<p>When each side of a particular square is lengthened by 2 inches, the area of the square
increases by 32 square inches. What is the length in inches of a side of the original
square?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8</p>
<p>A triangle is to be changed by increasing the length of its base by 40% and decreasing
the length of its height by 40%. Which of the following must be true about the area
of the triangle?
(A) It will increase by 40%.
(B) It will increase by 16%.
(C) It will not change.
(D) It will decrease by 16%.
(E) It will decrease by 40%.</p>
<p>Any way to solve that quickly with algebra without having to test it with a hypothetical triangle?</p>
<p>Alright, I’ll try to rephrase. Decreasing the base by 40% would give you a base that is only 60% of the original, therefore this can be written as .6b. Same deal with the height, except it’s increased. Now that you have the values of the new triangle, you plug it into the formula. </p>
<p>A = .5<em>.6b</em>1.4h</p>
<p>I factored out the .5<em>b</em>h, because those were the values of the original triangle, and substituted a singular variable A in its place. .6<em>1.4</em>A is equal to .84<em>A, and .84</em>A is 84% of A (if this doesn’t make sense, substitute a number such as 100 in the place of A). The original triangle is by definition 100% of A, so a drop from 100% to 84% is a decrease of 16% in the area of the triangle.</p>
<p>If you want to stay away from abstract approach (which is nonetheless fast and accurate,) you can quickly build a triangle with HELPFUL numbers. Since the problem is stated in percentage, build one that has an area of 100. </p>
<p>A super easy one would be to use a base of 20 and a height of 10. So far so good, the area is 100. Now change the numbers as suggested in the problem. Base becomes 28 and height 6. The area is thus 1/2 * 6 * 28 … or 84. Now compare 100 and 84, and that is a 16 reduction from 100. Or a 16 percent reduction. </p>
<p>Please note that the explanation is lenghty but practice makes those manipulations easy. After a while, it becomes second nature to plug in easy numbers for triangles.</p>
<p>I would believe that the long way is the only approach so this is how you do it</p>
<p>sqrt(x^2-t^2) = 2t-x
(sqrt(x^2-t^2))^2 = (2t-x)^2 square each side to remove the sqrt sign on the left side.
x^2-t^2 = (4t^2 - 4tx + x^2) expand the right side.
0 = 5t^2-4tx cancel the x^2 on both sides and add t^2 to both sides.
0 = t(5t-4x) factor.
t = 0 which cannot be true and then 5t-4x = 0
5t = 4x add 4x to both sides.
5 = 4(x/t) divide both sides by t.
5/4=x/t divide both sides by 4.</p>
<p>Ty, do you by any chance know any ways to kind of catch up on math? I have the blue book and it doens’t really show you very well how to solve things. Should I just start doing the practice tests and that’s how I will learn the math?</p>
<p>Brolex, the questions you are asking are from Algebra 1.</p>
<p>So my advice is start there and work your way up. Yes, you can buy the blue book but it looks like you need the basic algebra foundation first. Otherwise you’re going to be learning this piecemeal. That said, I haven’t used a prep book to learn algebra, so it might work.</p>
<p>I just saw that algebra was a long time ago for you. It was 21 years ago for me, last math class was 15 years ago, so I know how you feel. However… I still think you need a tutor. Because once you remember the concepts, it should all suddenly appear fairly do-able. If that is not happening you need more than superficial review.</p>
<p>I talked to my calc teacher today and he explained most of the stuff. I’m going to buy the Grubbers math book, I’ve heard that’s the best one at least. Any other books I should consider in addition?</p>
<p>Its free, so it might be a good resource. Another good strategy might be to obtain a textbook on Algebra and do a few problems from each chapter, especially harder ones. If you have access to one, IB Math Studies textbooks are pretty good for SAT Math review (if nothing else is available). They’re about equivalent to Algebra 2, Geometry, Trig, and some Stats. Its analogous to Pre-Calc, actually.
Getting a good (older ones usually have better problems, IMO) Algebra textbook plus using Khan Academy for explanation if necessary is probably a good strategy.</p>
You guys are amazing. Sorry if the questions I’m asking are easy, I’m trying to review math before I take my first practice test.</p>
Those are the only things that change so you’ll have .84 of the original area.</p>