As trembling hands turn the first page of the SAT, the heart of students drops like a rock. This first problem is a WORD problem and word problems are IMPOSSIBLE! The student drops his or her head. How can the test begin with such a hard problem? Be of good cheer young test taker, not all word problems are created equal.
In fact, many word problems that appear at the beginning of the SAT are easier than they seem. The math and writing questions on the SAT are set up in order of difficulty. This means that the problems in the beginning are on the easier side, and the problems towards the end are more difficult. This can be used to your advantage if you do not get overwhelmed by what the problem APPEARS to be, and, instead, focus on what the problem is. Let’s take a look at a question from the beginning of an SAT.
Three times some number is equal to twenty seven times one over that number. What is the number?
This may seem overwhelming at first, but our order of difficulty should lead us to believe that this is a relatively simple problem. The best thing to do with word problems of this sort is to start translating. “Some number” generally can be translated to “a variable” or “x”, so “Three times some number” can be translated to “3x”. “Twenty seven times one over that number” is just “1/x” times “27”, so our translated equation is “3x = 27/x”. This is an equation that is not too hard to solve. Start by dividing both sides by 3
x = 27/3x
This reduces to:
x = 9/x
After multiplying both sides by x, this becomes
x^2 = 9
and taking the square root gives an answer of
x = 3
This question was a pretty simple algebra question, and all it required was a bit of translation. This is very much in contrast to the hard problems which generally require a little more thought and do NOT have an obvious answer. Here is an example:
A square of length 10 units is broken up into 2 by 2 unit squares. Four points are drawn in the center of the four corner squares of this figure and a circle is drawn which goes through each of these points. What is the area of the circle (not shown)?
If this was at the beginning of the test, the instinct might be to assume that the circle simply touches the sides of the square, meaning the circle would have a diameter of ten and a radius of 5. This is, however, too simple for the end of the test, and if the answer feels too simple at the end of the test, then it likely is. Let’s try to draw this circle.
It is clear that the radius of the circle is not simply the length of the square. What seems to be more important is the diagonal! Now, since we are such great SAT students, we remember that a square is just two 45-45-90 triangles stuck together, which means that the diagonal of the square is the side of the square times the square root of two. Thus, the diagonal of the circle is 10√2. This is the diameter of the circle plus the two halves of the diagonals of the smaller squares.
Each smaller square is 2 units (10/5), so the diagonals of these squares will be 2√2. The total diameter is equal to the total diagonal of the large square minus two half-diagonals from the smaller square, which is the same as one diagonal from a smaller square, thus the diameter is 10√2 – 2√2 = 8√2 and the radius is half of that or 4√2. We can now easily find the area by plugging this into the area formula which is π r². π(4√2)² = 32π. VOILA!
Order of difficulty can be a guideline to help students figure out if their approach is too simple or too complex. Though order of difficulty is less useful with medium problems, it can be very helpful in determining if students are working too hard on a problem or not hard enough. Happy preparations!
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David Greenslade is a Veritas Prep SAT instructor based in New York. His passion for education began while tutoring students in underrepresented areas during his time at the University of North Carolina. After receiving a degree in Biology, he studied language in China and then moved to New York where he teaches SAT prep and participates in improv comedy. Read more of his articles here, including How I Scored in the 99th Percentile and How to Effectively Study for the SAT.