<p>A and B ran a race of 480 m. in the first heat, A gives B a head start of 48 m and beats him by 1/10 th of a minute. in the second heat, A gives B a head start of 144 m and is beaten by 1/30th of a minute. What is B's speed in m/s?</p>
<p>A: 12 m/s
B: 14 m/s
C: 16 m/s
D: 18 m/s
E: 20 m/s</p>
<p>i got C (there is no answer key)
but i only reached it after plugging in 16 for B's speed, along with the fact that the distance ran by B in heat 1 was 432; heat 2: 336</p>
<p>that means he ran 27 seconds in heat 1 and 21 seconds in heat 2
which also means that A ran 480 meters in both heat 1 and heat 2, and 21 seconds and 23 seconds, respectively.</p>
<p>then the speed of A is about 22.9 (i rounded to 23; you are not </p>
<p>the very fact that i had to round (not to mention the 4 minutes it took me to do this problem) led me to doubt the efficiency of my method</p>
<p>I also had to plug in choices; if i had started with A to work my way to C, it wouldve taken me even more time</p>
<p>any other ideas as to how to solve it?</p>
<p>and this one?</p>
<p>a certain quantity of a 40% solution was replaced by a 25% solution such that the final mix was a 35% solution</p>
<p>what fraction of the 40% was replaced?</p>
<p>1/4
1/3
1/2
2/3
3/4</p>
<p>i got 1/3rd because i again plugged in numbers</p>
<p>i actually tested all of the choices (took me 5 minutes to solve this problem)</p>
<p>say we have a 40g/100ml solution
if we take away 1/3rd, we’ll have a 66ml solution that contains about 26g of solute</p>
<p>again, i had to round</p>
<p>if we replace that with 33ml of 25% solution, we’ll be adding about 8 and a little bit more to the 26g, which is closest to a 35% solution</p>
<p>any other ways to solve this question?</p>
<p>i suck at setting up equations; my methods are, in my opinion, intuitive but obviously not efficient</p>
<p>nvm my solution in qusetion 1 was completely wrong because A would have to run the course in the same time for both</p>
<p>so i used this method:</p>
<p>432=(Ta-6)Rb
336=(Ta+2)Rb</p>
<p>by setting both to zero, we get</p>
<p>TaRb-6Rb-432 = TaRb+2Rb-336</p>
<p>by subtracting the 2nd from the 1st, we eliminate Ta</p>
<p>and get that -8Rb=-96</p>
<p>divide by -8</p>
<p>B’s rate was 12?</p>
<p>damn i was so far off
1x so far :(</p>
<p>I solved the 1st one in a similar fashion and I got that A ran in 30 seconds, and that B’s rate was 12 m/s. This is how I did it:</p>
<p>I know that velocity is measured in distance/time. I found the distance ran by B in both heats and the time it took (relative to time of A). These are equal to each other. I solved for time of A and substituted back into wither one of the equations and found that velocity of B is 12.</p>
<p>Velocity of B equals both:</p>
<p>(432)/(Ta+6) AND (336)/(Ta-2)</p>
<p>Cross-multiply both sides and solve for Ta (30 seconds) and substitute back in. (336/28) = (432/36) = 12.</p>
<p>For the 2nd question, I did get 1/3 by setting up a system of equations:</p>
<p>1st, I assumed that there are 100 mL of solution (you can pick any number you want)</p>
<p>Then, I set up this system (x being quantity of 40% and y being quantity of 25%):</p>
<p>x + y = 100
.4x + .25y = 35</p>
<p>The 1st equation shows that the total amount of 40% and 25% is 100mL.
The 2nd equation shows that the concentration of the 40% and of the 25% averages out to be 35% of the total solution (the rest is water). (Note that 35 comes from .35 multiplied by 100 (quantity). If you had picked 50 mL as your amount of total solution, you would multiply .35 by 50 instead of 100).</p>
<p>x is 66.66666666 and y is 33.3333333333</p>
<p>Before the 25% was added, x would be 100. 33.3333333333 divided by 100 is .333333333333333, or 1/3.</p>
<p>hmmm i think for the 2nd one</p>
<p>your equation is a bit flawed because as it stands, you are saying that x is the amount of the 40% solution you ahve after removing that certain quantity</p>
<p>which doesnt help you solve for the quantity that you removed</p>
<p>the equation should look like this</p>
<p>40g/100ml - Xml(40g/100ml) + Xml(25g/100ml) = 35g/100 ml</p>
<p>moving things around, i got</p>
<p>40g/100ml - 35g/100ml = Xml(40g/100ml) - Xml(25g/100ml)</p>
<p>5g/100ml = Xml[(40g/100ml)-(25g/100ml)]</p>
<p>5g/100ml = Xml(15g/100ml)</p>
<p>dividing both sides by 15g/100ml, we get 5/15 = 1/3</p>
<p>1/3=xml</p>
<p>so we lost 1/3rd</p>
<p>did i make any sense?</p>
<p>Yeah, it makes sense. The reason that my system works is because there is always 100 mL of liquid. It’s as if we removed some of 40% and replaced it with 25%. Initially, x=100 and y=0, and after the dilution, x=66.6666666 and y=33.333333333.</p>