<p>Haha, the harmonic mean formula existed a long time before I was born. All I have done is trying to show that its simplicity works very well when applied to a couple of problems that keep on presenting problems to the average SAT taker. </p>
<p>And as we know, the formuls is derived directly from the basic d = rt. </p>
<p>I think that ETS might punt this type of question before any book would give the formula a silly new name.</p>
<p>
</p>
<p>None of us know what ETS will present in its next tests. </p>
<p>However, what we DO know is that this type of question HAS appeared in various forms (multiple choices/gridin) on past SAT. We also know how many students failed to answer this specific question and that it has caused many to lose precious points.</p>
<p>PS IUse the search this forum for “Esther” and you will see dozens of threads that contain"</p>
<p>“Esther drove to work in the morning at an average speed of 45 miles per hour. She returned home in the evening along the same route and average 30 miles per how many miles did Ester drive to work in the morning?”</p>
<p>Diana ran a race of 700 meters in two laps of equal distance. Her average speeds for the first and second laps were 7 meters per second and 5 meters per second, respectively. What was her average speed for the entire race, in meters per second?</p>
<p>Last question on a grid-in!</p>
<p>So it’s 5.8333333333333333333 right?</p>
<p>There’s no way ETS would put questions this easily solved on the SAT…wow.</p>
<p><a href=“College Board - SAT, AP, College Search and Admission Tools”>College Board - SAT, AP, College Search and Admission Tools;
<p>“So it’s 5.8333333333333333333 right?”</p>
<p>Since it is a grid-in, it is easier to leave it in fractional form:</p>
<p>2.5.7 / 12 or 5.7 / 6 or simply 35/6. </p>
<p>PS Realize this was Question 25, or the last question of a grid-in.</p>
<p>HOLD</p>
<p>****ING</p>
<p>EVERYTHING
Where did you get that test? Are there more? Can I has? I wish analogies were still on the SAT, I ** own ** those.</p>
<p>Let me just make sure I’m clear on what I was saying. </p>
<p>(1) I am NOT saying that this type of problem doesn’t appear on the SAT. It certainly does appear. Distance, rate, time problems appear fairly often - often they are level 1, 2, or 3 problems. Once in a while they show up as a level 4 or 5 problem (infrequently). What we’re dealing with here is a level 5 problem.
(2) Both my strategy and Xiggi’s strategy will work for most of the level 4 or 5 problems of this type. My strategy is more general and will work for many, many other types of questions.
(3) I DO recommend memorizing “Xiggi’s formula” if you want to. And IF you remember it during the test, then use it. BUT if you don’t remember it, it’s no big deal. Just use the strategy I gave. If you’ve practiced using mine just a few times you will ALWAYS remember it - there are no new formulas involved. YES, Xiggi’s is a bit faster, but if you are answering this type of question, then you are going for close to an 800, and the small time difference probably isn’t an issue.</p>
<p>From the depths of this forum. :)</p>
<p>This was again the last grid-in question and a Level 5 question.</p>
<p><a href=“http://talk.collegeconfidential.com/sat-preparation/877591-sat-math-question-january-2008-qas.html[/url]”>http://talk.collegeconfidential.com/sat-preparation/877591-sat-math-question-january-2008-qas.html</a></p>
<p>A car traveled 10 miles at an average speed of 20 miles per hour and then traveled the next 10 miles at an average speed of 40 miles per hour. What was the average speed, in miles per hour, of the car for the 20 miles?</p>
<p>=========
PS I do not disagree with Steve with his conclusions. I simply believe that when this type of question shows up on a tests, anyone who paid a bit of attention to a very simple formula will be ecstatic to apply it. Hard question = easy solution that requires minimum calculations … that is a winner in my book. Fwiw, CC readers must have been smiling when they saw that question in January 2008! ;)</p>
<p>And, fwiw, I never will say that this is more or equally important to knowing the d=rt equations, or picking numbers to solve abstract problems.</p>
<p>^ I would say the Harmonic Mean way is better, since it has much less room for error. The lengthier way requires a lot of calculator use and equation setting up. Since both ways are pretty quick though, I personally would just use your way first, then check it with the xiggi way to remove all possibility of error.</p>
<p>Can someone tell me exactly how the “Xiggi/Harmonic Mean” formula is derived from the “r=d/t” formula? I’m just curious as to how the former is formed(lol).</p>
<p>d = rt</p>
<p>10 = 20t
t = 1/2</p>
<p>10 = 40t
t = 1/4</p>
<p>x/2 + x/4 = 20 -----> d(x)/r + d(x)/r = total distance</p>
<p>3x/4 = 20</p>
<p>3x = 80</p>
<p>x = 80/3</p>
<p>I agree with you Xiggi that your formula is better cause its tougher to work through with d=rt. Its just easier for me and im pretty sure for some other people to use this method.</p>
<p>Humm, I think I have done this a few times. There is really not much to learn from seeing how this very effective SAT tool is derived directly from the well-known formula … d = rt.</p>
<p>As we know the total average speed = Total distance/Total time
This is s = d/t. For the people who prefer the d=rt, simply substitute r for speed. </p>
<p>A. Lets establish that
d = distance traveled in each direction
The total distance traveled is d+d or 2d </p>
<p>B. Lets establish that
t1 = time spent on first leg
t2 = time spent on return leg
Total Time = t1+t2 </p>
<p>C. By formula, speed = distance/time but also time = distance/speed</p>
<p>S1= d/t1 and t1 = d/S1
S2 = d/t2 and t2 = d/S2 </p>
<p>Now, lets start (following A and B above)
Total average speed = Total Distance/Total time or
Total average speed = d + d / t1+t2
Total average speed = 2d / t1+t2</p>
<p>Substituting according to C yields
Total average speed = 2d / (d/S1) + (d/S2)</p>
<p>Getting rid of the common “d” gives
2 / (1/S1) + (1/S2) or
2 / (1.S2/S1<em>S2) + (1.S1/S1</em>S2) or
2 / (S1+S2 / S1<em>S2) or
2 * S1</em>S2 / S1 +S2</p>
<p>And here we have it
Total average speed = 2 * S1*S2 / S1 +S2
From here, please remember to use your reasoning power.</p>
<p>Thanks very much</p>
<p>Let me just mention that I believe that students that are going for an 800 should be doing these derivations (like the one Xiggi did above). Doing so will give a deeper understanding of the mathematics and increase your mathematical maturity. This is very important if you are trying to achieve a perfect score.</p>
<p>Btw going back to the problem xiggi posted- if the answer is 26.666667, how do you grid that? The directions say not to truncate decimals like 2/3, but the only way to fit this in the grid is to put 26.67.</p>
<p>You grid it as 26.6 or 26.7. Either is correct.</p>
<p>The “do not truncate” comes into play with an answer like .3333—if you answer .3 or .33 you will be wrong. You have to fill up the boxes with .333</p>
<p>
</p>
<p>The best answer to this problem is 80/3. This is not the same as mixing integers and decimals, as in 26 2/3. When possible, the BEST entry is always a fraction, especially when the correct approach yields a fraction in the first place.</p>
<p>The issue of the decimals is best avoided altogether. This is also a problem that should not require the use of a calculator, and dealing with decimals should not even come up. Do yourself a favor and resist the urge of grabbing that calculator. There is a reason why ETS allows one on the test! :)</p>
<p>After all, the only one needs is to write down “2.20.40 / 60” and reduce it to 80/3 in one super simple step. </p>
<p>In the problems listed above, the answer should be entered as 35/6 and 80/3.</p>
<p>I always prefer either gridding the improper fraction (as Xiggi suggests), or a truncated decimal. For example 2.5789978 would be gridded as 2.57. (The rounded decimal 2.58 is also correct, but rounding requires too much “thinking” so why do it?).</p>
<p>I agree with Xiggi that fractions are nicer when possible because you won’t accidentally truncate too soon. I can’t tell you how many times I’ve seen students get a grid in wrong because they gridded something like .222222222 as .2 (which is marked wrong) instead of .222 (which is marked correct).</p>
<p>But I have no real objection if a student is more comfortable with gridding decimals, in which case I strongly suggest truncating (correctly) as opposed to rounding.</p>
<p>It should be noted that once in a while it is impossible to grid a fraction, and a decimal must be used.</p>