How many miles long is the route from Houghton to Callahan?

Hi Guys,

It's a DS question and we don't need to calculate the actual distance between Houghton and Callahan; just the analysis that the information provided in the statements is sufficient is enough to answer our question.

Analyze the Given Info:
The question asks us if the information provided in the statements is sufficient to calculate the distance between two places. We know that Distance = Speed * Time. So, if we are given the values of speed & time taken, we can find the distance between the places. Let's evaluate the information given in the statements to see if it provides us the required information.

Analyze statement-I independently
St-I gives us two scenarios of speed & time taken to cover the same distance. Let's write equations for both the scenarios:

a. Time taken to cover the distance at a speed of \(55 \frac{miles}{hour} = \frac{D}{55}\) assuming the distance between Houghton and Callahan to be D.

b. Time taken to cover the same distance at a speed of \(50 \frac{miles}{hour} = \frac{D}{50}\).

Since we know that the difference between the time taken is 1 hour, we can write \(\frac{D}{50} - \frac{D}{55} = 1\).
From this equation, we can easily find out a unique value of D. Hence, statement-I is sufficient to answer the question.

Analyze statement-II independently
St-II tells us that it takes 11 hours to cover half the distance at a speed of 25 miles/hour. So, we can write \(\frac{D}{2} = 11* 25\).
Again, this equation would give us a unique value of D. Hence, statement-II alone is sufficient to answer the question.

It's important to remember that DS question does not ask us to find the actual value of the unknown. Once we deduce that the information provided in the statements is sufficient to find out a unique value of the unknown, we should not be doing further calculations to save our time.

Regards
Harsh

Statement 1: Ratio of average rates for same distance: 11:10, therefore ratio of time taken will be 10:11. As it is also given that time difference is 1 hr, actual time taken is 10 hrs at 55 mph and 11 hrs at 50 mph. We can calculate distance, thereby making statement sufficient.

Statement 2: Sufficient as both rate and time are provided, we can calculate distance.

Answer: D

Hi chetan2u, VeritasKarishma, Bunuel,

Could one of you experts please explain why statement 2 is sufficient? We are only given about the first half the distance. If it takes 11 hours to travel the first half of the route at 25mph, it could take maybe 5 hours to travel to cover the second half of the route at 50mph. Wouldn't this make the statement insufficient?

How many miles long is the route from Houghton to Callahan?

(2) It will take 11 hours to travel the first half of the route at an average rate of 25 miles per hour

We need distance from H to C.

For first half of the distance, it will take 11 hrs at a rate of 25 mph. So distance travelled in first half = 25*11 = 275 miles
Since distance of first half is 275 miles, distance of second half will be 275 miles too.
So distance from H to C will be 275 + 275 = 550 miles

We know distance = speed * time.

(1) It would take one hour less time to travel at 55 mph than 50 mph.

50 = D / (T + 1) ---- plus 1 because it takes one more hour to travel at 50 mph
55 = D / T

50 (T + 1) = 55T
50T + 50 = 55T
50 = 5T
10 = T

Plug 10 = T into 55 = D/T
D = 550 miles

Statement 1 is sufficient.

(2) 275 is half the distance; thus, 275 * 2 = 550. Sufficient.

Answer is D.

How many miles long is the route from Houghton to Callahan?

(1) It will take 1 hour less time to travel the entire route at an average rate of 55 miles per hour than at an average rate of 50 miles per hour.

50t = 55(t - 1) ---> t = 11
50 mph x 11 hours = 550 miles

Sufficient.

(2) It will take 11 hours to travel the first half of the route at an average rate of 25 miles per hour

r x t = d --> 25 x 11 = 275 miles

275 x 2 = 550 miles

Sufficient.

D.

Hi Bunuel,
I understand this question is pretty straightforward and the maths is quite simple too.
My only concern here is in Statement (2), how can we assume that the average rate remains same in the second half of the route as well? I've seen a question where assuming average rate was marked as wrong and the explanation provided was that we can't say that the speed will remain constant throughout unless mentioned explicitly.
Does this not leave Statement (2) in an ambiguous state?

saghosh1, you're correct about statement 2 when you say we cannot assume the speed to remain constant, but we don't really care. We need to find the total distance. Using the info given in statement 2, we can easily find the half of the distance and hence the statement is sufficient. Hope this helps, but happy to discuss further if it doesn't.

Hi Brian123,
thanks for the clarification. I really "overthought" the second statement.

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