in a fraction A/B, the variable A is the
numerator.
Percent:
a special type of fraction where the
denominator is 100.
Proper Fraction:
a fraction where the numerator is less than
the denominator (i.e., p/q where p < q).
Reciprocals:
two numbers are reciprocals of one
another when their product is equal to
one.
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References
http://www.mathleague.com/help/fractions/fractions.htm#whatisafraction
http://www.aaamath.com/fra.html#topic1
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