


Fractions
The ability to efficiently and correctly manipulate fractions
is essential to doing well on the Math IIC test. A fraction describes
a part of a whole. It is composed of two expressions, a numerator
and a denominator. The numerator of a fraction is the quantity above
the fraction bar, and the denominator is the quantity below the
fraction bar. For example, in the fraction ^{1}
/_{2} , 1 is the
numerator and 2 is the denominator.
Equivalent Fractions
Two fractions are equivalent if they describe equal parts
of the same whole. To determine if two fractions are equivalent,
multiply the denominator and numerator of one fraction so that the
denominators of the two fractions are equal. For example,
^{1}/_{2} =
^{3}/_{6} because
if you multiply the numerator and denominator of
^{1}/_{2} by
3, you get
As long as you multiply or divide both the
numerator and denominator of a fraction by the same nonzero
number, you will not change the overall value of the fraction. Fractions
represent a part of a whole, so if you increase both the part and
whole by the same multiple, you will not change their fundamental
relationship.
Reducing Fractions
Reducing fractions makes life with fractions much simpler.
It makes unwieldy fractions, such as ^{450}
/_{600} , smaller
and easier to work with.
To reduce a fraction to lowest terms, divide the numerator
and denominator by their greatest common factor. For example, for
^{450}/_{600} ,
the GCF of 450 and 600 is 150. The fraction reduces to
^{3}/_{4} .
A fraction is in reduced form if its numerator and denominator
are relatively prime (their GCF is 1). Therefore, it makes sense
that the equivalent fractions we studied in the previous section
all reduce to the same fraction. For example, the equivalent fractions
^{4}/_{6} and
^{8}/_{12} both
reduce to ^{2}/
_{3} .
Comparing Fractions
When dealing with integers, large positive numbers with
a lot of digits, like 5,000,000, are greater than numbers with fewer
digits, such as 5. But fractions do not work the same way. For example,
^{200}/_{20,000} might
seem like an impressive fraction, but ^{2}
/_{3} is actually
larger because 2 is a much bigger part of 3 than 200 is of 20,000.
In certain cases, comparing two fractions can be very
simple. If they have the same denominator, then the fraction with
the larger numerator is bigger. If they have the same numerator,
the fraction with the smaller denominator is bigger.
However, you’ll most likely be dealing with two fractions
that have different numerators and denominators, such as
^{200}/_{20,000} and
^{2}/_{3} .
When faced with this situation, an easy way to compare these two
fractions is to use cross multiplication. Simply multiply the numerator
of each fraction by the denominator of the other, then write the
product of each multiplication next to the numerator you used to
get it. We’ll crossmultiply ^{200}
/_{20,000} and
^{2}/_{3} :
Since 40,000 > 600, ^{2}
/_{3} is the greater
fraction.
Adding and Subtracting Fractions
On the SAT II Math IIC, you will need to know how to add
and subtract two different types of fractions. The fractions will
either have the same or different denominators.
Fractions with the Same Denominators
Fractions are extremely easy to add and subtract if they
have the same denominator. In addition problems, all you have to
do is add up the numerators:
Subtraction works similarly. If the denominators of the
fractions are equal, then you simply subtract one numerator from
the other:
Fractions with Different Denominators
If the fractions do not have equal denominators, the process
becomes somewhat more involved. The first step is to make the denominators
the same and then to subtract as described above. The best way to
do this is to find the least common denominator (LCD), which is
simply the least common multiple of the two denominators. For example,
the LCD of ^{1}/_{2} and
^{2}/_{3} is
6, since 6 is the LCM of 2 and 3.
The second step, after you’ve equalized the denominators
of the two fractions, is to multiply each numerator by the same
value as their respective denominator. Let’s take a look at how
to do this for our example, ^{1}/
_{2} + ^{2}
/_{3} . For
^{1}/_{2} :
So, the new fraction is ^{3}
/_{6} . The same process
is repeated for the second fraction, ^{2}
/_{3} :
The new fraction is ^{4}
/_{6} . The final step
is to perform the addition or subtraction. In this case,
^{3}/_{6} +
^{4}/_{6} =
^{7}/_{6} .
If you think it will be faster, you can always skip finding
the LCD and multiply the denominators together to get a common denominator.
In some cases, such as our example, the product of the denominators
will actually be the LCD (2 3
= 6 = LCD). Other times, however, the product of the denominators
will be greater than the LCD. For example, if the two denominators
are 6 and 8, you could use 6 8
= 48 as a denominator instead of 24 (the LCD).
The drawback to this second approach is that you will
have to work with larger numbers.
Multiplying Fractions
Multiplying fractions is quite simple. The product of
two fractions is the product of their numerators over the product
of their denominators. Symbolically, this can be represented as:
Or, for a numerical example:
Dividing Fractions
Multiplication and division are inverse operations. It
makes sense, then, that to perform division with fractions, you
need to flip the second fraction over, which is also called taking its
reciprocal, and then multiply:
Here’s a numerical example:
Mixed Numbers
A mixed number is an integer followed by a fraction, like
1^{1}/_{2} .
It is another form of an improper fraction, which is a fraction
greater than one. But any operation such as addition, subtraction,
multiplication, or division can be performed only on the improper
fraction form, so you need to know how to convert between the two.
Let’s convert the mixed number 1^{1}
/_{2} into an
improper fraction. First, you multiply the integer portion of the
mixed number by the denominator and add that product to the numerator.
So 1 2 + 1 = 3, making 3 the numerator
of the improper fraction. Put 3 over the original denominator, 2,
and you have your converted fraction, ^{3}
/_{2} .
Here’s another example:
