4.1 Order of Operations
4.2 Numbers
4.3 Factors
4.4 Multiples
4.5 Fractions
4.6 Decimals
4.7 Percents
4.8 Exponents
4.9 Roots and Radicals
4.10 Logarithms
4.11 Review Questions
4.12 Explanations
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 cross-multiply 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 11/2. 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 11 /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:
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