Arithmetic Review: Fractions (Part 5) In this final video on fractions we are going
to go over how to compare fractions and work with fractions whose numerators or denominators
are not necessary integers. On occasions the test might present you with math problems
in which you are asked to compare two fractions and decide which fraction is larger. Say we
are asked to compare the following two fractions: 3/7 and 7/12 in order to make a comparison of two or more
fractions we need to rewrite both fractions so that they have a common denominator. Similar
to the way we added and subtracted fractions, so we essentially need to find a common multiple
of 7 and 12, the quickest way to find a common multiple is by multiplying the denominators
together in this case 7 times 12 which is equal to 84, so we multiply the numerator
and denominator of the first fraction by 12, and multiply the numerator and denominator
of the second fraction by 7, doing that gives us the following equivalent fractions.
36/84 and 49/84
Now that both fractions have the same denominator we can go ahead and compare the numerators
to determine which fraction is bigger or which fraction is smaller, in this case the second
fraction is larger when compared to the first fraction.
We can also compare more than two fractions, say we are asked to compare the following
three fractions: 3/7, 4/8, and 7/11.
One way in which we can compare all three fractions is to find the least common
multiple between the denominators of all the three fractions, you can always multiply all
three denominators to find the common denominator, then write an equivalent fraction by multiplying
each fraction by an integer that would yield the common multiple as follows.
At times this method will be very time consuming since you might need to multiply each fraction by
more than one number to obtain the common denominator. A better and slightly faster
way of comparing fractions is by comparing two fractions at a time. Say we wanted to
know which of these 3 fractions is the largest? We can start by comparing the first 2 fractions,
before we compare them we first need to rewrite them so that both fractions have a common
denominator, the easiest way to do this is by multiplying the denominator of one fraction
with the numerator of the other fraction, in this case we find the product of 7 and
4 which is equal to 28 this becomes the new numerator of the second fraction, in the same
manner we find the product of 8 and 3 which is equal to 24 this number becomes the numerator
of the first fraction, since all we need to compare is the numerator of the equivalent
fractions, we can determine by observation that the second fraction is larger than the
first since 28 is greater than 24. So we can now cross out or eliminate the first
fraction since we concluded that it was smaller than the second fraction, now we go ahead
and compare the second fraction with the third fraction. Same deal we multiply the denominator
of one fraction with the numerator of the second fraction, this number becomes the new
numerator that can be used to make a comparison. Carrying out the products we see that the
third fraction is larger than the second fraction since 56 is greater than 44, so we conclude
that the third fraction is the largest of the three.
When using this method, always multiply from bottom to top, in the direction of the arrows,
working in the other direction or assigning the new numerator to the wrong fraction will
give you the wrong answer. The last thing to keep in mind is to make sure you read the
question correctly and determine the type of comparison they are asking you to make,
they might ask which fraction is larger or which fraction is smaller.
The next idea you need to be familiar with is that numbers of the form a/b may contain
numbers in either a or b that are not necessarily integers, when you are face to face with these
types of fractions don’t panic these types of fractions can be manipulated just like
fractions. For example the numbers, pi over 2 and pi over 3 can be added together as follows,
the common denominator will be 6, so we go ahead and modify each fraction so that we
obtain the common denominator 6, this means that we multiply the numerator and denominator
of the first fraction by 3, and multiply the numerator and denominator of the second fraction
by 2, then it’s just a matter of adding the numerators in this case 3 pi plus 2 pi which
simplifies to 5 pi. In the same manner we can multiply these two numbers together pi over 2 times pi over 3.
Multiplying across gives us the final answer equal to pi squared over 6. Lastly we can also divide
these types of numbers together like in the following complex fraction, pi over 2 divided by pi over 3.
Recall that this is another way of denoting division, so we can rewrite the complex fraction horizontally
as follows, next we go ahead find the reciprocal of the second fraction and change the division
operator to a multiplication operator, then its just matter of multiplying across, notice
that the pi’s cancel out leaving behind the final answer equal to 3/2.
The final types of problems you need to be careful with are problems that require you
to perform a combination of the operations that were described in the previous videos.
The important thing to remember is that the goal of all these steps and operations is
to simplify the fractions, let’s go over the final example and simplify the following
expression. The quantity (3/4 + 1/7) times (-2/5)
Alright, we first want to add the fractions 3/4 and 1/7, so we go ahead and
rewrite both fractions so that they have a common denominator then we add the numerators
keeping the common denominator doing that we obtain the following fraction, now we need
to multiply these two fractions before we do that notice that we can reduce the numerator
25 with the denominator 5, in addition we can also reduce the denominator 28 with the
numerator 2, yielding the following expression, having reduced the fraction we go ahead and
multiply across which results in the final answer equal to negative 5/14. Alright and
there you have it these are the basic ideas that you need to keep in mind when dealing
with fractions in the GRE. In next series of videos we are going to review decimals.