Cars Y and Z travel side-by-side at the same rate of speed along paral

(1) this info is not sufficient, we need info about X
(2) In this case we have an Isosceles Triangle with ratio of sides d:d:d\(\sqrt{2}\) add. we know that both need the same Time-t and we have a proportion for distance, let's say the distance d=5 -> Ratio \(5:5:5*\sqrt{2}\) ~7
we can setup an equtation Rate Y= 7/t, Rate Z =5/t Rate Y/ Rate Z = 7/t*t/5*100 ~140%
Answer (B)

how time is equal for Y and Z? I'm not able to understand

Hi..

read the following context



this means both are moving in the same line when drawn perpendicular to the initial direction.
since the cars are moving continuously and none halts in between, so whenever you look at the distance at a certain point of time, the TIME would be same as both are moving till that point of time.

Posting my way of approach by applying basic trigonometry. Assume the point of meeting to be Q. So distance traveled by Y is PQcosX and distance traveled by Z is PQ. Also assume speeds of Y and Z to be y and z respectively. We know that the time for both should be equal. So, PQcosX/y=PQ/z----cosX/y=1/z----y/z=cosX. Thus we can see that all we need is the value of x.

This asks for a specific number for the speed of car Y (beyond point P) as a percent of the speed of car Z. Notice that, while this is a "what is the value" question, it is asking for a ratio rather than a value for the speed of Y. This means that you may be able to find the relationship between the two speeds by leveraging your assets even if you cannot find a specific speed for each car. Remember also that complex situations like this one are often abstractions of fairly straightforward ideas. The test will reward you for breaking down that abstraction into something more concrete.
You are given that the roads that cars Y and Z are parallel until point P. At point P the cars are side-by-side. Then car Y makes a turn of x degrees, speeds up, and continues to keep up with car Z. So even though they are going at different speeds, they are travelling to the right (in the figure) at the same relative speed.

Statement (1) gives you the speed of car Z, 50 miles per hour. This statement does not give any information as to the speed of car Y, or the value of the angle x. It should be clear that this statement is not sufficient on its own, and you can prove it by thinking conceptually.

If the two cars had been travelling side by side on the same road before car Y veered off, you would be able to recognize that they seem to form two sides of a triangle with the angle x between them at point P. If angle x is very small, car Y will not have to go much faster than car Z in order to keep up, since it is travelling in almost the same direction.

However, as x increases, the speed of car Y has to increase as well. Thus, without knowing anything about the angle X, you can't make any deductions about the ratio of their speeds.

Eliminate choices A and D.

Statement (2) tells you that the measure of angle x is 45 degrees. While this statement may not appear sufficient, remember that you don't need to know the speeds of the two cars, only the ratio between them.
Because the cars are keeping up with each other, the amount of time that each car has had to travel is equal. That means that the increased percentage of distance that car y has traveled will be the same as the increased percentage or the rate of car Y compared to car Z. In other words: determine the ratio of the distance traveled by car Y to the distance traveled by car Z, and you have enough information to answer this question.

Again imagine that cars Y and Z were instead traveling side-by-side on the same road. As with statement (1), when car Y veers off at point P, it forms a triangle whose three sides are the distance between point P and car Y, the distance between point P and car Z, and the distance between cars Y and Z. Because you are told that cars Y and Z remain even, you should recognize that this means that this is a right triangle.
Going back to the information given in statement (2), you are told that x is 45 degrees. That means you have a 45-45-90 triangle with the distance that car Y has traveled as the hypotenuse and the distance car Z has traveled as one of the sides.

The ratio of the sides of a 45-45-90 triangle is x:x:x\sqrt{2}
where x is a length of one of the legs. The ratio of the distances traveled for Y to Z is the same as the ratio of the length of the hypotenuse to a side, x\sqrt{2}:x
which simplifies to \sqrt{2}:1
Since you have established that the ratio of the distances must be the same as the ratio of the speeds, you know that statement (2) is sufficient. The correct answer is B.

Veritas Prep Solution

Once Car Y and Car Z hit Point P, they "continue to stay even" - in other words, they cover their respective distances in the SAME TRAVEL TIME.

Given Constant Time ----- the Ratio of the Distances 2 Cars cover will be Directly Proportional to the Ratio of the Speeds the 2 Cars are driving at

Q asks for: Speed of Car Y is what percent of Speed of Car Z?

Thus, if we know the Ratio of: (Speed of Car Y) / (Speed of Car Z) : we can answer the Question and the Statement is Sufficient.


S1: Speed of Car Z is 50 mph.

Not Sufficient. No information about Car Y's Speed.

S2: X = 45 degrees

This means the Straight Line that Y travels beyond Point P is the Hypotenuse of a 45/45/90 Degree Right Triangle.

further, Car Z will travel the Leg of this Triangle (from Point P to the Vertical Dotted Line) because we are told the 2 Tracks are Parallel.

No matter what the Actual Speed or Actual Distance is, in order for the 2 Cars to "stay even" from Point P to the Vertical dotted Line:

Over a CONSTANT UNIT OF TIME:

the Ratio of the Distances the 2 Cars will have to travel in order to stay even is:

Car Y will have to travel a Distance of: sqrt(2) mile
___________________________________________
while Car Z will have to travel a Distance of: 1 mil


Ratio of: (Speed of Car Y) / (Speed of Car Z) = sqrt(2) / 1

S2 is Sufficient Alone

B

1st Approach
Since the cars are keeping with each other, Car Y must cover the distance in the ratio of cos(x).
1. Z=50 --- Insufficient
2. x=45 ----Sufficient

2nd Approach:
Since the cars are keeping with each other, Car Y must cover the distance equivalent to hypotenuse of the triangle with angle of x. The ratio of hypotenuse (distance-Y) and base (Distance-X) be determined with angle x.
1. z=50 - Insufficient
2. x=45 - Sufficient.

I am still not able to understand why time is same?

It is possible that Y moving at a higher speed reaches the point earlier than Z.
If distance and speed of Y is different from Z, then time can be different.
Can someone please explain?

(1) The speed of car Z is 50 miles per hour.
Insufficient since we have no requisite information to determine the speed of P
Clearly insufficient

(2) x = 45
use 45:45:90=x:x:x√2 rule can be used to determine the distance
the distance travalled by Y becomes x√2.
Since, Z & Y both take the same time, the speed of Y will be x√2/x.
=>141% speed of Z clearly sufficient

Therefore IMO B

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