The Flight Of Smaug

We could go at this by trying to find directly how far Smaug flew by calculating the sum of an infinite geometric sequence, but that might be a little tricky considering the ugly numbers in the problem. Instead, we can find the amount of time it takes Gandalf and Radagast to reach each other and multiply it by how fast Smaug travels, using the equation $d=s\times t$ (distance $=$ speed $\times$ time).

Gandalf and Radagast are traveling at each other at a relative speed of $14$ leagues per hour. We can treat this problem as Gandalf riding at $14$ leagues per hour towards a stationary Radagast. It would take Gandalf $\frac{1890}{14}=135$ hours to reach Radagast. In this time, Smaug will have traveled $X=135\times17$ leagues.

$\frac{X}{5}=\frac{135\times17}{5}=27\times17=\boxed{459}$

I applaud these characters on their abilities to not sleep for more than $5$ days.

This is actually remarkably simple, once you think about it. There is a slight ambiguity in the question for readers unfamiliar with Tolkien: Radagast is approaching Dumbledore from Dol Guldur, which means they are moving towards each other along a straight line!

Now, consider the time taken for Gandalf and Radagast to meet:

$1890 \div (6+8) = 135$

Since the time taken for them to meet is 135 hours, Smaug has also been flying for 135 hours! But note that Smaug is flying at a constant speed of 17 leagues per hour...

Hence, the distance travelled by Smaug is:

$135 \times 17 = 2295 \Longrightarrow 2295 \div 5 = \boxed{459}$

Time taken by Gandalf to meet Radagast = 1890/(8+6)=135hrs Through this whole time Smaug was moving at the speed of 17 lgs/hr. So distance that smaug travels,X=17*135=2295 leagues Thus, X/5=459 Ans.

This is a pretty easy problem but the description is too complex..... just think Gandalf and Radagast run 8+6=14 leauges per hour.... so total 1890/14=135 hours of running. In this period Smaug fly at 17 leagues per hour. so answer is 17*135/5=459. Hope this helps... Thank you.

I can give a shortcut which seems to be amazing for this lengthy problem.

In this problem, we see time remains constant,

i.e. The time taken by Gandalf & Radagast to meet each other riding in opposite direction is same as the Smaug riding in this entire duration.

We know that when time remains constant ,

Distance is directly proportional to Speed which means that the speed ratio is same as
the distance ratio and are directly related.

speeds of (Gandalf(8) + Radagast(6)) and Smaug can be related,

 14:17

which is same as their distance ratio.

The distance 1890 leagues must have been covered by (Gandalf + Radagast) .

In this time, Distance covered by Smaug = (1890/14)*17 = 2295.

X = **2295**.

X/5 = **459**.//

Gandalf and radagast are approaching each other(opposite direction) with speed 8=6=14 leagues/hr total distance=1890 leagues so time taken by them to meet=1890/14=135 hrs IMPORTANT: for 135 hrs Smaug should be running back and forth continuously.So, his total distance traveled X= 135 x 17 =2295 leagues x/5=459

total speed = 14 kmph. Time at which they meet each other = 1890/14,and that is when Smaug stops. Distance = speed time taken = (1890/14) 17=2295 = X.X/5 = 459

As Radaghast and Gandalf are moving towards each other so the 1890 leagues / ( the sum of there speeds) gives us the time they ll meet I.e 135 hrs. Now multiply that with the speed of smaug and VOLA we have the distance travelled by smaug, X= 2295. And X/5 = 459

The combined velocity of Gandalf and Radagast is 14 l/hr so it takes them 135 (1890/14) hours to meet since Time = Distance / Velocty. In that time Smaug flies at 17//hr (assuming changes of direction are instantaneous) therfore covers 17*135 leagues (since Distance = Velocity * time). X= 2295. X/5 = 459

Time for Gandalf and Radagast to meet= 1890/(8+6)= 135 hours. All this while, Smaug is flying to and fro between them at a speed of 17. So, leagues flown by Smaug, X=135x17.

X/5=135x17/5=459

Statement : Gandalf & Radagast are riding towards each other at 8 and 6 leagues per hour respectively. Hence if we calculate the relative speed its 14 leagues per hour (8+6),

So to cover the distance of 1890 leagues they would require = 1890/14 = 135 hours. In 135 hours Smaug who is flying back and forth at 17 leagues/ hr. will cover 17*135 = 2295 leagues = X

X/5 = 459

speed of gandalf is 8leagues/h speed of radagast is 6leagues/h as they are approaching each other therefore speed adds up hence net speed is 14leagues/h distance is 1890 therefore time is 1890/14=135 hours distance travelled by smaug is 135*17 after dividing by 5 we get 459

Well I have used a little bit of physics here Let the distance travelled by Gandalf be x ,distance covered by Radagast will be 1890-x (x is the distance at which they meet) Let the time taken by them for meeting each other be t.

Now for Gandalf. x÷t=8.
x=8t.
(Distance ÷time= speed) For Radagast.
1890-x ÷t = 6 1890-x = 6t 1890 - 8t = 6t 14t=1890 t= 135

Now the total distance travelled by samug in 135 hrs = 135 × 17 = 2295 Therefore. 2295÷5 = 459

This question was totally easy. First we need to ignore how much time the smaug turns.

The most important is to find how much time did Gandalf and Radagast to meet.

so x(8+6)=1890 x=135 the number of leauge/135=17

hence the number of leauge =2295 2295*5=459

total time taken for gandalf and radagast to be at the same position = 1890/(8+6)=135hrs. number of leagues covered by Smaug = X = 135*17=2295. X/5=459.

let Gandalf=Y , Radagast=Z & Smaug=X First X+Y=1890 , Y=(4/3)Z , (7/3)Z=1890 , Z=810 leagues & Y=1080 The time elapsed = (1080/6)=135 Hours the time elapsed for X for one shoot = (1890/17)=111.1764 Hours then the total leagues for X=(135/(1890/17))*1890=2295 leagues divided by 5 = 459 leagues

After setting out from their respective locations Gandalf & Radagast met after 1890/(8+6)=135 Hours. At the same time Smaug travelled 17 X 135 leagues. So Ans (17 X 135)/5=459.

From Gandalf's perspective, Radagast is traveling at $8+6=14$ leagues per hour towards him. This means that it will take him $1890/14=135$ hours to reach Gandalf. During this time, Smaug was traveling at a constant 17 leagues per hour, thus traveling $135 \times 17=2295$ leagues. This is our desired quantity $X$ , and one fifth of it is $\boxed{459}$ leagues.

Distance covered by Gandalf in 1 hour = 8 leagues

Distance covered by Radagast in 1 hour = 6 leagues

therefore, distance covered by them in 1 hour = 8 + 6 = 14 leagues

Time taken for them to meet each other = $\frac{1890}{14}$

Therefore, Time taken for them to meet each other = 135 hours

X = $135 \times 17$ league

X divided by 5 = $27 \times 17$ = 459

Best problem i've seen on here. Total distance = 1890 leagues. Radagast-Gandalf combined pace=14 LPH. Take Dol Guldor as starting place and say radagast's pace is 6 LPH and a fraction of total pace is 6/14 so total distance of Gandalf/Radagast meeting spot (from Dol Guldur) is (6/14)x1890 which gives a distance of 810 L from Dol guldur. And using T=D/S sub in values of radagasts speed of 6LPH and distance of 810L and get 135 hours of total travel time. As smaug stops flying back and forth when the 2 wizards meet (after 135 hours). Smaug is flying for 135 hours. then use smaug's pace of 17LPH in the equation D=SxT to get a total distance of 2295L for X. X/5=459. Answer=459

Firstly, we need to assume that Radagast starts from Dol Guldur (not mentioned in the problem).

Lets say the time taken for Gandalf and Radagast to meet is 'T' hours. Then, 8 T+6 T = 1890, T = 135 hours

Smaug may be flying back and forth, but since he/she is flying at constant speed and distance covered in 'T' hours is, D = 17*T = 2295 leagues

(D/5) =459, which is the answer to this question.

The distance between The Lonely Mountain and Dol Guldur is 1890 leagues. The relative velocity of Gandalf and Radagast is 14 leagues per hour (8+6). Thus Gandalf and Radagast would meet after 1890/14 = 135 hours. In 135 hours, Smaug would travel 135*17 leagues, which is equal to X. Hence, X/5 equals 459.

Gandalf and Dol Guldur are $1890$ leagues away, and are approaching each other with a combined speed of $8 + 6 = 14$ leagues per hour. So they will meet after $1890/14 = 135$ hours.

Hence, Smaug will fly $135$ hours with a speed of $17$ leagues per hour. So he will fly $135 \cdot 17=2295$ leagues.

So the answer is $2295/5 = \boxed{459}$ .

Distance between the two points=1890, Speed of Approach = (8+6)=14 L/Hr. Hence time taken to meet =1890/14 = 135 hrs. So distance travelled by Smaug, X = 135*17 = 2295 L. Reqd. Ans =2295/5 = 459. (This is yet another unnecessarily complicated version, if I may say so, of the bird flitting between two approaching trains attributed to Von Neumann)

You can approach the problem by solving the time needed for Gandalf and Radagast to meet then multiplying to the speed of Smaug to get the total distance traveled.

With this, we could calculate that Gandal and Radagast are nearing each other at the rate of their combined speeds (8 + 6) = 14 Leagues per hour. The total distance in between is 1890 leagues. We then get the time needed by dividing the speed to the total distance (1890/14) = 135 Hours From this we multiply the speed of Smaug of 17 leagues per hour = 2295 leagues. Finally to get the number asked we divide it by 5 and we get 459.

Total Distance = 1890 Leagues. Time Taken for Gandalf & Radagast to meet = 1890 / ( 8 + 6 ) Hrs. Total League crossed in that time by Smaug = ( 1890 / ( 8 + 6 ) ) * 17 Leagues = X

As Smaug continues the rounds until Gandalf and Radagast meet,first calculate the time taken for them to meet which will be 1890/(8+6)=135hrs. Multiplying the time taken with Smaug's speed we get 2295leagues as the total distance traveled by Smaug. And 2295/5=459.

To find the number of leagues that Smaug has travelled, we need to find the total time needed for Gandalf and Radagast to meet, so

Number of hours = $\frac{1890}{8+6}$ = 135 $\Rightarrow X = 17 \times 135 = 2295$

Time required for Radagast and Gandalf to meet = distance / relative speed = 1890 / (8+ 6) =135 hours Therefore distance covered by Smaug till these two meet = 135*17 Therefore X/5=135 * 17/5 =459

the relative speed of gandalf and radagast is (8+6)=14L/hr..(since they r approaching towards each other).. so they will mt after 1890/14 =135hrs.. the speed of smaug is 17L/hr . so the distance it can cvr in 135hrs is 135 17L=X(given) so X/5=135 17/5=459...