singhall wrote:
Bunuel wrote:
Two hikers simultaneously start walking towards each other from Kensington, MD and Reston, VA, which are 30 km apart. Both hikers walk at a constant speed of 5 km/hour each. At the same time, a fly takes off from Kensington and flies towards the Reston hiker at a speed of 10 km/hour. When the fly reaches the Reston hiker, it immediately turns around and flies back to the Kensington hiker, and it continues to do so until the hikers meet. At the moment when the hikers meet, the fly lands on the shoulder of the Kensington hiker. How many kilometers has the fly flown in total?
A. 25
B. 30
C. 37.5
D. 45
E. 60
Please help me understand where I went wrong.
Two people moving at same speed towards each other would meet at the center. So they meet at point which is 15km away from both ends (where fly lands on shoulder).
Fly is flying at 10km/h and Robert hiking at 5km/h. So they would meet at distance covered by fly (say fd = 10t), Robert rd = 5t
here t is same.
fd/10=rd/5 -- i
fd+rd = 30
rd= 30-fd
in i
fd/10= (30-fd)/5
fd/2=30-fd
fd=60-2fd
3fd= 60
fd=20
so fly covered 20km before it met robert, from here it starts to fly back to the point where robert and Kensington met (which we calculated was center)
By this logic now fly has to cover 5km.
Total distance = 25km.
one fault in my logic is, when the fly flies back from here, it would reach the center before Robert. But I couldn't really understand how to accomodate that, so I just let the fly to the midpoint because thats where the hikers meet. Please help me with this.
The combined speed of the fly and the hiker from Reston is 10 + 5 = 15 km/h. Thus, they will meet after 30/15 = 2 hours.
In 2 hours, the fly would cover 2*10 = 20 km, and each hiker would cover 2*5 = 10 km:
K ----------(10 km)---------- (Fly meets Hiker 1) ----------(10 km)---------- (Hiker 2) ----------(10 km)---------- R
Therefore, the distance between the fly and the hiker from Reston would be 10 km, which they would cover in 10/15 = 2/3 of an hour.
In 2/3 of an hour, the fly would cover an additional 2/3*10 = 20/3 km, and each hiker would cover 2/3*5 = 10/3 km:
K -----------(40/3 km)--------------- (Hiker 1) ----(10/3 km)---- (Fly meets Hiker 2) -----------(40/3 km)--------------- R
...
Continuing this pattern, each time the fly covers a third of the distance it did previously, we must sum up the infinite sequence: 20 + 20/3 + 20/9 + ... For an infinite geometric sequence with a common ratio |r| < 1, the sum can be calculated as sum = b/(1 - r), where b is the first term. Therefore, the sum is 20/(1 - 1/2) = 30.
That being said, solving this question using the method shown in the official solution is much quicker. Thus, I suggest understanding that one.