<p>Basic Strategy: Change fractions to decimals</p>
<p>Description: Just perform the division in your calculator to change a fraction to a decimal. Decimals are much easier to compare than fractions, and, with the help of your calculator, computations are much easier as well.</p>
<p>Also in a free response question when you get an answer in decimal convert it to fraction to make sure it fits in the grid, if it doesn’t, find a new answer :P</p>
<p>@Tomer</p>
<p>Although you can grid in your answer as a fraction (in any form that fits in the grid), it is generally easier to just grid in the answer as a decimal - simply truncate the answer to fit.</p>
<p>For example, if your calculator gives a final answer of 2.379989999, you can simply grid in 2.37.</p>
<p>Note that you can also grid in the rounded decimal 2.38, but I prefer gridding the truncated decimal 2.37 since there is less to think about.</p>
<p>For all the information you need on gridding in, see my article “Overview of the Math Sections of the SAT.” It’s posted on CC - just do a search.</p>
<p>Here is an example of a question that is time consuming without this strategy, but can be done in just a few seconds when the strategy is applied.</p>
<p>What is one possible value of x for which 5/16<x<2/5 ?</p>
<p>Basic Strategy: Pick a number</p>
<p>Description: Pick specific values for any unknowns in the problem. Then solve the new easier problem. Use the specific values you ave chosen in the answer choices as well and eliminate any choices that do not come out correct.</p>
<p>Comments: This is a fantastic “fallback” strategy that works on a wide range of problems in all topics and all difficulty levels. Although this is usually not the most efficient way to solve a problem, it is a method that will most likely get you the correct answer. </p>
<p>In my opinion, picking numbers is an important strategy to become proficient in. I will write some follow up posts with more inormation on how to use this strategy effectively and how to avoid some common mistakes that are made when trying to implement this strategy.</p>
<p>Yeah, working with decimals is easy, but if for example your answer is 0.23 then that means its 23/100 and it doesn’t fit in the grid, so the answer is wrong. While an answer like 0.24 is equal to 6/25 that does fit in the grid, so such an answer will work.
Or i’m wrong by that? Is it possible to get a decimal answer that can’t fit in the grid as a fraction?
Thanks for these tips! :)</p>
<p>You can grid in decimals that will not fit in the grid as fractions. This is one of the reasons why I always prefer gridding decimals. If your answer comes out to 23/100, then yes, you can grid in .23.</p>
<p>Back to the strategy of picking numbers! One of the most important things to remember about this strategy is the following:</p>
<p>Most of the time picking numbers only allows you to eliminate answer choices. So don’t just choose the first answer choice that comes out to the correct answer. If multiple answers come out correct you need to pick a new number and start again. But you only have to check the answer choices that haven’t yet been eliminated.</p>
<p>For example, consider the following problem:</p>
<p>Which of the following is equal to (x+66)/22 ?</p>
<p>(A) (x+33)/11<br>
(B) x+3
(C) 3x
(D) x/22+3
(E) (x+3)/11 </p>
<p>Note that if you choose x=0, the answer to the question is 3. Plugging 0 in for x into each answer choice gives the following:</p>
<p>(A) 3
(B) 3
(C) 0
(D) 3
(E) 3/11 </p>
<p>Note that choices (A), (B), and (D) all come out correct. A common mistake would be to choose choice (A) without checking the other choices.</p>
<p>See if you can solve this problem by picking a better number. Then also try to solve it algebraically.</p>
<p>More on picking numbers:</p>
<p>(1) Be aware that if you pick a number that is very simple (such as 0 or 1), more than one answer choice may come out correct. You can usually save time by picking a slightly less simple number (such as 2)</p>
<p>(2) Try to avoid picking numbers that appear in the problem.</p>
<p>(3) When picking two or more numbers try to make them all different.</p>
<p>Picking numbers in percent problems: If the word “percent” appears in a problem it’s usually a great idea to choose the number 100. This often works even when there is no variable in the problem.</p>
<p>Here is a Level 5 example:</p>
<p>If Matt’s weight increased by 30 percent and Lisa’s weight decreased by 20 percent during a certain year, the ratio of Matt’s weight to Lisa’s weight at the end of the year was how many times the ratio at the beginning of the year?</p>
<p>Try to solve this problem by choosing both Matt’s weight and Lisa’s weight to each be 100 at the beginning of the year.</p>
<p>What happens if you pick numbers other than 100 here?</p>
<p>Also try to solve this algebraically without picking numbers.</p>
<p>Which method do you prefer?</p>
<p>A few more tips for picking numbers:</p>
<p>(1) If there are fractions in the question a good choice might be the least common denominator (lcd) or a multiple of the lcd.</p>
<p>(2) Don’t pick a negative number as a possible answer to a grid-in question. This is a waste of time since you can’t grid a negative number.</p>
<p>(3) If your first attempt doesn’t eliminate 4 of the 5 choices, try to choose a number that’s of a different “type.” Here are some examples of “types”:</p>
<p>(a) A positive integer greater than 1.
(b) A positive fraction (or decimal) between 0 and 1.
(c) A negative integer less than -1.
(d) A negative fraction (or decimal) between -1 and 0.</p>
<p>(4) If you’re picking pairs of numbers try different combinations from (3). For example you can try two positive integers greater than 1, two negative integers less than -1, or one positive and one negative integer, etc.</p>
<p>Intermediate Strategy: Recognizing Blocks</p>
<p>Description: Let’s define a block to be an algebraic expression that appears more than once in a given problem. In SAT problems we can usually treat blocks just like a variable. In particular, blocks should usually not be manipulated - leave them as they are!</p>
<p>Comments: Blocks often appear exactly the same in two places. But sometimes they can appear in two different forms. For example, (a + b + c)/3 can also be written as “the average of a, b, and c.” </p>
<p>When you see a block in a problem there are two ways to handle it:</p>
<p>(1) replace the block by a single variable.
(2) leave the block as is, but think of the block as a single unit that should not be manipulated.</p>
<p>CAn you give us an example or two of the block strategy? Thanks!</p>
<p>Yes. Let’s start with a simple Level 1 problem with a block:</p>
<p>If 3(a+b)-4=41, then a+b=</p>
<p>The block in this problem is a+b. </p>
<p>The most common error in this problem is to apply the distributive property to the left hand side of the equation 3(a+b)-4=41 to get 3a+3b-4=41. This would make the question more difficult. If you are always looking out for blocks, you will never make this mistake.</p>
<p>Note that this question has the same solution as the following question:</p>
<p>If 3x-4=41, then x=</p>
<p>In other words, we can substitute the block for a single variable, and we get an equivalent problem.</p>
<p>Here is a Level 4 SAT problem where the block is a bit harder to recognize:</p>
<p>If x=7[(a+b+c+d+e)/5], then in terms of x, what is the average (arithmetic mean) of a, b, c, d, and e?</p>
<p>(A) x/35
(B) x/7
(C) x/5
(D) 5x
(E) 7x</p>
<p>What is the block here? Note that once you recognize the block you can get the answer in just a few seconds.</p>
<p>The block is [(a+b+c+d+e)/5] which is the average of a, b, c, d, and e. So x=7(ave), and
ave=x/7</p>
<p>To me, the block method seems very intuitive. After learning u-substitution in calculus, I see it’s kind of the same idea for both processes.</p>
<p>@pmian</p>
<p>That’s exactly right. You can see that if you have just a little experience recognizing block, these things will start to pop right out at you, and you can answer questions like this in just a few seconds. </p>
<p>@johnstucky</p>
<p>The block method is identical to u-substitution in calculus. The only difference is that on the SAT you will at most have to perform a very simple algebraic manipulation with a block, whereas in Calculus you have to do much more, including “undoing” the block (because usually you want to find x and not u). Note a couple of things:</p>
<p>(1) Blocks appear throughout mathematics. For example, in prealculus you might be asked to solve this equation: x^4-x^2+6=0. Of course you will find them in Calculus, Differential Equations, and more advanced mathematics.</p>
<p>(2) Whatever subject you’re doing, once you get a little practice with blocks, you should begin trying to solve problems without formally doing a substitution. For example, note what pmian did above - he gave the solution without formally changing the problem to x=7u. This will save time. As a simple Calculus example, try integrating xe^(x^2) quickly in your head without formally substituting.</p>
<p>(3) As far as I know, I am the only one that uses the word “block” to describe these substitutions. But especially on the SAT I think this word helps to emphasize my point.</p>
<p>Intermediate Strategy: Try a Simple Operation</p>
<p>Description: When being asked to find an expression consisting of more than one variable, a single simple operation often gives the answer, or at least makes the problem much easier to solve. The four most common simple operations are addition, subtraction, multiplication, and division</p>
<p>Comments: If the expression you are trying to find is a sum or difference, then addition or subtraction usually works. If the expression you are trying to find consists of products or a quotient, try multiplication or division first.</p>
<p>Here is a Level 2 question where this last strategy is effective:</p>
<p>If x+7y=15 and x+3y=7, what is the value of x+5y?</p>
<p>Note that this question can be solved in several different ways:</p>
<p>(1) Guessing and checking until you find x and y.
(2) Solving algebraically for x and y by using elimination, substitution, or Gauss Jordan reduction.
(3) Using MATRIX functions in your calculator to solve by Gauss Jordan reduction.
(4) Finding the point of intersection of the graphs of the two equations.
(5) Trying a simple operation</p>
<p>I believe that (5) is the quickest and easiest method in this case.
(3) Sol</p>