<p>@DrSteve can you solve the Joseph traveling problem…I’ve always had trouble with these types of rate problems.</p>
<p>I think (B) is the answer.</p>
<p>Here’s a hint to the ratio way: look at the speeds – they are in a simple whole number ratio. And since you travel the same distance both ways, the times for each leg of the trip will be in the same ratio. So divide 180 minutes into two “piles” with that same ratio. The small pile of time will be for the fast trip, the big pile for the slow.</p>
<p>@myolie</p>
<p>I’d like to give some students a chance to solve it first. I will then post a few solutions as well in a day or so. </p>
<p>If you put up your attempt at a solution I can point out any mistakes you are making. In particular, using a “d=rt chart” is a way that I find particularly nice and will work nicely for most of these problems. See post #9 from this thread for an example.</p>
<p>I’d also like to see pckeller’s solution. I think I know what you have in mind, but I’m not certain (my bit of confusion comes from the fact when the distances are the same, rate and time are inversely proportional and not directly proportional).</p>
<p>@DivisionByZero</p>
<p>Can you explain your solution? Did you use one of the methods I or pckeller suggested?</p>
<p>Here’s how I did it:</p>
<p>Let ‘x’ be the distance from Joseph’s house to where he works.</p>
<p>Then (x/30) + (x/45) = 3
x = 54
54/45 = 1.2 hours = 72 minutes</p>
<p>@DrSteve</p>
<p>I got B) 72 minutes but it took me forever to make sense of this.</p>
<p>So what I did was set (d/30) + (d/45) = 3
I solve for d and got 54 miles. Since we’re finding from home to work, I plugged d=54 to d/45 and got 1.2 hours. I convert 1.2 hours to minutes by multiplying 60 and got 72 minutes. Is this correct?</p>
<p>@DrSteve:</p>
<p>When it’s only two quantities, it doesn’t matter if the ratio is direct or inverse when you are dividing up the piles. You just have to stop and think when you are done to make sure that you assign the piles correctly. </p>
<p>For example, suppose two pirates are 15 years old and 25 years old and they have $1000 to share. And suppose that they decide to split it proportionally with their ages. They would have to split 1000 into 15:25 parts. So divide 1000 by 40 to get the size of each part and then multiply by 15 and 25 respectively to get the shares with the older pirate getting more.</p>
<p>But what if the younger pirate convinces the older pirate that just this once, they should split the loot INVERSELY with their ages? You would have to divide 1000 into two parts that had the ratio 1/15 : 1/25 — but when you multiply through by (15x25), you find that the ratio is also expressible as 25:15 which is the same ratio you had before but with the parts swapped. So go ahead and divide proportionally, but then assign the loot the other way: younger pirate gets more loot. If you want to check your work, see if the product of the age and the loot comes out the same for both… it should (and it does).</p>
<p>@ Division and myollie</p>
<p>Both of you did it correctly. Here is the “d=rt chart” that you used:</p>
<p><strong><em>Distance</em></strong><strong><em>Rate</em></strong> <strong><em>Time
home to work</em></strong><strong><em>d</em></strong><strong><em>30</em></strong><strong><em>d/30
work to home</em></strong><strong><em>d</em></strong><strong><em>45</em></strong><strong><em>d/45
total</em></strong>_________________________________________ _____3</p>
<p>And I’ll just add some more details for those that are confused.</p>
<p>The last column gives d/30 + d/45 = 3
45d + 30d = 3(30)(45)
75d = 3(30)(45)
d = 3(30)(45)/75</p>
<p>We want the time it takes Joseph to drive from work to home, that is we want d/45. This is equal to d/45 = 3(30)/75 in hours. To convert to minutes we multiply by 60.</p>
<pre><code>d/45 = 3(30)(60)/75 = 72 minutes, choice (B).
</code></pre>
<p>@pckeller</p>
<p>That makes perfect sense now. I was just getting hung up on the language you were using. </p>
<p>Can anyone else solve it using one of the other four methods mentioned?</p>
<p>Not sure if anyone used this exact method yet. Here’s how I do it.</p>
<p>Driving any distance at 30 mph takes (45/30), or 1.5*, as long as driving the same distance at 45 mph. Therefore, we can see that the ratio of time driving a distance at 45 mph to driving a distance at 30 mph is 2:3. Knowing that 3 hours is 180 minutes:</p>
<p>Divide 180 by 5 to get 36. We can see that one trip therefore took 2<em>36 = 72 minutes, and the other took 3</em>36 = 108 minutes. We only need the first one, however, since we are looking for the faster time.</p>
<p>(B) 72 </p>
<p>This took me about 30 seconds originally. Also, now I just read up and saw that this seems to be the method pckeller was advocating.</p>
<p>@VSJon300</p>
<p>Yes, the answer is D. </p>
<p>Assume that the distance for each of the three segments is 60 km. Then compute the time for each leg: swimming: 20 hr; biking: 3 hour; running: 6 hour. Total time = 29 hour. </p>
<p>Total distance is 180 km.
Average speed = Total distance/Total time = 180 km/29 which is close to 180/30 = 6 km/hr. </p>
<p>No need for the harmonic mean formula. One could also assign x as the length of each segment and arrive at the same answer. </p>
<p>Complete question below:
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathletes average speed, in kilometers per hour, for the entire race?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7</p>
<p>Can someone show me the easy way to solve no.20 problem in BB practice test 7 section 7? I can’t go to book owner’s area due to a server problem.</p>
<p>Well the graph moved 3 units to the right so h is -3 and then it moves 2 units down so k is -2. Thus, hk is 6. </p>
<p>Its a pretty simple problem and you should be getting this if you’ve been scoring 800’s in math. Hahaha</p>
<p>I did the same thing but wasn’t sure as I don’t have access to explanation. Only answers.</p>
<p>:D</p>
<p>This last problem gives an example of another technique on my list of Advanced Strategies.</p>
<p>Basic Transformations:</p>
<p>Given a function f(x) and a positive number c, we have the following basic transformations.</p>
<p>f(x) + c : vertical shift up c units
f(x) – c : vertical shift down c units
f(x – c) : horizontal shift right c units
f(x + c) : horizontal shift left c units
-f(x) : reflection in x-axis
f(-x) : reflection in y-axis
cf(x) : vertical expansion if c > 1, vertical compression if 0 < c < 1
f(cx) : horizontal expansion if 0 < c < 1, horizontal compression if c > 1</p>
<p>Thanks a lot DrSteve. That helped a lot. Most of the mistakes i made are related to graph of function. After your help, I’ve got 3 800 in a row in practice test. So, thanks a lot.</p>
<p>That’s great drexter. As some of you (including myself) have pointed out, there is nothing in this thread that is absolutely necessary to get an 800. Nonetheless, I expect that many of you will find the strategies that we discuss here useful. Pick out the ones that you find most helpful. </p>
<p>Just one comment on the basic transformations: Note that the vertical transformations behave as you would expect, and the horizontal transformations behave opposite from how you would expect. For example replacing x by x-3 shifts the graph to the right 3. </p>
<p>I will post additional solutions to the rate problem shortly and then I’ll get back to counting problems. After that I’ve got a lot more good stuff to come…</p>
<p>Let me give a few more solutions to the rate problem I presented earlier. Here is the problem again:</p>
<p>Joseph drove from home to work at an average speed of 30 miles per hour and returned home along the same route at an average speed of 45 miles per hour. If his total driving time for the trip was 3 hours, how many minutes did it take Joseph to drive from work to home?</p>
<p>(A) 135
(B) 72
(C) 60
(D) 50
(E) 30</p>
<p>We have already seen solutions to this problem using a “d=rt chart” and pckeller’s method of ratios. Here are a few more solutions:</p>
<p>Solution using Xiggi’s formula: Average Speed = 2(30)(45)/(30 + 45) =36
So Total Round Trip Distance = r<em>t =36</em>3 = 108
Distance from Work to Home = 108/2 = 54
Time from Work to Home = distance/rate = 54/45 = 1.2.</p>
<p>Finally multiply by 60 to convert to minutes. 1.2*60 = 72, choice (B).</p>
<p>Solution by starting with choice (C): Let’s start with choice (C). If it took Joseph 60 minutes (or 1 hour) to get from work to home, then the distance from work to home is d = 45 miles. This is the same as the distance from home to work. Therefore, the total time for Joseph to get from home to work would be t = d/r = 45/30 = 1.5 hours. But that means that the total trip only took 2.5 hours. So we can eliminate choices (C), (D), and (E). Since Joseph is traveling faster from work to home, it should take him less than half the time to get home. So the answer is less than 1.5 hours = 90 minutes. This eliminates choice (A), and therefore the answer is choice (B).</p>
<p>Solution by estimation: 3 hours is the same as 180 minutes. If John was travelling at the same rate for the whole trip, it would take him exactly half this time to get from work to home, 90 minutes. Since Jon is travelling a little faster on the way from work to home, the answer will be a little less than 90, most likely 72, choice (B).</p>
<p>
</p>
<p>Why is it not 6 just like above permutations problem?</p>
<p>^because it asked for COMBINATIONS, not ARRANGEMENTS. “Permutation” is a synonym for “arrangement”.</p>
<p>There are 6 permutations but they can be written in pairs: 12, 21 and 13,31 and 23,32. So there are 6 permutations but only 3 combinations.</p>
<p>Also, this is one of those cases where it helps to realize that selecting combinations of two items from a list of 3 is the same as selecting the ONE item that you leave out. And there are clearly 3 ways to do that…</p>