Numbers & Operations
Fractions
Fractions are ubiquitous on the SAT Math section (there’s an SAT word for you to chew on). A fraction shows the relationship between a part of something and the whole of something. Here’s what a fraction looks like: .
A fraction is composed of two numbers, a numerator, and a denominator. The numerator (the part) is above the fraction bar, and the denominator (the whole) is below it. So in our example, 2 is the numerator, and 3 is the denominator.
What if we multiply both the numerator and the denominator of by 4? We would end up with:
Because we multiplied both parts by the same numbers, the two fractions ( and ) have the same part-to-whole relationship. These fractions are equivalent fractions, which means they equal each other.
Suppose we decide that is too bulky. We know that 8 and 12 have an LCM of 4—remember that term?—so we can whittle that down to a slimmer fraction:
Taking a fat fraction and dividing out similar terms from both the numerator and denominator is called reducing a fraction.
Reducing fractions and making equivalent fractions are necessary skills, because you often have to perform these tasks before adding, subtracting, multiplying, or dividing fractions. Don’t simply take our word for it. Review the following sections.
Adding and Subtracting Fractions
We’ll start simple. Fractions with the same denominators are the easiest. You need only add or subtract the numerators:
No rocket science there. Fractions with different denominators are one degree trickier. Before adding or subtracting them, you first need to make sure that the denominators are the same.
For instance, if you need to perform the operation , first you have to equalize the denominators of these two fractions. You do this by finding the lowest common multiple of denominators 3 and 15. This number is the lowest common denominator, or LCD.
Because 15 is a multiple of 3, the LCD is 15. So all you need to do is multiply 3 and 5 to get 15. However, if you increase the denominator by 5, you need to increase the numerator by 5 too. Remember: all fractions are the parts of a whole:
Now you can get your subtraction on:
Suppose the LCD is difficult to find, such as when you have 12 and 18 in the denominator. You could simply multiply 12 and 18 and use the product as a common denominator. This gives you some big fractions that you have to reduce. Working with big numbers increases the possibility of errors, so it’s best to find the LCD before proceeding.
Multiplying and Dividing Fractions
Compared to adding and subtracting with different denominators, multiplying fractions is easy. You just find the product of the fractions’ denominators and numerators:
Sometimes you can cross-cancel numbers and make multiplying even simpler. In the example below, the 5 in the numerator takes out the 5 in the denominator.
Division of fractions is just like multiplication of fractions with a little twist. When dividing, you flip the second fraction, then multiply:
One key concept to remember is: when you multiply a fraction by itself, the resulting fraction is less than the original fraction. For example:
The fraction is smaller than .
Weird-Looking Fractions
Not every fraction on the SAT is in the button-down conservative style. Many items are designed to freak you out. To do this, the SAT uses mixed numbers and complex fractions. We’ll cover complex fractions first.
For complex fractions, you have to calculate a fraction of a fraction:
See that bar between the 3 and the 1? It’s a fraction bar, but you can also think of it as a division sign. That bar holds the key to solving complex fractions. Take the fraction above the bar and divide it by the fraction below the bar. You just learned how to divide fractions, so this shouldn’t be tough:
Simple enough. On to mixed numbers.
A mixed number, or improper fraction, includes both an integer and a fraction. For example, is a mixed number. Mixed numbers are impenetrable to simple addition and subtraction, so you need to do one of two things before you work with them:
  1. Convert them into proper fractions.
  2. Use your calculator and change them to decimals.
Converting improper fractions is pretty simple. Start with a mixed number, such as . Multiply the integer 4 by the denominator 7, then add that product to the numerator 2 and place the new number over the old denominator:
Remember that on the grid-in section, you cannot write a mixed number as your answer because the computer reads as . That’s why converting mixed numbers becomes so important. Of course, you can also just convert the whole mess into a decimal.
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