There are many questions on the math section of the SAT / ACT which can be answered in a technically correct manner. However, glancing at the question and giving it some thought before your actual exam can help save you ever-precious time on the test itself.
One of the most difficult elements of the math section is time management. Many of the questions that students get wrong on the SAT / ACT could be solved if there was unlimited time. Often, the questions are designed this way. It's possible to answer them quickly and neatly if you have sufficient acumen - or long and painfully, if you're taking the direct route. This is the SAT / ACT's way of punishing you for not thinking innovatively enough.
The problem is, most students are taught to answer questions in a cookie-cutter manner. They go through all the steps of a specific process, without considering whether there's a better way. In some cases, this means the solution will take more time. In others, it means not solving at all.
The questions we'll look at today most often appear in straightforward algebra: solving for one or two variables, or solving for an algebraic phrase. We'll start by looking at a relatively simple question. By learning to recognize such questions ahead of time, you may save yourself a little time on the test. But you'll also be training yourself for questions where considerable time will be saved if you do them correctly. In addition, there are questions that actually can't be solved if you haven't caught on to the "trick." We'll look at all three of these question types today.
Before we start, please note: this is not a tutorial in isolating variables or solving for systems of equations. If you have trouble with these basic mathematical operations, please do some worksheets and sharpen your skills before taking the SAT / ACT. This article targets how to answer certain questions quickly and efficiently, without taking the long route.
Let's look at some examples.
Given: 6t + 2 = 7
What does 42t +17 equal?
The usual way students would solve this is through what they've learned in school: find t, and then find the value of the algebraic phrase:
Step #1: Subtract 2 from both sides of the original equation
6t = 5
Step #2: Divide both sides by 6
t = 5/6
Step #3: Multiply both sides by 42
42t = 35
Step #4: Add 17 to both sides
42t + 17 = 52
There are numerous steps here. And since this kind of question would probably appear on the non-calculator section, Step #3 might provide a bit of a hassle.
Note that Solution A first tries to isolate t, since this is probably what you've learned in school. But the question never asks you to find t. They ask you to find 42t + 17, which can be found in a simpler way.
Solution B takes advantage of the fact that 42 is a multiple of 6. That is, 6*7 = 42.
Step #1: Multiply the original solution on both sides, by 7
42t + 14 = 49
Step #2: Add 3 to either side
42t + 17 = 52
There are many questions like this on the SAT / ACT. You're given a single equation; instead of the variable 'x', 't', etc., you're asked to find an algebraic phrase. And it's usually easier to find the algebraic phrase than by way of first isolating the variable.
Now let's look at some questions where an alternative route would save considerable time.
3x + 13y = 19
4x -6y = 8
What does x + y equal?
You can solve this question in the usual process for systems of equations. Find one of the variables, and then plug in and solve for the other.
Let's look at an alternate solution.
Note that you haven't been asked to find either 'x' or 'y'. They're asking for the very specific algebraic phrase 'x+y'. This indicates that there may be a shortcut. And indeed there is: if you add the two equations, you'll get the same coefficient for 'x' and 'y'. That means that 'x+y' can be isolated, without knowing what either 'x' or 'y' by itself is.
Step #1: Add the two equations
7x + 7y = 27
Step #2: Factor out the 7.
7 * (x + y) = 27
Step #3: Divide by 7 on both sides:
x + y = 9.
The answer is 9.
Whenever you're asked to find a specific algebraic phrase instead of a simple variable, glance over the question to see if there's a shortcut.
The important thing to remember is that the SAT, or the ACT, is not about solving each question, step by step, in the manner you were taught in class. It's about getting the right answer as quickly and easily as possible. Always keep that in mind when solving!
Let's look at a similar question.
7x + y = 11
2x + 11y = 13
What does x - 2y equal?
In this case, adding the two equations is useless: we would get 9x + 12y, which is useless. Let's try subtracting the equations instead.
Step #1: Subtract the two equations
5x - 10y = -2
Step #2: Factor out 5 from the left side
5 * (x - 2y) = -2
Step #3: Divide both sides of the equation by 5
x - 2y = -2/5
-2/5 is the answer.
Please note that the very fact of asking for an algebraic phrase, instead of a variable, does not necessarily mean that there's a shortcut. But it does mean that you should at least look.
In all the questions so far, there's a way of solving the question in the straightforward manner you were taught in class, even if it does take a long time. But there are certain questions that are impossible to solve if you go about them directly, and you must learn to think a little differently. Let's look at a few such examples.
We'll start with a simpler one:
5*(a + b) + 4 = 9
What does a + b equal?
After the previous questions, this one should be easy. The key is to notice that the question isn't asking for a, or b; in fact, it's impossible to isolate either of these variables from this equation. But if you think of 'a + b' as one variable - say, 's = a + b', you can solve this question easily:
5s + 4 = 9
By subtracting 4 and then dividing by 5 on both sides, we'll get that s = a + b = 1.
2*(s/t) + 7 = 3
What does t/s equal?
In order to answer this questions, two things must be taken into account:
1) The question doesn't require t, or s, but rather t/s
2) t/s is the reciprocal of s/t. That is, t/s = 1 / (s/t)
So, to solve the question:
Step #1: Subtract 7 and then divide by 2 on both sides of the original equation:
--> s/t = -2
Step #2: Take the reciprocal of s/t
s/t = -2 --> 1 / (s/t) = t/s = -1/2
Ergo, the answer is -1/2
If you understand that you're not being asked to solve for a or b, the questions starts looking a lot more approachable.
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