<p>Hey guys, I'm doing some tests and have encountered some problems which I'm not sure how to do. I've seen a cr thread were people post their questions and help each other understand why an answer is right, so I thought we could do the same for math :)</p>
<p>I'll start out by posting some of the ones I got wrong, and hopefully you guys can answer and post your own too! Also, when you answer, try and explain the steps so that others can understand the process, after all the main point of this thread is to help each other out! :D</p>
<ol>
<li><p>What is the greatest possible area of a triangle with one side of length 7 and another side of length 10? </p></li>
<li><p>If a positive integer n is picked at random from the positive integers less than or equal to 10, what is the probability that 5n + 3 ≤ 14?
I put 3/10 because I thought that the only numbers that could be used were 0, 1 and 2, but the correct answer was 1/5... Is 0 not counted as an integer or something ??</p></li>
<li><p>If t is a number greater than 1, then t^2 is how much greater than t?
The correct answer is t(t-1), however I'm not sure what that means... </p></li>
<li><p>The height of a right circular cylinder is 5 and the diameter of its base is 4. What is the distance from the center of one base to a point on the circumference of the other base?</p></li>
<li><p>If j, k, and n are consecutive integers such that 0 < j < k < n and the units (ones) digit of the product jn is 9, what is the units digit of k?
I don't get what "units digit" means..? </p></li>
</ol>
<p>Rather than completely explain the questions, I’ll point you in the right direction. Putting in some effort to figure out the questions may help them stick better.</p>
<ol>
<li><p>Look up the Triangle Inequality Theorem. It’s very likely to show up on the test.</p></li>
<li><p>0 is neither positive nor negative</p></li>
<li><p>Trying picking a number for t. It will always be true as long as t > 1 as the problem says.</p></li>
<li><p>This is a Pythagorean Theorem question is disguise. Using the points indicated in the question, you can draw a right triangle.</p></li>
<li><p>Units digit is the ones place.
I.e. 5,234,239 has a 9 as its units digit.</p></li>
</ol>
<p>However, regarding the first one, can you show me how to apply the Triangle Inequality Theorem? I used the formula (b<em>h</em>sinx)/2 to get my answer, and since the maximum value of sin is 1, I got my answer which is 35. I’d like to know how to do it the other way too though :)</p>
<p>The way you obtained the answer is fine. Sketch the triangle, and note that the area is 1/2 * base * height. A sketch of possible triangles would show that a right triangle with one base 7 and the other 10 has the maximum area so you don’t really need to memorize the formula you used. You can quickly derive it.</p>
<p>I don’t see how the triangle inequality theorem helps in this problem.</p>
<p>Here’s a quick way to visualize the answer to number 1 without having to draw anything:</p>
<p>Put the fingertips of your right hand to the palm of your left hand forming an L (your left hand fingers should be pointing straight up, your right hand palm down). This represents a right triangle with base = right hand, and height = left hand. </p>
<p>Now if you move your left hand in either direction you get a triangle with the same base but a smaller height. Thus the right triangle gives you maximum area.</p>
<p>Remark: If you move your left hand to the left (or counterclockwise) you get an obtuse triangle. Note that the altitude of an obtuse triangle doesn’t touch the base, but if you extend the base to a line, the altitude does hit that line.</p>
<p>Thanks for the answers! Heh, yeah ACTSecrets that sometimes happens to me too. Sucks to get a wrong answer because of that :/</p>
<p>Here’s another one! </p>
<ol>
<li>The least and greatest numbers in a list of 7 real numbers are 2 and 20, respectively. The median of the list is 6, and the number 3 occurs most often in the list. Which of the following could be the average (arithmetic mean) of the numbers in the list?
I. 7
II. 8.5
III. 10</li>
</ol>
<p>I know the answer is all of them. And that the set is 2, 3, 3, 6, …, …, 20. I just don’t know how to get those 2 last numbers. Do I have to choose numbers at random? I’ll be on time constraints and don’t think I’ll be able to do that… Is there a trick or something?</p>
<ol>
<li><p>A telephone company charges x cents for the first minute of a call and charges for any additional time at the rate of y cents per minute. If a certain call costs $5.55 and lasts more than 1 minute, which of the following expressions represents the length of that call, in minutes?
A) 555-x / y
B) 555+x-y / y
C) 555-x+y / y
D) 555-x-y / y
E) 555 / x+y</p></li>
<li><p>Marbles are to be removed from a jar that contains 12 red marbles and 12 black marbles. What is the least number of marbles that could be removed so that the ratio of red marbles to black marbles left in the jar will be 4 to 3?</p></li>
<li><p>For what positive number is the square root of the number the same as the number divided by 40?
Any “quick” way to do this? I spent too much time trying to plug in random numbers and gave up…</p></li>
</ol>
<p>No kidding: hilight the text of your questions and right click with your mouse. Choose “Search with google” and then select from lots of great explanations – no waiting!</p>
<p>1) (C) ; I do these problems straight up without any algebra. As x = 155 to make y a nice integer. That leaves us with 400, and assume y = 200, which makes the call 3 minutes.
Substitute and you’ll find (C) as the answer. Mathematical? Not really, but good for the SAT? Yes!</p>
<p>2) 3 ; 4:3 indicates that 12:x should correspond. To make a 4:3 ratio, remove 3 black marbles (12:9) </p>
<p>3) 1600 ; sqrt (x) = x/40, put everything to the second power and you’ll get 1600.</p>
<ol>
<li><p>C. I usually just substitute numbers, it works quicker and is more accurate (at least for me).</p></li>
<li><ol>
<li>Remove 3 black marbles and you have 12:9, or 4:3.</li>
</ol></li>
<li><p>Well if you think about it the only number it could logical be would be the square of 40, 1600. Can’t really explain further on this one…</p></li>
</ol>
<p>I’m unsure about the first one still. I understand that you guys assume what x and y are, but I don’t get how you decided which numbers to use. </p>
<p>This is how I did it:
Option A) is the amount of minutes the answer lasted without counting the first minute.
Therefore, the real amount of time the call lasted would be 1 + 555-x / y, which would be y/y + 555-x / y, so 555-x+y / y would be the correct answer? Is this ok?</p>
<p>Anyway I can replace the x and y (how can I choose the numbers?) to know if my answer is correct?</p>