Are you looking to amp up your SAT math score? In this blog, we’ll show you some shortcuts for the SAT math section.
The Ratio Shortcut
For the first question, we’ll learn the Ratio Shortcut. This is the idea that the ratio of sides, areas, and volumes can be calculated between each other really quickly and easily. For example, if you have two squares where the smaller square has a side length of x and the larger square has a side length of y, you can quickly find the ratio of the areas of the squares by squaring the ratio of the side lengths. Then you can also find the ratio of the volumes by cubing the side length ratio.
Now let’s see this shortcut in action with a couple of problems. With the first problem, we don’t actually have to use it to be as fast as possible, but we’ll try using it anyway. In the video, Brooke takes us through solving this problem, using the given ratio of side lengths and squaring it to find the ratio of the areas. For this problem, it was probably just as easy to solve it the old-fashioned way, but for the next problem, we’re going to see that this shortcut is going to come in handy.
In the second problem, we have two boxes: one is a big box and has a volume of 96 cubic meters, and the other is a smaller box and has a volume of 12 cubic meters. Brooke shows us in the video that we can solve this by cubing the side length ratio and setting that equal to the volume ratio that is given. Then, we can cross-multiply and solve for the missing side length. The thing about this problem is that if you didn’t have this idea of the ratio shortcut, it gets a little tricky because you only know one of the three sides, so understanding ratios is important to getting this right.
Use the Graph
The next shortcut is a graph shortcut, and that’s basically to use the graph! Trust the graph on the SAT, but make sure you look at the labels because they’re super important.
The next problem has a really ugly system: it has a quadratic and an exponential function. So you can see that there are two intersection points on the graph; one is approximately at (2, 6) and the other is at about (5, 4.5). So we can use these two points and compare them with the answer choices to see which one is close. The first answer choice is (2, 6) reversed, which we can cross out, and the third is (4, 4), which is not close to the two points we have. If we go back up to the graph and draw a line up from where x = 4.5, we can see that it’s not the point of intersection. Therefore, the only remaining answer choice is (2, 6). Yes, you can try solving this algebraically, but the equations are ugly to get into algebraically, so just use the graph! The farther you are from linear equations, the more we recommend using the graphs. And even if they don’t give you a graph for a system, you can try sketching it out to help you out. If you do want to double check this at the end, you can plug in (2, 6) into both of the equations and make sure everything is correct.
Make Exponents Zero
The last tip is to make exponents zero. Whenever you have exponential questions, sometimes a good technique is if you make the exponent zero, the whole thing becomes 1.
For example, in this problem, we know that (0, 2) is going to be on this graph by using that method. Then, we can eliminate answers really quickly with this by seeing which graphs don’t contain that point.
In the next question, we can get all the way to the answer with this trick. We have a graph of this equation, and we need to find the value of a. So what we can do is to pluck a point from the graph, and the easiest point is going to be where x = 0, which is (0, –3). In the video, Brooke plugs this point into the equation and solves for a, and we end up with a = 4. And we’re done!