<p>This is a difficult SAT problem that I can't figure out. Help me. Here it goes: </p>
<pre><code> Gwyneth rode her bicycle from her house to school at an average speed of 8 miles per hour. Later that day, she road from school back home along the same route at an average speed of 12 miles per hour. If the round trip took her one hour, how many miles long is the round trip?
Thanks.
</code></pre>
<p>10 miles ?</p>
<p>From xiggi's post up in the forum: </p>
<p>"A girl rides her bicycle to school at an average speed of 8 mph. She returns to her house using the same route at an average speed of 12 mph. If the round trip took 1 hour, how many miles is the round trip. </p>
<p>A. 8
B. 9 3/5
C. 10
D. 11 1/5
E. 12 </p>
<p>PR offers this solution: First the problem is a hard problem (level 5). TCB assumes that the common student will not attempt to solve the problem and pick the trick answer of 10 since it represents the average of 8 and 12. The common student second choice will be to pick a value that is stated in the problem: 8 or 12. PR provides the strategy to eliminate those Joe Blogg answers. Again, the conclusion of PR is to end up with two choices and pick between B and D. In their words, the student will be in great shape! </p>
<p>What's my issue with this? In my eyes, a 50-50 chance is really not good enough. When you consider how this problem can be solved, the recommendation to guess becomes highly questionable. </p>
<p>What could a student have done? Use a simple formula for average rates -an opportunity that PR strangely forgets to mention. Is this formula really complicated? I could detail the way I developed it while working through similar problems, but the reality is that millions of people have seen it before. Im absolutely convinced that many good tutors teach it, but you wont find it in the typical help book. Here it is: </p>
<p>[2<em>Speed1</em>Speed2] / [speed1 + Speed2] or in this case:
2* 8 * 12 / 8 + 12. </p>
<p>Most everyone will notice that the answer is 2*96/20 or simply 96/10. This yields 9.6 or 9 3/5."</p>
<p>No, that is not the answer. The right answer is 9 3/5. Does anyone know how to get to this answer?</p>
<p>I still don't get it. Can someone else explain it to me?</p>
<p>9.6 mi (4.8 mi each way)</p>
<p>Work on it, and if you still don't get it, I'll explain</p>
<p>Ok, this is a pretty simple question.
We are looking for:
(X-Xo); distance</p>
<p>We know:
ttotal= 1 hour (60 mins)
V1= 8mi/hr (to school); slower and should take more time
V2= 12mi/hr (to home); faster and should take less time</p>
<p>With this in mind, we know that the rate from the school, V2, is simply 1.5 times faster than the initial rate, V1.</p>
<p>(V2 = 1.5(V1) or V2/V1 = 1.5; 12/8=1.5).</p>
<p>Therefore, the time should be the same proportion since the distance is constant and equal in both directions. What I mean is, the time home should be 1.5 faster at 12mph than the 8mph because the distance is constant for both routes.</p>
<p>This will give us
t1 (time to school) and t2 (time from school)
Remember that t1 + t2 = ttotal</p>
<p>With the total time ttotal= 60min we know that the route to school should be 1.5 times slower than the route back back (this is the main logic for this problem), or</p>
<p>t2 + 1.5(t2) = ttotal ; where t2 is the time from school to home as defined above</p>
<p>We know ttotal=60mins</p>
<p>by simple algebra</p>
<p>2.5(t2) = 60 mins and
t2=24 mins
so if it took 24 mins back and the total time was 60 mins, then</p>
<p>60-24 = 36 mins to the school</p>
<p>Remember that 12mph is faster and should require less time than 8mph;</p>
<p>Unit Conversion:
24 mins = 0.4hr
36 mins = 0.6 hr</p>
<p>So we know V1, V2, t1, t2. Now it is just the simple distance formula:</p>
<p>Velocity = distance/time
v1=(X-Xo)/t</p>
<p>8mi/hr = (X-Xo)/.6 hr
(X-Xo) = 4.8 mi (to the school)</p>
<p>To check we look at the other direction which should also equal 4.8:</p>
<p>12mi/hr = (X-Xo)/.4 hr
(X-Xo) = 4.8
They are equal. Yay.</p>
<p>Therefore the roundtrip would be 4.8(2) = 9.6 or 9 and 3/5</p>
<p>Let me know if that was clear.</p>
<p>I forgot to mention that t1 = 1.5(t2);</p>
<p>This is an inverse proportion problem. You need to see that as the velocity increases, the time decreases. </p>
<p>V1(t1) + V2(t2) = (X-Xo); roundtrip</p>
<p>uhhhh...oooookk
let me give a much simpler explaination
let x=time it took to travel from house to school
distance=rate multiplied by time
break the trips into halfs
going to school: 8x=half of roundtrip
school to house: 12(1-x)=the other half of roundtrip
since distances equal each other because she travels the same distance you can say that 8x=12(1-x)
8x=12-12x
20x=12
x=3/5
8*(3/5)=half the roundtrip=4 and 4/5
double that to get the value of the roundtrip which is 9 and 3/5</p>
<p>
[quote]
Most everyone will notice that the answer is 2*96/20 or simply 96/10. This yields 9.6 or 9 3/5."
[/quote]
</p>
<p>
[quote]
No, that is not the answer. The right answer is 9 3/5. Does anyone know how to get to this answer?
[/quote]
</p>
<p>Doesn't 9 3/5=9 3/5?</p>
<p>hey hannana, I was trying to give a comprehensive view of the problem and a step by step solution (it must be the tutor in me speaking).</p>
<p>I only did that because the problem was quite easy, and since the previous posters couldn't get it after multiple tries, a more detailed solution would have probably served them better as far as understanding is concerned. Your logic is a little more compact, but didn't really explain why. Either way, 9.6 is correct.</p>
<p>yeah yeah they messed up....whatever</p>
<p>You guys are all making it too complicated. You want the average speed, which is given by this formula:</p>
<p>(2<em>speed1</em>speed2)/(speed1+speed2)</p>
<p>Plug in and solve, you get 9.6 miles per hour. The trip took one hour, so 9.6 is the answer. Beware if it had said the roudn trip was 2 hours, you would need 9.6 times two.</p>
<p>Yeah, that formula is called the harmonic mean. It is the best way to solve those type of q's.</p>
<p>Thanks, guys. I understand this problem now.</p>
<p>Can you explain how the harmonic mean formula is derived? I don't understand why you multiply by two, etc.!?</p>
<p>Since I just learned the harmonic mean formula, I can't really explain the derivation of harmonic mean forumula to you. For more info, just do a simple google search.</p>
<p>Well, the general formula for harmonic mean is:</p>
<p>n/(1/n1) + (1/n2) + (1/n3)....</p>
<p>In this situation, the value of n is 2. Let us represent speed 1 as x and speed 2 as y. So, average speed = 2/(1/x) + (1/y). This simplifies to <a href="2xy">color=blue</a>/(x+y)<a href="After%20finding%20common%20denominator%20on%20bottom%20and%20simplifying%20complex%20fractions">/color</a></p>
<p>As for the derivation.. the distance traveled to and back is the same. The units on (1/n1) and (1/n2) etc. become hour/mi, since 1/(mi/hr) simplifies. Back to the original problem, hr/12 mi + hr/8 mi = 5 hr/24 mi. The numerator, n is 2. So when u put the formula together, 2/(5 hr/ 24 mi) = 48 mi/5 hr, or 9.6 mi/hr.</p>
<p>I will now show the derivation with variables:
Let t1 be the time spent traveling at speed 1 and t2 the time at speed 2.
Let d be distance and r be rate.
The total time = t1 + t2.</p>
<p>Because d1 = (r1)(t1) and d2 = (r2)(t2)
t1 = d1/r1 and t2+ d2/r2</p>
<p>Let r be the average speed overall.
2d = r(t1+t2)
And, 2d = r(d1/r1)+(d2/r2)</p>
<p>Divide both sides by d.
2 = r/r1 +r/r2</p>
<p>Solve for r.
2=(r)(r2)/(r1) + (r)(r1)/(r2)
r = (2)(r1)(r2)/(r1+r2)</p>
<p>Thus the harmonic mean formula is derived.</p>
<p>Lol, I was afraid of the day someone would post the words "harmonic mean". :) </p>
<p>It is EXTREMELY doubtful that ETS will ever go beyond presenting problems with very simple data such a 1 hour roundtrip and very basic rates. There is really no reason to explore the formulas for harmonic means beyond its simplest form. </p>
<p>While it is rather easy to show how the formula is derived from the basic d = r*t,<br>
you can safely assume it works without having to know its entire origin. </p>
<p>When this type of problem shows up on the SAT, it takes two quick steps:</p>
<ol>
<li>Recognize the CORRECT problem and verify the variables<br></li>
<li>Use the quick and dirty formula or know how to pick the answer that ALWAYS works if the questions is a MC. </li>
</ol>
<p>Do not make it harder than necessary,</p>
<p>True, you only need to remember the simple formula for the SAT. However, the topic had progressed beyond use on the SAT to derivation of the formula. </p>
<p>P.S. Ziggi- what did u get on the SATs?</p>