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Test Yourself Now!  
This is the spot where we post a new, super-challenging problem every week or two. These problems are more difficult than those on the actual SAT, but serve to really get the brain churning on complicated math and English topics. Enjoy this problem and explanation by instructor Kiet Vo!



Q: Ansel and Bob live d miles apart along a straight road. To celebrate the arrival of spring, they decide to walk along the road towards each other until they meet. Ansel walks a miles per hour and Bob, who is slower than Ansel because of his peg leg, walks b miles per hour. At the same time that Ansel leaves his home, his pet falcon flies out the door and down the road towards Bob, who left at the same time as Ansel. Once the falcon reaches Bob, it turns around and flies back towards Ansel, and continues to fly back and forth between the two walkers at c miles per hour until Ansel and Bob meet. At the moment that Ansel and Bob meet, how many miles will the falcon have flown?

A) d
B) c
C) cd / (a + b)
D) d(a-b) / c
E) d(a+ b) / c


A: This problem seems daunting at first! As the falcon flies back and forth, how are we to calculate the ever-shrinking distance between Ansel and peg-leg Bob? Is the falcon's distance flown going to be the sum of geometrically decreasing terms (and how many are there?!#@!), each of which is arrived at through brute, finger-numbing calculations? Hardly. As with many problems that seem conceptually difficult and time-consuming on the SAT, there exists a simpler, quicker, and more elegant approach than one that would consume sheets of your scratch paper. So just stay cool, step back, and look at what you know.

Since we're given a distance (d) and various rates at which the participants travel (a,b,c), we know that this problem is clearly a distance equals rate times time problem (d = r x t).

We're asked for the distance that the falcon will have flown by the time the two friends meet. Well, again, distance equals rate times time. And we know the falcon's rate/flight speed, c, since it's given to us. So all we need then is the time that the falcon has flown. How long is that? Well, the falcons flies from the moment that both friends start walking until the time that they meet. So we need to figure out how long it takes for Ansel and Bob to meet.

Again, we use what we know: d = r x t. We want t , or how long it takes Ansel and Bob to meet. We know the total distance that they traveled (cleverly labeled d). We need r. If Ansel walks a mph and Bob walks b mph, what's the appropriate expression for how much ground they cover together? How about a + b? Does this makes sense? Let's see. If for example, they walk for one hour, Ansel will walk a miles and Bob will walk b miles. So together, they cover a + b miles in one hour and their combined rate would be a + b mph. So, now that we know the distance and the rate, we can solve for the time it takes for the two to meet:

Distance = Rate x Time (Ansel and Bob)
d = (a + b) * t
t = d / (a + b)

The time for the two to meet is the same time that falcon has flown, so the last step is to plug this t back into the falcon's distance equation:

Distance = Rate x Time (falcon)
Distance = c * d / (a + b)

We scan the answer choices, and voila, we select (C). See? That wasn't so taxing...definitely not as taxing and time-consuming as walking bd / (a + b) miles with a peg leg!

 
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