Being able to efficiently and correctly manipulate fractions is essential to doing well on the Math IC test. A
fractiondescribes a part of a whole. It is composed of two expressions, anumerator and a denominator. The numerator of a fraction is thequantity above the fraction bar, and the denominator is the quantitybelow 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 theydescribe equal parts of the same whole. To determine if two fractionsare equivalent, multiply the denominator and numerator of one fractionso 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
samenonzero number, you will not change the overall value of the fraction.Fractions represent a part of a whole, so if you increase both the partand whole by the same multiple, you will not change their fundamentalrelationship.
Reducing Fractions
Reducing fractions makes life with fractions a lot simpler. It takes unwieldy fractions such as 450 /600 and makes them into smaller, easier-to-work-with fractions.
To reduce a fraction to its lowest terms, divide the numerator and denominator by their GCF. For example, for 450 /600, the GCF of 450 and 600 is 150. So the fraction reduces down to 3⁄4.
A fraction is in reduced form if itsnumerator and denominator are relatively prime (their GCF is 1). Thus,it makes sense that the equivalent fractions we studied in the previoussection all reduce to the same fraction. For example, the equivalentfractions 4/6 and 8/12 both reduce to 2/3.
Comparing Fractions
When dealing with integers, large positivenumbers with a lot of digits, like 5,000,000, are
greater than numberswith fewer digits, such as 5. But fractions do not work the same way.For example, 200/20,000 might seem like a big, 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 fractionscan be very simple. If the denominators of two fractions are the same,then the fraction with the larger numerator is bigger. If thenumerators of the two fractions are the same, the fraction with thesmaller 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 twofractions is to utilize cross-multiplication. All you have to do ismultiply the numerator of each fraction by the denominator of theother, then write the product of each multiplication next to thenumerator you used to get it. We’ll cross-multiply 200/20,000 and 2/3:
Since 40,000 > 600, 2 /3 is the greater fraction.
Adding and Subtracting Fractions
On
SAT II Math IC, you will need to know howto add and subtract two different types of fractions. Sometimes youwill be given two fractions with the same denominator, and other timesyou will have two fractions with different denominators.
Fractions with the Same Denominators
Fractions can be extremely easy to add andsubtract if they have the same denominator. In addition problems, allyou have to do is add up the numerators:
Subtraction works similarly. If thedenominators of the fractions are equal, then you simply subtract onenumerator 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 makethe denominators the same, and then to subtract as described above. Thebest way to do this is to find the least common denominator (LCD),which is simply the LCM of the two denominators. For example, the LCDof 1/2 and 2/3 is 6, since 6 is the LCM of 2 and 3.
The second step, after you’ve equalized thedenominators of the two fractions, is to multiply each numerator by thesame value as their respective denominator. Let’s take a look at how todo 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 canalways skip finding the LCD and multiply the denominators together toget a common denominator. In some cases, such as our example, theproduct of the denominators will actually be the LCD (2
3 = 6 = LCD). But, other times, the product of the denominators will begreater 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 and reduce your answer in the end.
Multiplying Fractions
Multiplying fractions is quite simple. Theproduct of two fractions is the product of their numerators over theproduct of their denominators. Symbolically, this can be represented as:
Or, for a numerical example:
Dividing Fractions
Multiplication and division are inverseoperations. It makes sense, then, that to perform division withfractions, all you have to do is flip the second fraction , which isalso 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 11/ 2.It is another form of an improper fraction, which is a fraction greaterthan one. But operations such as addition, subtraction, multiplication,or division can only be performed on the improper fraction form, so youneed to know how to convert between mixed numbers and improperfractions.
Let’s convert the mixed number 11 /2into an improper fraction. First, you multiply the integer portion ofthe mixed number by the denominator, and add that product to thenumerator. So 1
2 + 1 = 3, making 3 the numerator of the improper fraction. Now, simplyput 3 over the original denominator, 2, and you have your convertedfraction.
Here’s another example:
