SAT algebra and fractions are no more complicated than any other kinds of algebra and fractions. Many beginning SAT students struggle with algebraic fractions, making mistakes that they would themselves find laughable if they were working with numbers rather than variables.
You will struggle to do well on the SAT math exam if you are at all uncertain about how to deal with fractions, whether they be algebraic or numerical. This post will help bring more clarity to SAT algebra and fractions.
Ultimately, it is important to recognize that a fraction is simply another way of representing division. Writing '2/5' simply means '2 divided by 5,' which is exactly equal to 0.4. Sometimes it is more convenient to write a quantity as a fraction rather than a decimal because you intend to do other operations with it.
Another important consideration is that fractions are exact, whereas decimals need not be. While 2/5 is exactly equal to 0.4, 1/3 is NOT exactly equal to 0.333. Regardless of how many significant figures you use, you will never be able to write a decimal that is exactly equal to 1/3. We will discuss significant figures at greater length in another post.
A decimal such as 0.4 could be exactly 2/5. On the other hand, it may mean 0.43 expressed to 1 significant figure. This depends entirely on the intentions of the author, which may or may not be made explicit.
Therefore it is good practice to leave fractions as fractions unless you are explicitly asked to express them as decimals.
Much of the motivation for students unnecessarily converting fractions to decimals stems from their desire to use their calculators rather than multiplying or dividing in their heads. KEEP THE USE OF CALCULATORS TO A MINIMUM. If you know your multiplication tables, it is much faster and more accurate to multiply and divide small numbers mentally than to punch them into your calculator, which additionally admits the possibility that you will enter a number wrongly.
In simplifying fractions, you have undoubtedly come across the idea of cancelling common factors from the numerator and denominator. Bear in mind that this act of cancelling is simply dividing both numerator and denominator by the common factor.
For example, if we have a fraction like 2/6, we can cancel (or divide out) a common factor of 2 from both numerator and denominator: 2/6 = 2/(2.3) = 1/3.
However, if we have something like 2/(5+2), we CANNOT cancel the 2's – they are not common factors. Try it with your calculator if you don't believe me: 2/(5+2) = 2/7 ≠ 1/5.
Similarly, if we have an algebraic fraction like a/(a + b), we cannot cancel the a's: a/(a+b) ≠ 1/b.
This is one of the most common and frustrating mistakes that SAT students make. Please do not join the herd.
The rules are exactly the same whether we are working with numbers or variables. If you are really in doubt, try substituting numbers for the variables and see if what you are doing makes sense.