Arithmetic Review: Fractions (Part 1) In the following series of videos we are going
to review the concepts associated with fractions let’s first start with the basic definitions
and terminology. A fraction is a number of the form a/b, where
a and b are integers and b does not equal 0 since division by zero is undefined.
In essence you can think of a fraction as another way of writing division for example
the fraction 4/5 is just another way of writing 4 divided by 5, the fraction bar is pretty
much a shorthand way of denoting division. In general the integer on top of the fraction
bar is called the numerator, and the integer underneath the fraction bar is called the
denominator. For example negative 5 over 3 is a fraction in which -5 is the numerator
and 3 is the denominator. These types of numbers are also referred to as rational numbers.
Now let’s talk about the various properties of fractions.
If both the numerator a and denominator b are multiplied by the same nonzero integer,
the resulting fraction will be equivalent to a/b. For example, say that I have the fraction
negative 5 over 3, if we multiply both numerator and denominator by the integer 4 we would
obtain the following fraction, -20/12 this fraction is equivalent to negative 5 over 3 or -5/3.
In the same manner if we multiply the numerator and denominator by negative 1,
we would obtain the following fraction, 5/-3
The result of these two examples illustrate a couple of important properties
that you should keep in mind when dealing with fractions. If both the numerator and
denominator have a common factor, then the numerator and denominator can be reduced into
an equivalent fraction. For example, if we were to take the fraction 40/72, we can rewrite
the numerator 40 as 8 times 5, in the same manner we can rewrite the denominator as 8
times 9. Since both the numerator and denominator have 8 as a common factor you can cross out
or cancel this integer from both numerator and denominator, this process leaves you with
an equivalent fraction in this case 5/9. Fractions of this form are said to be reduced in simplest form.
Let’s go over another example, say we had
the fraction 16/20, there are two ways you can reduce fractions, the first requires you
to write the prime factorization of the numerator and denominator, the prime factorization of
16 is 2 times 2 times 2 times 2, and the prime factorization of 20 is 2 time 2 times 5,
now it’s just matter of crossing out or canceling factors that both the numerator and denominator
have in common. In this example both the numerator and denominator have the factor 2 times 2
in common so we go ahead and cancel them out, doing that we are left with the following
factors in the numerator and denominator, since there are no other factors that are
common to both the numerator and denominator we go ahead and simplify the expression obtaining
the equivalent fraction 4/5. You can achieve the same result by dividing the numerator
and denominator by the factors that are common to both. In this example, 4 is a factor of
both 16 and 20, this means that both the numerator and denominator can be divided evenly (without
remainder) by 4, doing this yields the equivalent fraction 4/5.
So either the prime factorization method or the common factor method can be used to reduce
a fraction into an equivalent form. If you can figure out the common factor to both numerator
and denominator by just looking at them then go ahead and divide this common factor to
both numerator and denominator, on the other hand if you are having trouble finding the
common factor then you always have the option of writing the prime factorization and proceed
canceling factors in this way. When you are face to face with fractions always reduce
them before doing anything else. The last property I want to go over is the
use of negative signs on a fraction. A fraction with a negative sign in either the numerator
or denominator can be written with the negative sign in front of the fraction; for example
the fraction, negative 5/3, is equivalent to 5 over negative 3 which is also equivalent
to negative 5 thirds, all 3 fractions are equivalent, regardless of where the negative
sign is located, it can be on the numerator, denominator or in front of the fraction all
of these fractions describe the same fraction. A fraction that contains a negative sign in
the numerator and denominator such as -5/-3 is equivalent to positive 5/3, since a negative divided
by a negative equals a positive. Alright these are the three properties of
fractions I want you to be familiar with. The first one was the idea of generating an
equivalent fraction by multiplying both numerator and denominator by an non zero integer, the
second property involved reducing a fraction into simplest form by crossing out or canceling
a common factor, and lastly a fraction with a negative sign in front is equivalent to
a fraction that contains a negative sign in either the numerator or denominator.
These properties are going to be important when we start reviewing operations with fractions.
In part 2 of this video series we will review how to add and subtract fractions.